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-'■ E-AouTT \5--Sr,31.1-'J3
TUFTS COLLEGE.
THE GIFT OF
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ELEMENTS
or
ALGEBRA,
BY
BOURDON,
TRANSLATED FROM THE FRENCH
rOS THB
USE OF COLLEGES AND SCHOOLS.
BOSTON:
UILLIAKi), GBAT, LITTLE, AND WILKIN3.
183 1.
Entered aecordio^ to Act of Congiefic, in the year 1831, by HilUard, Gray, dt Co., in
the Clerk^s Office of the District Court of Massachusetts.
CAMBRIDGE : E. W. METCALF AND CO.,
Printen to the UiiiTeraity.
ADVERTISEMENT.
The following translation extends to the General Theory of
Equations^ and 'comprehends a little more than one half of the
original work. Tn preparing it for the pmss considerable use has
been made of a translation by Augustus De Morgan, Professor of
Mathematics in the University of London, extending to about one'
fourth of the original treatise. Questions, intended as an exercise
for the learner, are occasionally introduced. They were selected
from the celebrated collection of Meier Heirsch.
CamMidge, My, 1881.
s.
*1A
\
ERRATA.
Page 11,
13,
17,
20,
30,
30,
35,
40,
43,
46,
61,
52,
67,
74,
91,
94,
97,
« 103,
« 104,
« 109,
119,
((
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/or 130,63 read ISa^P.
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factions
+ 6
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more
them
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a — 36.
fractions.
— 6.
z into.
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greater,
themselves.
3a«.
3 7 3.
TT
89
TT'
1
(12)2'
0,01.
— a\/c.
n^
(a + x).
Vtf's — p. [ary.
unauxili4zry known read t£nA:nou;n auxUi-
formula rea^ formula (1).
fo rmula " formula (7).
^np + m^ — m) read {\<np+m^ — wi).
2 f' read 2 i''^
6 pieces " 26 pieces.
a (a — 6 t).
tt
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dele a.
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for dy
" 300
cc 17 5
CC
read y.
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«
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VI
Errata,
Page 211, line 21, for the product read be the product.
« 216,
« 230,
« 17, " =6
" 19, « (3 a 68
+ 5.
(Sabf.
" 233,
" 1, " V
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« 237,
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(3" - 1).
CONTENTS.
Introduction 1
Of Algebraic Operations 7
Of Algebraic Addition , . . , 11
Of Algebraic Subtraction 12
Of Algebraic Multiplication 14
Division 20
Of Algebraic Fractions 36
Elementary Theory of the Greatest Algebraic Connnon Divisor 37
Examples in the Reduction of Fractional Expressions to their
Simplest Terms 45
Problems of the First Degree 46
Preliminary Notions on Equations ibid.
Of Equations of the First Degree with One Unknown Quantity 48
Of Equations and Problems of the First Degree with Two or
more Unknown Quantities . • , 61
Elimination ^ • • • • 62-
Problems which give rise to Negative Results. Theory of
Negative Quantities 71
Discussion of Problems of the First Degree, containing One or
more Unknown Quantities 79
General Investigation of Equations and Problems of the First
Degree 85
Resolution of Problems and Equations of the Second Degree 106
Formation of the Square and Extraction of the Square Root of
Algebraical Quantities ibid.
Calculus of Radicals of the Second Degree 112
Problems and Equations of the Second Degree . . . . 116
General Discussion of the Equation of the Second Degree • • 126
Of Transformations which may be performed upon Inequalities 145
Questions concerning Maxima and Minima. Properties of Tri-
nomials of the Second Degree 150
Propertiesof Trinomials of the Second Degree .... 154
viii Contents.
. On Equations and Problems of the Second Degree, containing
Two or more Unknown Quantities . . . . . 159
Extraction of the Square Root of an Expression which is partly
Rational and Partly Irrational 163
Indeterminate Analysis of the First and Second Degree 167
I. Equations and Problems of the First Degree with Two Un-
known Quantities ibid.
IL Of Equations and Problems with Three or more Unknown
Quantities . . 188
III. Of Indeterminate Analysis of the Second Degree . . 197
Of the Formation of Powers and Extraction of Roots of
any Degree whatever . . . . . 203
I. Binomial Theorem of Newton and the Consequences which
are derived from it 204
Consequences of the Binomial Formula and of the Theory of
Combinations 214
II. Of the Extraction of the Roots of Particular Numbers . . 216
Of the Extraction of Roots by Approximation .... 225
III. Formation of the Powers and the Extraction of the Roots of
Algebraic Quantities. Calculus of Radicals .... 228
Calculus of Radicals 232
IV. Theory of Exponents of whatever Nature .... 243
Of Progression by Differences and by Quotients . . 248
I. Progression by Differences ibid.
Of Progression by Quotients . . ' 254
IL Of the Theory of Exponential Quantities and Logarithms . 265
Theory of Logarithms 271
III. Application of the Theory of Logarithms . . . . 277
J^ote . 293
Questions % 297
ELEMENTS OF ALGEBRA.
INTRODUCTION.
1. Algebra is the part of mathematics in which symbols are
employed to abridge and generalize the reasonings made use of ia
the resolution of questions relating to numbers.
We distinguish questions into two leading classes, theorems and
problems, A theorem has for its object to demonstrate the exist*
ence of certain properties of known and given numbers. In a
problem it is proposed to determine certain numbers from other
numbers that are known, and which have with the first relations
indicated by the enunciation.
2. The principal elements used in algebra to arrive at this dou-
ble object are 5 t,
(1.) The letters of the alphabet, which serve to designate the
numbers which form the subject of the reasoning. Their use is
necessary, not only to abridge, but also to generalize the reason-
ing ; for by them it is more clearly seen that a property belongs at
the same time to several numbers ; or, if the question be a prob^
lem, that the method of finding an answer to it is the same, what-
ever be the particular numbers stated in the question.
(2.) The sign +, which denotes the addition of two or rnore
numbers, and is called plus. Thus 25 -|- 36 is twenty-five plus
thirty-six, or twenty-five added to thirty-six. Also a -{- 6 i& called
a plus &, and js the number signified by a, augmented by the num-
ber signified by b.
(3.) The sign — , which is called minusj and denotes x the
subtraction of one number from another. Thus 45 — 24 is called
45 minus 24, and is the difiference between 45 and 24 ; a — i is
called a minus &, and is a diminished by b.
Bour. Alg. 1
2 Elements of Algebra.
(4.) ^he sign of multiplication, which is X 9 or a point placed
between the two quantities. Thus 36 X ^^j or 36 . 25, denotes
36 multiplied by 25, or the product of 36 by 25. When the num-
bers which are multiplied together are designated by letters, it is
usual to write them one after the other without the interposition of
any sign.
Thus a b means the same thing as a X b ox d .b} a & c is the
same as a X ^ X c, or a . 6 • c.
It must be understood that the notation ab or a be, which is
more simple than aXb or aXbx c, can only be used when the
numbers are designated by letters. For if we wished to represent
the product of 5 multiplied by 6, and we wrote it for the sake of
brevity 56, we should confound this product with the number fifty-
six in the decimal system of nqtation.
This remark is very important to beginners.
(5.) The sign of division, -r- or — , the former being placed
with the dividend preceding and the divisor following it, and the
latter with the dividend above and the divisor below it.
Thus 24 -r- 6 or V signifies 24 divided by 6, or the quotient of
24 divided by 6. ^ means a divided by 6, and is often expressed
by saying a by b.
(6.) The vincuiumj — or ( ), the former being drawn over a
quantity consisting of two or more letters, and the latter being made
to enclose the compound quantity. Thus (a + 6) X c signifies the
product of the quantity {a + b) by c, and not the product of b by
c added to a, as we might suppose, if no such sign were used.
(7.) The coefficient^ which is employed when a number, desig-
nated by a letter, is to be added several times to itself. Thus*
fl-|-fl-|-a-|-a-|-a, which represents the number a added four
times to itself, is written 5 a. Also II a represents a added to
itself 10 times; 12 a b represents the product ab added to itself
11 times.
The coefficient is the particular number written before a quantity ^
designated by one or several letters^ which denotes the number of
times plus one that this second quantity is added to itself.
(8.) The exponent^ which is used when a number is multiplied
several times by itself. Thus, instead of writing
aXaXaXaXa, or a. a. a. a. a.
Introduction. 3
it 19 usual to write o^, which signified a'raised to the ifth power, or
the product of a multiplied four times by itself. Thus b^ is the
same thing ^s b ,b ,b ,b .b ,b»
The exponent is a number written on the right of a Utter and a
little above it, and denotes how many times plus one the quantity^
designated by the letter, is to be multiplied by itself, or how many
^ times this letter is a factor in a product.
We give the name of power to the result of the muhiplication
of a number several times by itself, and the degree of the power,
or the exponent, is the number of times which the letter is repeated
in the expression of the power written at full length.
To show the importance of the exponent in algebra, suppose it
were desired to express that a number a is multiplied three times
by itself, that the pro4uct is then multiplied three times successively
by 6, and that this new product is multiplied twice successively by
€.' This is written simply d* b^ c*. If the last result is to be added
six times to itself, or multiplied by 7, it is written thus, 7 a* b^ A
This give^ an idea of the conciseness of algebraical language.
(Sf.) 'the sign \/, which is placed before a number when the
3
extraction of any root of that number is indicated. Thus \/a is
4
cftUed the third ot cube root of a ; \/b is called the fourth root of b.
The second, third, &£C. root of any number is the number which,
when raised to the second, third, &;c. power, reproduces that
number.
(10.) The sign by means of which it is expressed that two quan-
tities are equal to one another. This sign is =, and is read thus,
is equal to, or equals.
Thus the way of expressing that the difference of 36 and 25 is
11, is 36 — 25 = 11 ; that is, 36 minus 25 equals 11.
(11.) The sign of inequality >, which is used to express that
one quantity is greater or less than another.
Thus a ]> 6 signifies that a is greater than b, and a <Cib signi-
fies that a is less than b. The opening of the sign is always turned
towards the greater quantity.
From what has been said we may evidently regard algebra as a
species of language formed of a number of signs, by the aid of
* which the connexion of ideas is perceived with great facility in the
reasoning which it is necessary to adopt, either to demonstrate the
existence of a property, or to find the solution of a problem. A
4 Element^ of Jllgebra.
better idea will be formed as to tbe utility of algebraical symbols
from the following questions.
FIRST Q,UE8TI0N. PROBLEM.
3. The sum of two numbers is 67, their difference is 19 ; what
are these two numbers ?
We must establish, by tbe help of the signs, which have been
defined, a connexion between the known and unknown numbers in
tbe statement of the question.
If the less of the two numbers had been known, the greater
might have been found by adding 19 to the other. Call the less
of the two 0?, the greater is then- a? +19, their sum is a? -|- a? + 19,
or 2 a? -f- 19. Now this sum is 67. We have therefore the equa-
tion 2a? + 19= 67.
Now, if 2 a? augmented by 19 gives 67, 2 a? is less than 67 by
19, or 2 a? = 67 — 19 = 48.
Therefore a? is half of 48 ; that is, a? = V = 24 ; and as the
less number is 24, the greater, which is a? + 19, is 24 + 19, or
43. We may easily see that 43 + 24 = 67, and that
43 — 24 = 19.
The algebraical' operations contained in tbe above solution are
merely the following. Let a? be the less number. Then a? + 19
is the greater. And, by the question, 2 a? + 1 9 = 67.
Therefore 2 a? = 67 — 19 z= 48, and a? = V = 24.
Consequently a? + 19 = 24 + 19 = 43.
Another Solution.
<
Let X be the greater number.
Then x — 19 is the less, and 2 a? — 19 = 67.
Therefore 2 a? = 67 + 19 = 86, and a? = y = 43.
X — 19 is therefore 43 — 19, or 24.
The student may now see how all the reasonings necessary for
the solution of a problem may be contained in a very small space ;
though in this case, had they been written in ordinary language,
they would have occupied more than a pagCt
I •
Iniroduetian.
General Solution of this Problem.
4. The sum of two numbers is a, and their diflference 6. What
m
are the two numbers ?
Let X be the less number.
Then x -{- bis the greater,
and 2 a? -f- 6 = a.
Therefore
a — b a b
2a? = a — 6, and a? = — 5 — = 5 — g*
Consequently
As the form of these results is the same, whatever value be
given to a and 6, it follows that the greater of two numbers is
equal to half their sum plus lialf their difference^ and that the less
is equal to half their sum minus half their difference.
Thus, let the sum of two numbers be 237, and the difference
d9 ; the greater will be
237 ,99 237 + 99 336 ,^q
-2-+T"^— 2 "^•2- = ^^®'
and the less will be
237 99 138 -^
^-2-°'-r = ^^-
Indeed
168 + 69 = 237, and 168 — 69 == 99.
From tliis solution is seen the utility of representing by letters
the given quantities in a problem. As the arithmetical operations
upon the letters can only be indicated, the result at which we arrive
retains indications of the several operations which must be per-
formed on the known quantities in order to find the unknown ones.
The expressions
a j^ 6 J a b
2 + 2 and ,j — g,
obtained in the preceding problem, are called in algebra formulaSf
because they may be regarded as containing the solutions of all
questions of the same nature, in the enunciation of which the only
things which vary are the numerical values of the given quantities.
Elemenii of Algebra.
SECOND QtJESTION. — THEOREM.
5, The sum of two numbers multiplied by their difference is equal
to the difference of the squares^ or second powers of those numbers.
Let 12 and 9 be the two numbers; their sum is 21, and their
difference 3 ; and the product 21 X 3, or 63, is equal to 144, the
square of 12, diminished by 81, the square of 9.
But to make this property known, whatever be the two numbers,
let us represent them by a and 6.
Their sum will be a + &, and their difference a — 6. To form
•the product of these two expressions, first multiply the sum a -[- 6
by a. This product will be a X a + i X a, or, more simply,
€? •\' ab\ for we most take each of the two parts of which a + ^
is composed, as many times as there are units in a, and add
together the two products. But it is not by the whole of a that we
are to multiply, but by a diminished by 6 ; therefore the product
a^ -(- a 6 is too great by (a + ^) X 6 ; that is, by a 6 + ^^- W®
must then subtract a 6 + 6^ from a^ + a 6, the algebraical repre-
sentation of which is a^ + a 6 — ab — 6^ . a^j gg -[- a 6, — ab
cancej each other, the product required is a^ — b^.
This result being the same, whatever values afe given to a and
i, it follows that the theorem is true for any numbers whatever.
THIBB QUESTION. THEOREM.
6. If to the two terms of a proper fraction^ that isj of a nunAer
less than unity, the sam^e number be added, the resnUing fraction is
greater than the first.
Let Y^ be the proposed fraction ^ add 3 to each of its terms, and
it becomes ^j. These two fractions, when reduced to a common
denominator, become ^/^ and yVy. Now the second fraction is
evidently greater than the first.
To ascertain whether this theorem is true, whatever be the pro-
posed fraction, lei us designate this fraction by -r, supposing a < 6.
Let m be the number to be added to both terms of the fraction ;
we shall have
a-^-m
6-(- »i*
Algebraical Operations. 7
To compare these two fractions, it is necessary to reduce them
to a common denominator. For this purpose, multiply the two
terms of the first fraction by b -}- m, and of the second by ft.
Now, to multiply a by & -|- m is to take a as many times as there
are units in b, and again as many times as there are units in m,
which gives ab -{■' am. Also, the product of b hy b -^-m is
6* -4- 6 m, which eives ,« ; t — for the first fraction. In the
' ° 0'' -f- om
3ame way, if we multiply the two terms of the second fractioo
« + »« 17-1 ab + bm
r-\ — by 6, it becomes ,q , / — .
The two numerators a 6 -f- a m, a 6 + 6 », have a common part
abf and the part 6 m of the second numerator is greater than the
part a w of the first, because 6 > a. Therefore the second fractioxi
is greater than the first, which was to be proved.
We may see also by the foregoing reasoning that t must be a
proper fraction, in order that the theorem may be true ; for other-
wise the contrary of this theorem would be true, since we should
then have a6 + 6m<^a6 + am.
7. By. reflecting upon the solutions which have been given of the
preceding question, it will be perceived that the employment of
algebraical signs gives rise to rules common to several questions.
Thus, for instance, in the second and third questions, we have
been led to perform the multiplication of a sum a -j- ^ by a number
a, of a sum a + m by 6, and of a number a by a sum b -y^m. It
appears then, that by establishing general rules for finding the
results of all the operations which we have occasion to perform on
algebraical quantities, we arrive at settled methods for resolving
all questions relative to numbers by means of algebraical symbols.
This part of algebra may be entitled, the method of performing
all the operations of arithmetic on algebraical or literal quantities ;
that is on numbers represented by algebraical symBols.
Of Algebraical Operations,
8. Every quantity written in algebraical language, is called
an algebraic quantity^ or the algebraic expression for the proposed
quantity.
8 Elements of Algebra.
Thus 3 is the algebraic expression for three times the number
a, 5 a^ is the algebraic expression for five times the square of a»
2fl9 — Sab -\- 4b^ is the expression for twice the square a, dimin-
ished by three times the product of a and b, and then augmented
by four times the square of b.
We give the name of monomial or simple quantity to a quantity
consisting of one term only, or which is not joined to any other by
the sign of addition or subtraction ; and that of polynomial to a
quantity consisting of more terms than one, or an expression com-
posed of several parts separated from one another by the signs -f-
or — •
Thus 3 a, 6 a*^, 7 a^ b^, are monomials ; S a — 6 6, 2 a^ — Sab
-f- 4 6® are polynomials. The first of these two expressions is
called a binomial, because it is composed of two terms. The
second is called a trinomial, because it is composed of three terms.
9. The numerical value of an algebraical expression is the num-
ber which would be obtained, if particulaj" values were given to the
letters which it contains, and all the arithmetical operations indi-
cated, were performed. This numerical value evidently depends
on the particular values which are given to the letters, and gener-
ally varies with them. Thus 2 a^ has for its numerical value 54,
when a is made equal to 3, for the cube of 3 is 27, and twice 27
is 54. The numerical value of the same expression is 250" when
a is made equal to 5, for the cube of 5 is 125, and twice 125 Is
250.
We have said that the numerical value of the expression will
generally vary with ^ those of the letters composing it; in some
cases, the numerical value of an algebraical expression may remain
constant, although the values of the letters which compose it vary.
Thus, in the expression a — 6, as long as values are given to a
and 6, which increase, by equal degrees, the expression will not
change.
For example, let a = 7, 6 = 4, then a — 6 = 3.
Let a = 7 + 5, or 12, and 6 = 4 + 5, or 9, still a — 6 = 3,
&c.
The numerical value of a polynomial does not change when the
order of the terms is altered, provided care be taken to preserve to
each term its proper sign. Thus the polynomials 4 a' — 3a^6
-h 6 flc*, 6 fl i? — SaH + 4 a^ and Aa^ + ba (? — Z a^b, have
Algebraical Operations. 9
the same numerical value. This is an evident consequence of the
nature of arithmetical addition and subtraction. This observation
will be very useful in what follows.
10. Of the different terms which compose a given polynomiali
those which are preceded by the sign -{- ^1*6 called positive terms,
and those which are preceded by the sign — , negative terms.
These denominations are improper, but custom has sanctioned
them.
The first term of a polynomial is not usually preceded by any
sign, and then it is supposed to have the sign -}-.
11. Each of the literal factors which compose a term, is called
a dimension of that term ; and the degree of a term is the number
of these factors or dimensions. The coefficient is not counted as
a dimension. Thus 3 a is a term of one dimension, or of the first
degree ; 5 a & is a term of two dimensions, or of the second degree ;
1 c?h(?i being the same thing as 1 aaahcc^ is of six dimensionsi
or of the sixth degree.
In general, the degree, or number of dimensions of a term, is
the sum of the exponents of the letters which compose it. It may
be remarked also, that from the definition of an exponent, a letter,
which has no exponent, is considered as having unity for its
exponent. Thus the degree of the term % c?h cd^ \^
2+ 1-1-1+3, or 7.
A polynomial is called homogeneous when its terms are all of
the same degree.
3a — 26-f c,3a2 — 4a6 + 6^6a2c— 4(?4-2c8rf,
are all homogeneous ;
8a^ — 4a i + c,
is not homogeneous.
12. Similar terms are those which are composed of the same
letters with the same exponents. Thus 7 a &, 3 a i, are similar
terms, and 4 a^ 6^, 6 o^ P, are also similar ; 8a^b and 7 a 8^ are not
similar terms ; for though they are composed of the same letters,
each letter has not the same exponent in both. It often happens
that a polynomial contains several similar terms in its expression,
and in that case it admits of simplification.
Let the polynomial be
4a^b — 2a^c + 9a(^ — 2a^b+7a^c — 6P'
Bour.Alg. 2
10 Elementi ofMgebra. !
Hiis can (9) be written thus,
4o«6— 2a»i + 7a»c — 3a«c + 9aci — 6
Now4o®6 — 2a^b is evidendy 2a^6; la^c — 3o"cis4a*c.
Then the polynomial itself becomes
2o36 + 4a»c + 9fl(?— 66»-
Let there be in any polynomial the terms
+ 2anc», — 4 a»6c8, + 6 aHc8, — Son (?, + lla56 A
First, the sum of the positive terms
+ 2o3 6 (? + 6 o^i (? + 11 fl'ftc?,
is equal to + 19 a' 6 <? ;
the sum of the negative terms
_4a»6(?— 8 a'ic^, is— 12 o»6 A
The result of the five terms is therefore
19a3jc* — 12a»Jc", or 7a»ftc».
It may happen that the sum of the negative terps is greater
than that of the positive. In this ease, subtract the positive
coefficient from the negative one, and place the sign — before the
result. Thus, if 5 a^ & be the sum of the positive terms,* and
— Qa^b of the negative ones; since — Qa^b is equivalent to
— 5a«6 — 3a^ 5, then 6a«6 — 8a*6
is equivalent to
6 a« 6 — 6 a« 6 — 3 a« ft, or to —Sa'b.
Whence this rule may be deduced ; — To reduce several similar
terms to one, form one positive term of all the positive similar
termSf which is done by adding the coeffi4dents of these terms, and
affionng to their sum the literal part common to them all. Form,
by the same means, one single negative term of all the terms which
have the sign — ; subtract the less sum from the greater, and give
to the rmdt the sign of the greater.
It is^important to observe, that the reduction affects only the
coefficients, and never the exponents.
We shall find, from this rule, that
Qa^b —Sa*b — 9a^b +I5a^ b — a^b = + 3a«6
7a6c« — ab c^ — 1 abc^ — Sabc^ +4abc^ = — 5a6c'.
The reduction of similar terms is a species of operation peculiar
to algebra^hich appears in algebraical addition, subtraction, mul-*
Mgebraic Addkion. 1 1
tiplicatioD, and diyisiop. These operations we now proceed to
develope*
Of Algebraic Addition.
13. If the quantities 3 a, 5 &, 2 c be added, the result of this
addition is denoted thus, 2a-{'5b-\'2c; an expression which
does not admit of being simplified.
The result of the addition of the simple quantities 4 a' &',
2aS6», and 7a« 6^ is4a« J« +2a« 6» + 7a" iS which when
reduced becomes 13 a, i,.
Take the polynomials
3a« — 4a6, 2a« — 3a6+6S2a6 — 6 6«.
To form one polynomial which will express the sum of thesCi
observe, that to add to the number expressed by 2 a' — 4 a i,
that expressed by 2 a^ — Sab -{-b^iis the same thing as adding
the difference between the number of units in 2a^ -f- i', and the
number of units in Sab, which could easily be done if particular
values were given to a and b. But as this cannot be done in
the actual state of the quantities, it will amount to the same thing
to add 2 a' +b' to Sa' — 4a&, and then to subtract Sab^
which gives
Sa*—4ab + 2a' +bl — Sab'y
or, changing the order of the terms (9),
Sa'—4ab + 2a'—Sab + b'.
In like manner, to add 2ab — 5 b' to this last expression, it is
sufficient to write
Sa' —4ab + 2a' — Sab + b'+2ab — 5ib'.
It now remains to reduce the similar terms (12), and the result is
6o« — 6a6 — 46».
Since analogous reasonings may be applied to all polynomialsi
we may deduce this general rule for the addition of two or mor9
polynomials: — fVrite the given polynomials one qfter wbe . oiberi
preserving to the terms which compose them their respehme tigns^
and reduce the similar terms if there are any. f
The following are examples ; it is required to add *
I. 3a«— 4a6 — 2i*
ba' +2a6 — 6«
Sab — 2b' —Sc'
The result is, 8a' +ab — 5b' —SC. "*^
IH^ Elements (^Algebra.
4a>& — 8c» +9 6«c — 3d^
The result of the addition is,
6a«6+6a6c — 36^0— 14c3 + 2cd^ — 3c^^
Id practice, the given quantities are usually placed under each
other, as in these two examples ; the reduction of the similar terms
is then made, and the results written down with their respective
signs*
Thus, in the first example, as the term 3 a^ in the first line is
similar to the term 5 a^ in the second, write 8 a^ as the result of
the reduction of these terms. Pass then to the term — 4 a 6, and
reduce it with the terms '^2ab and 3 a 6, which gives + a 6.
Write this by the side of 8 a^. Continue the operation thus until
all the terms have been reduced.
Of Algebraic Subtraction.
14. The result of the subtraction of 46 from 6 d, is 5 a — 45.
In like manner the difference between 7 a* 6 and 4a»ftis
7a*6 — 4a»6, or 2a^b.^
Let it now be required to subtract 2 6 — 3 c from 4 a. The
result may be expressed in this form, 4 a — (2 6 — 3 c), by putting
the quantity to be subtracted between pareitheses, and writing it
after the first quantity with the sign — . But it is often necessary
to form a single polynomial from this expression; and in this
consists, for the most part, the rule for algebraic subtraction.
Accordingly, observe, that if a, 6, and c, were given numericaUy,
and the operations indicated by 2 6 — 3 c performed, the result
must be subtracted from 4 a. But as this cannot be done in the
actual state of the quantities, we begin by subtracting 2 6 from 4 a,
which gives 4 a — 2 6. But in taking away 2 6 units, we have
taken awpr a quantity too great by 3 c units ; the result must then
be rectid^ by adding to it 3 c. Thus 4a — 26-{-3cisthe result
of the subtraction required.
Let it be required to subtract 5 a* *— 4a6+35c — 6* from
8 a' — 2 a 6 ; the operation may be indicated by the following
expression,
8a« — 2a6 — (6a3 — 4a6 + 3 6c — 6*).
To reduce tBli^ expression to a single polynomial, observe, that to
Algebraic Addition. 19
subtract 6a* — 4^6 + 3^^ — ^* 's equivalent to subtracting the
difference between the sum & a* -{-^bc ofthe positive terms, and
the sum 4 ab -{- b^ of the negative ones. We first take away
6 fl* -|- 3 6 c, which gives S a' — 2ab — 5 a* — Sic; but as
this result is necessarily too little by 4 a & -f- &*, we must add this
last quantity ; it thus becomes
Sa^—2ab — 5a^ —3bc + 4ab + b^^
or 8a*— 2aJ— 6a« +4a6 — 36c+6*,
if the terms are placed in their order. Reduced this becomes
3a* + 2aft — 36c — i«.
From what ^as been said, the following rule may be deduced ;
To svhiract one polynomial from another j toriie the polynomial
to be iubtracted ctfier the other, changing the sign of each of its
terms, and reduce the similar terms of the resulting expression, if
there are any. Examples ;
I. 6a> — 4a«i + 36«c ^ - 7^3 _7 ^^ 5 j. ii Ja .
— (3a 6— 2a» —Sb^c)^-^^ ^^ b + iio c.
-(6a6-4crf + 36»+3a«)5= — ^* + ^'^~** •
15. By this rule, certain transformations may be made in
polynomiak.
For example, •
6a^—2ab + 2b*—2bc
is reduced to
6a« — (3oJ— 26«+2 6c)5
and, in like manner,
7a» — 8a«6— 45«c + 65»
is reduced to
7 a » — (8 a« 5 + 4 6« c — 6 6»),
or 7a» — 8a«6 — (4iac — 66').
These transformations, which consist in decompoang a polyoo^
nual mto two parts separated from one another by the siga «^, are
very useful in algebra*
14 Elements of Algebra.
Of Algebraic Multiplication.
16. We take for granted a principle which is generally admitted
in books on arithmetic, that the product of two or more numbers is
the same, whatever be the order in which they are multiplied*
(Lacroix^s AritL 2&.)
Let us consider first the case where a simple quantity is to be
multiplied by another simple quantity ; let it b6 required to multiply
7a' 6* by 4 a* 6. The expression of the product is written thus,
7 a* J* X 4 a* 6. But this may be simplified, since by the pre-
ceding principle, and the signification of algebraic symbols (2), it
is the same as 7 x 4X a aaaabbb. Now, as the coefficients
are particular numbers, they may be formed into one by multiplying
them together, which gives 28 for the coefficient of the product.
The product aaa a a is a', and the product bbb isb^ ; so that
the final result is 28 a'^ &*•
Let 12 a* 6* c* be multiplied by 8 a* 6* d«, the product is,
12 X 8 X aaaaabbbbbbccdd, or 96 a^ b^c' d'.
From these we may see that, to multiply two simple quantities by
one another, we must,
,(1.) Multiply the two coefficients together ;
(2.) TVrite after this product all the letters which enter into the
multiplicand and multiplier^ giving to each letter an exponent
equal to the sum of the corresponding components in the two factors;
(3.) ]^a letter enter info one only of the factors, unite it in the
product with the exponent which it has in that factor. The, rule
relative to the coefficients is attended with no diffimlty.
The reason for the rule respecting the exponents is, that any
number a ought to be as many times a factor in the product as it
is in both the multiplier and the multiplicand. Now the exponent
of a letter equals (2) the nuniber of times which it enters as a
iectpr ; therefore the sum. of the two exponents of the same letter
markv the number of times which it ought to be a factor in. the
product required. We shall find from the preceding rule that
8 a* 6 c* X 7 a J c d* = 56 a* J* c» rf«,
2la^b^ cd X Qabc^ =; 168a* 5» c* c?,
4abc X Tdf =28 ah cdf.
17. We now come to the multiplication of polynomials.
JllgAraic MuUipUcaiian. 15
Let there be two polynomiab, a + b -{-c, and d +/ composed
of terms which are all positive or additive, the product may be
expressed ia this form, (a + 6 + c) (d +/)• But it is oftea
necessary to form a single polynomial of the product thus indicated,
and in this consists the rule for the multiplication of algebraical
quantities.
It is evident that the multiplication ofa4-^+ cby(I-|"/is
the same thing as taking a + i -{- c as many times as there are
units in c2, then as many times as there are units in fj and addbg
together the two products. But to multiply a -^ b -{' chy d^ is to
take each of the parts of the multiplicand d times, and to add these
partial products, which gives ad -{•• bd -^ cd* In like manner, to
multiply a '\' b + chjfi is to take each part of the multiplicand
/times, and add the partial products. In this way we find,
{a + b + c){d+f) = ad + bd + cd + af+bf+cf.
Thus to multiply two polynomtaU^ composed of positive terms, mul-
tiply successively each of the terms of the multiplicand by each of the
terms of the multiplier ^ and add all the products.
If the terms have coefBcients or exponents, follow the rules laid
down (16) for the multiplication of simple quantities. For exam-
ple, (3 a^ -j- 4 a6 -{- 6^) (2 a + ^ i) gives for its product
6o3 + 8a»6 + 2aJ» + 15a«6 + 20aft« + 66»,
or by reducing
6 a3 + 23 a« 6 + 22 ai^ + 66'.
To take a more general case, we must begin by observing, that
if tiie multiplicand contains negative as well as positive terms
this factor expresses the difference between the number of units in
the sum of its positive terms and the sum of its negative ones.
And the same holds true of the multiplier. It follows that the
multiplication of any two polynomials is reduced to the multiplica-
tion of two binomials, such as a — 5, c — d, where a denotes the
sum of the positive terms, and — b of the negative terms of the mul-
tiplicand, and c and — d stand for the same things in the multiplier.
We can now effect the multiplication expressed by (o — b) (c — d).
To multiply a — 6 by c — d is evidently to take a — & as many
times as there are units in c, and to subtract from it a — &, taken
as many times as there are units in d, in other words, we are to
multiply a — 6 by c, and to subtract a--^b^ multiplied by d. But
(a — 6) X c is a c — 6 c, and (a — b) X <i is a rf — 6 d;
16 EkmmU of Algebra.
and since the 9ecoQd product is to be subtracted from the first, we
must (14) change the signs of ad — bd, and write them after
a c — i c, which gives
(tf — 6)(c — d) =: ac — be — ad •\^ bd.
The least reflection on the manner in which this product has been
formed, will show that, in all cases of multiplication, each positive
terni of the multiplier must be multiplied by each term of the mul-
tiplicand, and each partial product must have the same sign as the
corresponding term of the multiplicand ; that each negative term of
the multiplier must be multiplied by each of the terms of the mul-
tiplicand, and each partial product must have the opposite sign to
that of the corresponding term in the multiplicand. In the partial
multiplications of each term of the multiplier by ^ach term of the
multiplicand, the jrules laid down in article 16 must be observed.
For example, multiply * >
4o> — 6a«6 —Sab^ + 2b^
by/ :2a'—2ab—4b^
8 a» — 10a* 6 — 16 a« 62 + 4 a* 6'
' —12a* J + 15a^b^ + 24an^ — 6ab'^
— I6a^b^ +20 a^b^ + 22 ab* —Sb'f
8 a« —22 a* 6— 17 o^ b^ +48 a^ J^ + 26 a i* —8b'.
After having placed the polynomials under one another, multiply
each of the terms of the first by the term 2 a' of the second, which
gives
8 a« — 10 a* 6 — 16a»6« +4a''b^,
whose signs are the same as those of the multiplicand. Pass then
to the second term 3 a & of the multiplier ; and as this has a nega-
tive sign, in multiplying all the terms of the multiplicand by it, take
care to give each product the opposite sign to that of its correspond-
ing^ term in the multiplicand, and we shall have
— 12 d^ b + 15 a^ b^ + 24 a^b^— 6 ab*.
Write this under the first.
The same operation, with the term 4bfj which is negative, will
give
-_'16 a^ 62 + 20 a^ i^ + 32 a 6f — 8 6«.
By reducing the similar terms we shall have for the product,
80^-22 a^J — 17 a^b^ + 48 a^b^ + 26 aJ* — 86^.
Algebraic MuMpUeation, 17
Tlie rule for the signs which it is most important to remember in
the multiplication of polynomials, is the following. Whenever two
terms of the mtdtipUer and multiplicand have the same sign^ the
corresponding product has the sign -f* ; and when they ha/ve dif-
ferent signs the product has the sign -— .
This is expressed in algebraical language by saying that + multi-
plied by +, or — multiplied by — , gives + ; and that — multi-
plied by +> or + multiplied by — , gives — . But this rule, which
is unintelligible by itself (because it is not known what is meant by
the multiplication of two symbols, not of quantities, but of arithmet-
ical operations), ought only to be regarded as an abbreviation of
the preceding one. This is not the only case in which algebraists,
to save words, have employed expressions which are incorrect, but
which have the advantage of fixing rules in the memory.
The following are examples of the rule ;
I. 2a^ — 5bd + cf
— 5a^ + 4bd—Qcf
— 15a* + 37o26d — 29a^cf— 20l^a^ + 44bcdf— 8cy»
II. 4a^h^— ^b^c+ %€?b<?— Sa^(? — 7ab<?
2ab^ — 4abc — 26c» + <?
Sa^b^ — I0a%^c + 28aW(? — 340^6 V
— 4a%^(? — I6a^bh + 12a%c* + ^a^^c'*^
+ 14a%c« + 14ai9c« — 3aV — 7aic».
18. There are several important remarks to be made on alge-
braic multiplication.
First. If the polynomials, whose product is required, are homo-
geneous (11) (and the greater part of the questions in which the
aid of algebra is wanted, particularly those of geometry, lead to
similar expressions), the product of the two is also homogeneous ;
this is an evident consequence of the rules relative to letters and
exponents in the multiplication of simple quantities. Moreover, the
degree of eachnterm of the product wiU be equal to the sum of the
degrees of any two terms of the multiplicand and multiplier. Thus,
in the first of the two preceding examples^ all the terms both of the
multiplicand and multiplier being of the second degree, all the
terms of the product are of the fourth degree. In the second, the
multiplicand being of the fifth degree, . and the multiplier of tb^
Bour. Alg. 3
{8 J^eamUs ofAlgthra.
thirdi the product is of the eighth degree. This t emark eenres in
practice to discover any errors of calculation in the exponentd^
For example^ if in one of the terms of a product which ought td
be homogeneous, the sum of the exponents is eqpial to 6, while in
all the rest it is 7, there is an error in the addition of the expo*
nentSi and then we must go back to the multiplication of the two
terms which produced that partial product.
Secondly. When there is no reduction of sinl^ilar terms in tlie
product, the whole nun^er of temu in the prodmt is equal to ikt
product of the number of terms in the multiplimnd multiplied by the
number of terms in the multiplier. This is a consequence of the
rule (17). ^
Thus, when there are five terms in the multiplicand, and four in
the multiplier, there are 5 X 4 or 30 in the product. In geiEeral^
if the multiplicand is composed of m terms^ and the midtiplier of n,
the product will contain m X n terms*
Thirdly. When there are similar terms, the whole number of
terms in the reduced product may be much less.. But among the
diflerent terms of the product^ there are those whjeh cdnnoC be
reduced with any others. These are,
(1.) That which arises from the multiplication of Aa term of the
multiplicttndf which has the highest^ exponent of one of the letters^
by that term of the multiplier which has the highest exponent of the
same letter.
(2.) Tluit which arises from the inultiplication of the two terms
which have the hast exponent of the same letter.
These two products contain that letter with a higb^ «id Itfweib
exponeni: than can be found in any other term, consequently (hey
cannot be similar to any other term. This remark, the truth of
which may be seen from the rule of the exponents^ will be of greirt
use* in divit^on.
19. To finish die subject of algelnraical multiplicatioo, we slnA
subjob some results of frequent use in algebra*
(1.) The square, or second power of the binomial «» -f>> &* On
ihe preceding pnnci{des,
((J + 6)« = (a + 6) (a + 6) = rf» + « a & + fca ;
that is, the square of the sum of two quantities is composed of the
square of the first added to the square of the second^ added to tufice
^ product of the first by the second.
MMpUaUion. 1 9
The square of 6 a^ + ^ A* ^ ^^ b® f<Mnd to be,
(2.) The square of a — ft.
(a — ft)' = (a— 6)(a— ft)z=:a» — 2aft + ft»;
that is, the square of the difference of two quantities is composed of
the square of the first added to the square of the second^ minus tvnce
the product of the first and second. Thus,
(7a«ft« — 12aA»)« = 49o* ft* — IGSa' ft» + 144o«ft«.
(3.) The product of a + ft and a -~ ft.
(a + ft)(a — ft)=:a« — ft*;
that is, the sum of two quantities multiplied by their difference^ gives
the difference of' their squares. This is the theorem demonstrated
m article 5. Thus,
(8 a» + 7 a ft*) (8 a» — 7 a ft*) = 64 a« — 49 a* ft*.
By combining the3e results the products of certain polynomials
may be found more readily than by the ordinary methods. For
example, to multiply
5a*— 4aft + 3ft* by 6a* — 4aft^3ft*.
The first of these two quantities is the sum of the two numbers
5 a* — 4 « ft and 3 ft*, aqd the second is their difiTereoce* Their
product is therefore
(6 a* —4 ab)' -^(3ft*)*, or 25 a* ^ 40 a» ft + 16 a* ft* ~9ft«.
20. It will be observed that in the result of mukiplieatien which
have been obtained, their composition, or the method of obtaining
them from the muhiplieand und multiplier, is independeat of the
particular values of a and ft, the letters which enter into the factors.
The manner in which an algebraical product is formed from its
two factors is catted the law of that product j and this law is always
the samCj whatever he the values of the letters whidi enter into the
factors.
21. The factors of a given polynomial are sometimes evident on
inspection, and the decomposition is often useful.
Let the polynomial be 25 a* ^^20a^ b^l5a* ft*, it is evident
that the factors $ md a* enter into each of its terms. Thus, the
polynomial can be put under the form
5a*(5a*— 6aft + 3ft*).
20 Elementi ofAlgdn-a.
Id like manner, 64 a* b* — 25 a* b* maj be traDsfornoed into
(8 o« ft' + 5 aft*) (8 A« ft' — 6 a ft*).
For, as 64 a* ft • and 25 a* ft« are the squares of 8 a* ft' and 5a ft*,
it follows that the expression is the difference of two squares, and
(19) that it is the sum of the roots of these squares multiplied by
their difference.
Examples in Muliiplicaiion of Compound Quantities.
1. (3a + 3ft + 4c)x (3a + 3ft — 4f)
= 9 a' + 18 a ft + 9 ft' — 16 c'.
2. (4a + 4ft — 3c — 6rf) X (4a + 4ft + 3c + 6rf
= 16 a' +32 aft + 16 ft' — 9 c' — 36 c d — 36 d'.
3. (5a«— 3aft + 7ft') X (3a — ft)
=:15a' — 14a'ft + 24aft'— 7 ft'.
4. (5aft + 3ac — 4ftc) X (7 aft — 18ac + 2ft c + d)
= 35 a' ft ' — 69 a' 6 c — 18 a ft' c + 5 a ft d — 54 a« c'
+ 78aftc' + 3acd— 8ft' c'— 4ft erf,
5. (a + ft + c + d) X (a — ft — c — d)
= a'— ft'— 2ftc — 2ftd — c'— 2cd — d'.
6. (_2a+3ft — c') X ( — 3/— 7a + c»)
= 6tt/— 9ft/+3c'/+ 14a«— 2Jaft + 5ac'
+ 36c'— c*. '
7. (3a+4c — 5d)H-(6ft — 7n— 6m) X 3aH-4c
_5d)— (6 ft — 7n — 6 m)
= 9a' +24ac — 30ad + 16c'— 40crf+ 26d«
— 36ft'+84ftn— 49n«+72m6— 84nm — S6m'.
Diinsion.
22. Algebraical and arithmetical division have the same object^
viz. when the product and one of the factors is given, to find the
other factor.
Let us consider first the case of two simple quantities.
. To divide 72 a ^ by 8 a' the result of which is expressed thus
-g^. A third quantity is wantetl, which, when multiplied by the
second, will produce the first. Now, from the rules for the mul-
tiplication of simple quantities, the quantity sought is such that its
Diviiian. 21
coefficient nuiltiplied by 8 gives 73, and the exponent of a added
tb 3, its exponent in the divisor, gives 5, the exponent of the divi-
dend. This quantity is therefore found by dividing 72 by 8, and
subtracting the exponent 3 from the exponent 5, which gives
8a3 "" ^'
and it is plain that 8 a' X 9 a* = 72 a*.
By the same rule
and
labx 6a^bc=:25a^h^c.
Hence it will be seen that to divide one simple quantity by another,
it is necessary,
(1.) To divide the coefficient of one by that of the other ;
(2.) To write after this quotient the letters which are common to
the dividend and divisor^ giving each an eocponent equal to the
excess of its component in the dividend above that in the divisor ;
(3.) To write after these the letters which enter into the dividend
and not into the divisor ^ irnth their respective exponents.
Hence
23. The division of one simple quantity by another is impossible^
(1.) If the coefficient of the one be not divisible by the coefficient
of the other ;
(2.) ^ the exponent of any letter be greater in the divisor thfn^
tn the dividend ;
(3.) If the divisor contain one or more letters which are not in
the dividend.
If either of these three things happen, the quotient remains under
the form of a fraction, that is, an expression in which the algebraical
»gn of division necessarily enters, but which may often be simplified.
For example, let it be proposed to divide 12 a* b^ c d by
I^A^bc^,
We canqot gjjj^e this quotient the form of a simple quantity, tliKt
is,^of a quatjitity freed from the sign of division, because 12 is not
divisible by 8^ and the exponent of c is less in the dividend than in
' 12 a* 6^ c (f
the divisor. So the quotient must have .the form g aA 2 * ^^^
ff9 Element qf Algebra.
liw pay be smpliSed ; for, smce 4, a>, &, and (s am comtiMi to
lK>tb tonns of the fractipo, they may bo canoellod, and the restdt
. Sa^bd
becomes — ^ •
To simplify a fraction, the terms of which are simple quantities,
we must,
(1.) Divide the two coefficients hy their greatest common factors
(2.) Take the least of the two exponents of the same letter from
the greatest f and write the letter j mth this different for its ex^
ponentf in that term of the fraction which has th(i greatest exponent;
(3.) Write those letters with their respective exponents which are
not common to both terms j in the terms to which they beiong.
Thm we find thai;
48q3 62c d!3 _ 4a^ ^ S7al^c^d _ Slbf^c ^ 7 (fib _ _1_
Ip the l^st example^ as all the factors of the dividend are found
in the divisor, both terms of the fraction may be divided by the
numerf^tpr, and the numerator is reduced to unity«
34» It often happens that the exponent of certain letters is the
same in the dividend and divisor.
For example, in dividing 24 a^b^ by 8 a' 6', as the letter 6 has
the same exponent in both, the quotient must be without b ; and
-g- — ^ == 3 a. But this quotient may be put in such a form as
to preserve traces of the letter i, which has disappeared in the
reduction.
If we agree to apply the rule of exponents (22) to the expression
^ it becomes g = b^^ = 6®. Thb new symbol b^ indicates
(2) that the letter enters times as a factor into the quotient, or,
which is the same thing, that it does not enter at all ; but it indicates
at the same time that it did enter both into the dividend and divisor,
and that it has disappeared in the division.
This symbol has the advantage of preserving traces of a num-
ber which made part of the problem to be solved, without changing,
on that account, the value of the result ; for, since b^ came from
so
jg, which is equal to uni^, it follows that 3 at* b equivalent to
3 a X 1) o' to 3 a. In like manner
Dwirim. 33
«
As it is inqfiortaiit lo have exaet notions ^oooernnig thd origiQ nil
signification of the symbols used in algebra^ we shall show tlmt gen^'
erally any quantity a, with an exponent 0^ is equivaleat to^anityV
that is, a® = 1. .
This expression arises ftom a having the same exponent in the
dividend and divisor of the division. So we have a" = — (where
m denotes the yrhole number which is the exponent of a). But
the quotient of a quantity divided by itself is I. Therefore
-- = a" = 1.
a*
We repeat that the symbol a® is employed ^mveniihnaUf ik
order to preserve a letter which entered into the eaunciatkni of if
question, bul which must disappear in the division ; and it is often
oecessaij to preserve this mode of expressbn.
Dkuian tf 2W Polynomials^
25. It is required to divide
bla^i^+ 10a* — 48iP^6— .166* + 4a6»
in order to Mow th^ steps of the ptocess easily, they are iBs*
posed thus ;
61a«6» + 10rf« — 46(r^b — lU* + 4ai* ) 4fl& — 5a» ^- 8i^
8a% — lOa^ + 6a«6a j ^2a» + 8a4 — 5fi*
57a269_ 40a34 _ 1564 ^ 4^3
— S2a^b? + 40a^b — 24at»
26a«6»— 156* — 20ai*
20aft8 — 25a2J»+15ft*
' The object of this operation is, as we have already said (22),
to find a third polynomial^ whichy being mtdtipUed by the second^
shaU produce the first.
From this definition and the rule for the multiplicaticjn of poly-^
nomials, it appears that tlie dividend is the sufn after addition and
reduction^ of the partial products arising from multiplying each
term of the divisor by each term dt the quotient sought. U then
we can discover in the dividendl, a tei^m which afosot vfUhaut rednc^
f4 i ElemenU of Algebra.
lum^ from the muhaplicatioo of one of the terms of the divisor by
one of the terms of the quotient, then, by dividing the term of tbe
dividend by that of the divisor, we may be certain of obtaining one
term of the quotient sought.
Now, from tbe third remark of article 1 8., the term 10 a^ contain-
ing the highest exponent of the letter a arises without reduction
from the multiplication of those two terms of the divisor and quo-
tient which have the highest exponent of the same letter. Then
the division of 10 a^ by — 6 a^, certainly gives one term of the
quotient sought. But here a difficulty presents itself, which is, to
determine the sign which this term of the quotient ought to have.
Thi^ we may not be hereafter detained on this subject, we proceed
to establish a rule for the signs in division.
As in multiplicatioti the product of two terms of tbe same sign
has the sign -f-, and the product of two terms of contrary signs has
tbe sign — , we may conclude,
(1.) That if the term of the dividend has the sign -^^ and the
term of the divisor has the sign -f- alsOj the term of the quotient
must have the sign -|- ;
(2.) Jff^the term of the dividend has the sign .+, and that of the
divisor the sign — , the term'of the quotient must have the sign — >,
hecause there is no sign hut — , whichj when combined with the sign
— of the divisor J will give the sign -{-ofthe dividend ;
(3.) J^the term of th^ dividend has the sign — i and the term of
the divisor the sign -^-^the quotient must have the sign — ;
(4.) Jifthe dividend has the ^^n— —, and the divisor the sign — ,
the quotient must have the sign 4"* ,
That is, if the terms of the dividend and divisor have the same
sign, the quotient must have the sign -f«, and if they have contrary
signs, the quotient must have the sign — . We may abbreviate the
rule by saying,
like signs give -|-,
unlike signs — .
In the proposed example, 10 a* and -^ 5 a^ have unlike signs,
and their quotient must have the sign — ; moreover 10 o*, divided
by 5 a^j gives 2 a^ (22) ; then — 2 a^ is a term of the quotient
sought. Write this term underneath the divisor, and multiply each
term of the divisor by this term, then subtract the product
Division* 26
from the diyid^Dd, which is done by writing it with contrary signs
uoderaeatb the dividend, and performing the reducdon. The first
partial operation gives
57a2 62_40aH — 15 6^ + 4ab\
This resuh is made up of the partial products of each of the
terms of the divisor, by each of the remaining terms of the quo-
tient. We may then regard it as a new dividend, and reason on
it as on the former one. We must then take, in this result, the
term — 40 a^ 6, which has the highest exponent of a, and divide it
by the same term — 5 a^ of the divisor. Now, by the preceding
principles, — 40 a^ J divided by — 5 a^ gives -|- 9 a 6, a new term
of the quotient, which is written by the side of the first. Multiply
each of the terms of the divisor by this term^ and write the product
with contrary signs, underneath the second dividend. Make the
reduction, and the result of this second operation is
26 a\b^ — 15 b* —20 ab^.
Divide 25 o* 6' by — 5a^, the result is — 5t*, which is the
third term of the quotient. Multiply the divisor by this terra,
write the terms of the product with contrary signs underneath the
last dividend, and make the reduction ; the result obtained is 0.
Then — 2 a« -f 8 a 6 — 5 6«, or 8ab — 2a^—bb^, is the
quotient required, which may be verified by multiplying the divisor
by this polynomial ; the product will be equal to the dividend.
If we reflect on the preceding reasoning, we shall perceive that,
as in eaoh partial operation, it is necessary to find that term of the
dividend which has the highest exponent of one of the letters, and
to divide it by that term of the divisor which has the highest expor
nent of the same letter, the trouble of jooking for them might be
avoided by taking care at first to write down ^he terms of the
dividend and divisor, so that the exponents of the same letter should
diminish as we go from right to left. This is called arranging the
dividend and the divisor according to the powers of the same letter.
By this preparation, the first terms of the dividend and divisorj on
the left hand, are always those which must be divided by one an-
other, in order to obtain one of the terms of the quotient; and it is
the same in all the following operations, because the partial quo-
tients, and the products of the divisor by those quotients, are always
arranged.
The following is an exhibitioa of the preceding division, after the
arrangement of the two polynomials.
Bour. Alg. 4
36 Elements of Algebra*
10(1* — 48a% + ola^i' + 4ab^ — 166* > -^ 5a» + 4fl& + 3^
— 10£i*.+ 8a36 +6a26» O — 2a2 + 8a6 ~ 662
"^ _ 400^6 + b7a%^ + 4ab^ — 156*
_j- 40a36 — S2a%^ — 24ab^
"^ 25a«62 _ 20a63 — 156*
— 25a36» + 20a63 + 156*
" ' ■■ T ' I ■■■■-!■ .■■ ■■. -
26. From this we may deduce the following rule for the division
of polynomials ; arrange the dividend and divisor according to the
powers of the same letter y divide the first term on the left of the divi-*
dend by the first term on the left of the divisor, and the first term of
the quotient is found ; multiply the divisor by this term, and subtract
the product from the dividend. Divide the first term of this re-
mainder by the first term of the divisor, and the second term of the
quotient is found ; multiply the divisor by this second term, and
subtract the product from the result of the first operation. Con-
iinue the same series of operations until the result becomes ; in
which case the division is said to be exact.
When the first term of the arranged dividend is not exactly
divisible by the first terra of the arranged divisor, it is a sign that
the division is impossible, that is, that there is no polynomial, which
when multiplied by the divisor can produce the dividend. And, in
general, the divi^on is impossible when the first term of one of the
partial dividends is not divisible by the first term of the divisor.
27. Though there is some analogy between arithmetical and
algebraical division, with respect to the manner in which the opera-
tions are disposed and performed, yet there is this essential differ-
ence between them, that in arithmetical division the figures of the
quotient are obtained by trial, while in algebraic division the quo-
tient obtained by dividing the first terna of the dividend by the first
term of the divisor, is always one of the terms of the quotient
sought. If these two terms are not divisible by one another, we
may conclude immediately that the division is impossible. Besides^
there is no reason why the operation should not commence from
the right instead of the left, and it would then begin with those
terms which have the least exponent of the letter, with reference to
which the whole has been arranged. In arithmetical division, the
quotient can only be found by beginning from the left.
Division, 37
Sach is the independence of the partial operations belonging
to this process, that after having subtracted from the whole
dividend, the product of the divisor by the first term of the quo-
tient, we may, in the second operation, divide by one another
those two terms of the new dividend and divisor, which have the
highest exponent of a different letter from that which was first
taken ; and one of the terms of the quotient which remained to be
determined is thus found. If we retain the same letter, it is be-
cause there is no reason for changing, and because the two polyno-
mials being already arranged with reference to the powers of one
letter, the first terms on the left of the dividend and divisor will
give a terra of the quotient ; whereas if the letter were changed, we
should have to search again for the terms which contain the highest
exponent of the new letter.
28. To divide
21 x^ y^ + 25 a?3 y^ + 6Sxy* — 40 y» — 56 a?* — 18 a?* y
by 5 y^ — 8x^ — 6 a? y.
The operations are as follows, after arranging the whole with
reference to the powers of y.
— 40y5 +6833^ + 253:^ + 21ar^—18a:4y-«56r5> 5y ^-.6xy — 8ja
-|-40^^_4ary4-„64a:3j/3 S --8y3+4a^-^3xS^+7x*'
— 20iy4 + 24a;2y3 + 32x3y2
— l5arV + 531^2 _i8j:43^ — 56x6
+ 15x22^ — 18a%2 — 24x4^
35x3y3 — 42r4y — 56x5
— 35a^2+42ar4y + 56r5
As it is important for beginners to make themselves familiar/ with
algebraical operations, and, above all, to calculate with readiness,
we shall give this last example again with some simplifications, which
it is convenient to introduce.
-^0yS+68xy4+25x2ya+21x»yg--18x4y— 56x5 > 5yg— 6xy— 8x2
Ist rem. 20:^4l-39a;8y3_|^ix3^«--i62:4y— 56x5 J — 8y3+4xy®--3x^+7x3
2d rem. —15xy+53xV— 18^:^—56x5
3d rem. 35x3y«— 42x4y— 56x5
X Elements of Algebra.
First, divide — 40 y^ by 6 y^ , and the quotient is — 8 y^. Mul-
tiply 6y« by — 8 y^, and we have — 40 y*, which, with its sign
changed, is 40 y^, and destroys the first term of the dividend.
In like manner — 6xy X — 8y^ gives + 48 a? y\ and with its
sign changed — 48 a? y^, which reduced with 68 cry* t^ives 20 a? y*.
Lastly, — 8 a?a x — 8 y^ gives, after its sign is changed, — 64 a* y*;
which, reduced with 25 x^ y^/is — 39 a?^ y^. The result of the first
operation is, therefore, 20 a? y* — 39 a?* y', followed by those terms
of the dividend which have not been reduced with the partial pi'o-
ducts already obtained.
The new dividend is ti^eated after the manner of the fi:rst, anrf
so on.
To divide 95 a — '7Scfi+ 56 a* — 25 — 59 a^ ,
by _ 3 a« 4- 5 _ 11 a ^ 7 ^3.
56a*— 59a3— 73a* + 95tf — 25 ^ Ta^ — 3a«_ 110 + 5
Istrem. — 35a3+ 15a* + 5^« — 25 > 8a — 5
2d rem.
29. It is possible that one oi^ both of the given polynomials may
contain several terms which have the same exponent of that letter
according to powers of which the whole is arranged.
How, in this case, is the division to be performed ?
Suppose it were required to divide
Ila*6 — 19 a6c + 10 a^— 15 a^cH- 3 a52+ 15 5 c* — 5 iV
by 5 a^ -\- Sab — 5 6c.
The two terms 1 1 a* 6 — 15 a'c can be put in the form
(11 6 _ 15 c) a', or 116
— 15c
by writing down a* once, and placing on its left, in the same vertical
column, the aggregate of the quantities by which it is multiplied.
This polynomial multiplier may be called the Coefficient of a.
This seeond method of bringing together the terms which oo]>
tain the same power, is preferable to the first, on two accounts ;
(1.) because, if there are many terms in the dividend and divisor,
the same horizontal line will hardly contain them all ; (2.) because,
isince the coefficient of each power ought itself to be arranged in
powers of one of its letters, we are obliged, in employing the first
method^ to make a modification, when the first term is negative,
which may lead to error.
o%
«%•
For elatnple^' in fhe expression
— 16 6*fl' + rihca^ — B<?a\
the modification consists in putting this expression under the form
_(15J«_75c + 8c*)o*,
which, by ttie second method, is written thus, — 15 6'
+ 76c
— 8c*
and this metliod has the advantage of preserving to each term the
sign which it had at first.
In like manner — 19 a J c + 3 tf 6^ is written +3 6' a.
— 19 6c
The operation is performed thus^
lOa^ 4- 116 I a^ + 36' a — 56'c + 156c' ) 5a' + 36a — 56c
— 15c — I96c
2a + b —3c
1st rem. 56
— 15c
a'+ 36' a — 56'c+156c'
— 96c
2d rem. ,
First, divide 10 a^ by 5 a', and the quotient is 2 a. Subtract the
product of the divisor by 2 a, and the first remainder is obtained.
Divide the part of^this remainder, which contains a', by 5 a', and
the quotient is 6 *>— 3 c. The subtraction of the divisor, multiplied
by 6 — 3 c, gives no remainder, therefore 2 a + 6 — 3 c is the
iquotient required.
To explain generally the preceding method; which is the most
complicated in division, let the dividend be
A a^ + B a^ + c it* + D a + E, and the divisor a^ o' + b' a + c'.
(It is customary in algebra, when there are a great number of
iquantities in the same question, to designate si bertain number of
them by dijSerent letteirs ; and, in order to avoid having too many
letters, the others are designated by the same letters accented.)
In these two polynomials, ^ach of the coefficients, a, b, c, d, e,
a', b', c\ is the aggregate of several terms. Thus, a a^ represents
all the part of the dividend which contains a\ Since the highest
exponent of a is 4 in the dividend, and 2 in the divisor, it most be
2 in the quotient, which is therefore of the form
To determine that part of the quotient which has the highest power
of a^ observe that the product of the two parts a' a' &nd a" a*,
30
JSlemenis of Algebra.
cannot undergo any reduction with the other terms of the product
of the divisor and quotient (Remark 3d^ art. 18), and therefore
must be equal to the part a a^ of the dividend which has the high-
est power of a. If then we divide a a^ by a^ a*, we shall have the
part of the quotient A^^a^y which amounts to dividing a by a^', since
a^, divided by a*, gives a^. If a and a' ar« themselves polynomials
of one or more letters, the operation of division is performed, which
requires that the polynomials should be arranged according to the
powers of some one of their letters ; and this is the reason for what
is said ahove, that in writing the terms which are multiplied by one
power of some letter, it is necessary to arrange them according to
the powers of some other letter. If several terms in one column
contain the same exponent of the second letter,, they must be
arranged according to the powers of a third letter.
Having obtained a^^ a^ multiply the divisor by it, and subtract
the product. The first remainder is thus obtained, and the same
operation is performed on it as on the dividend.
The following are examples of this method. The partial divis-
ions which the principal operations require, are annexed.
12J*
a^ + 2363
a* -1- 1064
a ^
— 296c
— 316*c
66V
f
+ 16c*
— 96c*
r
+ 16c»
)
1st rem.
+ 1563
a* + 1064
a
— 256*c
— 66*c*
— 96c*
•
+ 15c3
36
— 5c
+ 2i»
46
— 3c
a
+ 56*
— 3c*
a
2d rem.
First partial division.
12 68 — 296c+ 15c3>36
1st rem.
2d rem.
96c + 15c* )4b — Sc
\ Sb —
)4b —
— 5c
Ist rem.
Second partial division.
15 63 — 25 6* c — 9 6c* + 15 c3^ 36 _^
3c>
— 96c*+ 15c^
> 36 —
Ub' —
Diviium,
31
i
e6tfC-76*
—10
+236
— ^
+22b*
—316
a»+463 ja^jt
— 96' —26
+ 66
— 6
36
5
a+h*
—261
2a'— 36
+4
—1
a+1
1st rem. — 96*
+276
—20
2d rem.
3d rem.
4th rem^
(1.)
(2.)
a
3*
126'
—236
+ 5
a
36
—5
a
Partial divisions,
66 — 10 ^ 36 — 5
0)2
— 9 6' + 27 6 — 20 > 3'6 — 5
126
— 20 )
— 20 5 —
36 + 4
(3.)
12 6* — 236 + 5>36 — 6
) 36 — 5
546 — 1
(4.)
— 36+5
3^_— 6)36 — 5
30. There is another important case of algebraic division;
\?faere the dividend contains one or more letters which the divisor
does not contain. The division may be made in the ordinary
manner, but there is a much more simple method of finding the
quotient. Suppose, for example, that the dividend contains various
powers of the letter a, and that this letter does not enter into the
divisor (which is then said to be independent of a). Arrange the
dividend according to the powers of a, and let it be
A a* + B a' + c a* + D a + E,
4 being the highest exponent of a, and a, b, c, d, e, simple quan-
tities or polynomials not containing a. Let m be the divisor, which
is independent of a.
Since the divisor, multiplied by the quotient, must reproduce the
* The remainders are not written down at full length, as it would
only be repeating the terms of the dividend.
3S Element ofMgebra.
dividend, and tsince the divisor m does not contaio a^ it is dear that
the quotient is a polynomial, which contains the satge powers of a ^
as are in the dividend. The quotient is tl^erefore necessarily of
the form
A' a* + B' a^ + c< a* H- 1/ a + b^
Imagine that this quotient is found, and that the divisor is multi-
|)lied by the several parts Af a\ b' a?, &c. The producte will be
A^ M a\ B^ M a^ c^ M a', &c. ; and as these are not capable of any
reductioji with one another, since they contain different powers of a^
they must be respectively equal to the terms a a^, b a^ &.C., of th^ -
dividend.
We h2\ve therefore,
a' M = A, or Af == -,
' M
b'jm =: B, or b' = -,
Sec. &c.
and so on ; whence this general proposition is deduced ;
J^ a polynomial^ arrmged according to the powers of a certain
letter^ be exactly divisible by a polynomial which is independent of
that letter^ each of the coefficients of the various powers which enter
into the first polynomial^ must be divisible by the second. Th^
coefficients of the various powers of that letter in the quotient are
quotients arising from the division of the coefficients of the dividend
by the divisor.
To divide
3a363 ^ 2ab(? — 2b^(? + b^— Sa%<? + 3ab^c—a?(? + bc^ + a%^e
hy b' — c\ '
The dividend, arranged with reference to a, is
(3J3 + b^c — 3i(? — (?)a^ + {Sbh — Sb(?)a +b^ — 2b^(? + be*.
The three partial divisions are
3fe3.^fe2c — 35cg — c8 3 63c — 35 c3 , b^ — 2b^c^ + bt*
6a_^2 ' 62_c2 ' ^^" 62_e2 '
the quotients of which are 3 6 + c, 3 6 c, and 6^ — bc^; therefore
the whole quotient is
{Sb + c)a' + Sbca + P — bc\
The two last quotients can be obtained more easily than by the
ordinary method, if we observe that
Divitian. 3S
, 36V — 3ir» = 36c(6« — c•),^
and ^
b^ — 2P<^ + bc^=ib {JA—2V e + e') = h (&' — c*)' (19).
We may observe, while on this subject, that though there are
general rules for performing all these operations, they can very
often be simplified, and these simplifications must not be neglected
when there is an opportunity of making them.
31. Among the different examples of algebraic division, there is
one of such remarkable application, and so often met with in the
resolution of problems, that algebraists have made a kind of theo-
rem of it. We have seen (5 and 19), that
(a + J) (a _ J) = a« _ J« .
fj^ 59
hence, — , - gives a -{-b iox its quotient.
Divide c? — 6^ by a — 6, and the quotient is a* -\- ab -{' J*.
In like manner, divide a* — 6* by a — 6, and the quotient is
These results are obtained by the ordinary operations of division ;
and analogy would lead us to conclude that, however great the
exponents of a and i, the division leaves no remainder ; but anal-
ogy cannot lead to certainty or rigour.
To be certain on this point, call m*the exponent of a and &,'and
begin the division of a* — 6* by (a — 6). First divide a* by a,
the quotient is a"*"^ (22). The product of a ' — b and a*~^, being
subtracted from the dividend, leaves 0"*^* 6 — 6** for the first
remainder, which may be put under the form b (a**"^ — 6*^^).
Whence it is plain, that if a*~^ — 6**""^ be exactly divisible by
a — 6, so also is a* — 6" ; that is, if the difference of the same
powers of a certain degree of two quantities is divisible by their
difference, the difference of the powers of the next greater degree
flS 52
IS also divisible by that difference. Now, — , is exacdy divisi-
ble, and gives the quotient o + J ; whence — ~-j- gives an ex-
act quotient, equal to
o' + ^.J,
'a — b '
or to a* + (o + b) 6, or to a* + a 6 + 6*.
Bour. Alg, 5
34 Elements ofAlgebra^
ff* — fc*
For the same reason ^ . gives an exact quotient equal to
* a — d '
or to a' + 6 (a* + aJ + V\ or to a^ + a* J + a 6» + V.
J 11 fl"* — 6"* . ,
sina, generally, — zzriT S*^^^ ^" exact quotient equal to
a»-i ^ a'^s^ + a"^6* + + o J—* + ft*-!.
This proposition may be verified a posteriori, by performing the
multiplication
(o*"-! + a'^-sj ^ am-3 J ^ ^ + a6«-2 + 6«-i) (a — J).
We niay see that the partial products o" and — t" are the only
ones which are not destroyed in the reduction. For example, the
multiplication of a"^^ 6 by a, gives a"^^ 6, and the multiplication of
Qin-i fjy — J gives — a"*~^6, which destroys the preceding pro-
duct. The same holds of the other terms.
Beginners should reflect well on the preceding method of demon-
stration, as it is often employed in algebra*
32. We have given (23 and 26) the principal characteristics by
means of which it appears that the division of simple quantities or
polynomials cannot be exactly performed ; that is, that qo third
algebraic quantity can be found, which, multiplied by the secondi
will produce the first*
Wo may add that, in polynomials, simple inspection often shows
that one is not divisible by another. When the polynomials con-
tain two or more letters, before arranging them acccording to the
powers of one of the letters, compare the two terms of the dividend
and divisor which contain the highest exponents of each letter. If
the terms containing the highest exponents of any one of these let-
ters are not divisible by one another, an exact division is impossible.
The same remark applies to each of the operations which the pro-
cess contains.*
* These remarks are only applicable to all cases in which the
coefficients are whole numbers. For it may happen that two polyno-
mials, one or both of which contain fractional coefficients, may be
divisible by one anotl^er. For instance
^a'+l^«6 + |6»,dividedby|a + i6,
gives the quotient a ^ H" 7 ^*
Division. 35
Suppose it were required to divide
iia* — ^a*b + 7 ab* — 11 b^
by 4a»_8a6 + 36*.
With the letter a the division appears possible, but od looking at b
it is shown to be impossible, since — 11 6^ is not divisible by 3 6*.
The following considerations will conclude this subject ;
(1.) A polynomial cannot be divided by another polynomial con-
taining a letter which is not in the dividend ; for it is impossible that
a third quantity, multiplied by a second, which contains a given
letter, can produce a result independent of that letter.
(2.) A simple quantity is never divisible by a polynomial, because
(18) the product of two polynomials has always two terms at least
which cannot be reduced.
(3.) A polynomial is not divisible by a simple quantity, unless that
simple quantity divide each term of the dividend exactly, and the
quotient is found by separating the common factor from all the
terms. ,
Examples in Division of Compound Quantities.
1. (4ac — 2arfc)-f-2a = 2c — 2de.
2. {Sa* — 6ab)-, 2a= — 4a + 3i.
3. (a 6 — a c) -T- (6 — c) =z a,
4. (ac — be + ad — 6 rf) -r- (a — 6) = c + rf.
6. 4a' + 6a6 — 4aa? + 9 6a?— 16x')-r.(2a+ 3a?)
= 2a + 36 — 6a?.
6. (4a^ + 4a?* — 29a? + 21)-r-(2a? — 3) = 2x»+ 5a? — 7.
7. (36a'— 12a6 + 46* — 36ac+ 126 c + 9c») -r-( 6a
_26 — 3c)
= 6a — 26 — 3 c.
8. (a« — 46*) -r- (o* + 2 6') = a* — 2 6*.
9. (a4_ gi j« _ 6 a 6 c' — c*) -r- (a* _ 3 a6 — c')
= a* + 3a6 4- c*.
10. (64 a' + 64 a 6 + 16 6" — 9 d* — 48d — 64) -r- (8a
+ 46 + 3rf + 8)
=:8a + 46 — 3d — 8.
11. (32a5 + 65)-r-(2a + 6)
=r 16 a* — 8 a3 6 + 4 a* 6* — 2 a l» + 6^
36 Elements of Algebra*
12. (18a» + 33a6 + 42ac— 12ad — 306' + 1246c
-}-86£/— 16 c* — 32c£?)-5-(6a+ 16 6— 2c — 4ii)
=:3a — 26 + 8c.
Of Algebraic Fractions.
33. Algebraic fractions are the same in their nature as arith-
metical fractions } that is, the unit is conceived to be divided into
as many equal parts as there are units in the denominator (this
denominator being either a simple quantity or a polynomial), and as
many of these parts are taken as there are units in the numerator*
Hence, addition, subtraction, multiplication, and division of fractions,
are performed by the rules which are laid down in arithmetic. But
in all applications of these rules, the foregoing methods for treating
whole algebraical quantities must be attended to. It would be
needless to detain the student on this subject ; the sequel will afford
abundant opportunities for the practice of these rules.
Nevertheless, the reduction of algebraic fractions to their most
simple expression is entitled to particular notice.
When the division of simple quantities cannot be exactly per-
formed, it is indicated by the common sign of division ; and. in this
case the quotient takes the form of a fraction, which we have
already shown how to simplify (23). Such polynomial fractions as
the following are also very easily reduced. As a first example, take
gg — 68 ■
a2 — 2 a 6 + 62*
This fraction (19) can be put under the form "^
(a-6)(a + fe)
(a — 6)9 '
suppress the factor a — 6, which is common to both terms, and the
result is
a -|- 6
Again, take
a— 6'
6a3_ 10025^.5^52
8a3 — 8a2 6
This expression is equivalent to
5a(a8 — 2q6 + 6g) 5a(fl — 6)g
8a9^a— 6) . ' ^^ ^^ Qa^\a—hy
Elementary Theory of the Oreatest Algebraic Common Divisor. 37
Suppress the common factor a {a — i), and the result is
5{a — b)
8a '
These particular cases which we have examined are those in
which the terms of the fraction contain the product of the sum and
difference of the two quantities, or the square of their sum and
difference ; practice will enable the student to effect these and
similar decompositions with ease, when they are possible.
But the two terms of the fraction may be more complicated
polynomials, and then their decomposition into factors is no longer
80 easy, and recourse must be had to the process of finding the
greatest common divisor.
Elementary Theory of the Greatest Algebraic Common Divisor,
34. The greatest common divisor of two polynomials is the geat"
est polynomial, with reference to the exponents and coefficients which
divides exactly the two proposed polynomials.
The distinguishing property of the greatest common divisor is,
that if we divide the two proposed polynomials by it, the quotients
which result are prime to each other^ that is, they do not contain
any common factor.
This proposition is evident. For let A and B be the given
polynomials, D their greatest common divisor, A^ and B' the quo-
tients, we have necessarily
A=i A' X D nni B z=iB' X D.
Now, if A', and B^ had still a common factor (2, it would follow
that d X D would be a divisor common to the two polynomials,
and greater than X), either with reference to the exponents or to
the coefficients, which would be contrary to the definition of D.
36. We have seen {Arith. 61),
(1.) That the greatest common divisor of two whole numbers
contains as factors all the particular divisors common to the two
numbers, and can contain no others ;
(2.) That the greatest common divisor, of two whole numbers is
the same as that oj the smaller number and the remainder cfter their
division.
The theory of the greatest algebraic common divisor depends
equally upon these two principles.
38 Elements of Algebra.
This being admitted, suppose in the first place that it is required
to find the greatest common divisor of the tWo polynomials
a^ — a^b + 3a63 — 363 and a^—5ab + 4bK
First Operation,
a3~ dfib + Sal^ — Sb^ ^ flSi— 5 a i + 4^
+ 4a^b— ab^ — Sb^ Sa + Ab '.
1st rem. I9ab^—19b^
or i9P{a — b)
Second Operation.
a9 _ 5 a 6 + 4.6» > a— b
— 4ab + Ab^\a — 4b
which gives a — 6 for the greatest common divisor.
We commence by dividing the polynomial of the higher degree
by that of the low^r ; the quotient is, as we have seen above,
a '{' 4b*j and for a remainder we obtain . • . 19 a b^ — \9b\
It follows from the second principle, that the greatest common
divisor sought is the same as that of the remainder, and the poly-
nomial which we have taken as a divisor.
But since 19 ab^ — 19 6^ may be put under the form . . . .
19 6* (a — 6), we see that the factor 19 b^ divides this remainder
without dividing c? — 5 a 6 -f- 4 6^ ; then, according to the first
principle, this factor cannot enter into the greatest common divisor ;
so that we can, without inconvenience, suppress it, and the question
is reduced to seeking the greatest common divisor of
a^ — 6 a 6 + ^^ ^^^ ^ — ^'
Now dividing the first of these two polynomials by the second,
we have for ah exact quotient, a — 4 6 ; then a — i is the great-
^t common divisor of the two proposed polynomials.
.Let us take again the same example, having arranged it with
reference to i,
— 36' + 3a6» — a«6 + a^ and AV—bab + a\
Elementary Theory of the Greateet Algebraic Common Divisor. 39
First Operation.
— 12 ¥ + \2ab^— 4a^b + 40? ^ 4 ja — 5 f/6 -^ gg
1st rem. — Sab^— a^b + 4a^)—3b , — 3a
— 12at»— 4«»6 + 16a3
2d rem. — 19a^b + 19 a'
or 19 a^{—b + a)
Second Operation.
4ja_6a6 + a^\ — b + a
— ab + d*} — 4&-f~^
which gives — b '\- a or a — i for the greatest commoD divisor.
At first it is difficult to perform the division of the two po]]|rno-
mials because the first term — 3 6' of the dividend is not idivisible
by the first term 4 b^ of the divisor. But if we observe that the
coefficient 4 is not a factor of all the terms of 4 6* — 6 a i -|* a',
and that thus, according to the first principle, 4 cannot be a part of
the greatest common divisor, we can without any inconvenience
introduce this factor into the dividend which gives
— 12P+ I2a}^ — 4a^b + 4a^;
then the division of the two first terms becomes possible.
Performing this division we find for a quotient — 3 &, and for a
remainder — Sab' — a' 6 -|- 4 a'.
As in this remainder, the exponent of b is still equal to that of
the divisor, we may continue the division *by multiplying anew this
remainder by 4, in order to render the division of the two first
terms possible.
This being done, the expression becomes
— 12 ab' —4a'b+ 16 a',
which, divided by 4 6* — 6 a J -f- ^*> g'^^s for a quotient — 3 a
(which we separate from the first by a comma, as having no con-
nexion with it), and for a remainder — I9a' b -{* 19 a^.
This last remainder can be put under the form 19 a^ ( — 6 + a).
We then suppress the factor 19 a^ as making no part of the great*
est common divisor, and the question is reduced to finding the
greatest common divisor of46* — 6a6 + a^ and — b + a.
40 Elements of Algebra.
Dividing these two polynomials by each other, we find for an
exact quotient — 4 6 + a ; then — b -^ a or a — & is the greatest
common divisor sought.
36. In this example, as in all others where the exponent of the
principal letter is greater by unity in the dividend than in the divi-
sor, we can abridge the operation by multiplying each term of the
dividend by the square of the coefficient of the first term of the
divisor. We see indeed, by this means, that the first partial quo-
tient which we obtain, ought to contain the first power of this coef-
ficient. By multiplying the divisor by the quotient, and making the
reduction, with the dividend thus prepared, we have a result which
ought still to contain the coefficient as a factor, and the division can
be continued until we obtain a remainder of a lower degree than
the divisor, with reference to the principal letter.
The following are the operations ;
First Operation.
Having multiplied by 16 or by the square of 4;
— 48P + 48 ab^ — 16 a^b + IGc^^ 4 fe^— 5 at + o»
— I2ab^— 4a^b + l6a^) 126 — 3o
1st rem. — 19 a^b + 19 o^
or 19a»(— 6 + a)
Second Operation,
4 68_6a5-|-a»> — b + a
— ab + a^ y — 46-|-a
N. B. If the exponent of the principal letter in the dividend
exceeds the exponent of the same letter in the divisor by two or
tliree units, we must multiply the dividend by the third or fourth
power of the coefficient of the first term of the divisor. This is
easily perceived.
37. Let there be for a second example,
16 a« + lOa^J + 4a^b^ + 6a^P — Sab*
and 12 o^ b' + 38 a« P +Uab^— 10 6«.
Before proceeding with the division of these two polynomials, we
begin by observing that the first has a common facter a in all its
terms ; and since this factor is not found in the second polynomial,
we may suppress it, as making no part of the common divisor.
A^Araic Fractions. 41
For the' same fea^a the factor 2 8*, cdinmbri to dll the tetini of
the second polynomial, and not found in the first, may be sup-
pres^d. So that the question is reduced to seeking the greatest
common divisor of the polynomials
15 a* + 10 a^b + 4a« 6* + 6a63_3j4
and 6a^ + \9a^b+ Sab^ — 6R
ftrst Operniion,
30tf* + 200^6+ &a%^ + \2ab^ —6b* I 6a^+19d^+S'ab^—&l^
— 75a% — 32a»ft» + ^lab^ — 66* >• fid, —'2bh
_ 1 50a»fi — 64a36« + Uab^ — 1 26*
1st rem. +41 \a%^ + 274aA3 — 1376* ;
or J 3762(3(1? + 2a6 — 63).
Second Operation,
6c?+\9aH+ Bab^'-^6l?l 3c^+2ab — i^
+ 16a26 + 10fl6« — 56')2a +66
•**
Then 3a* + 2a6 — 6* is the greatest common divisor.
By following the same method as in the preceding example, wer
must multiply the whole dividend by the coefficient 6 of the first
terra of the divisor, or rather by the square of six ; but as 15 and 6
have a common factor 3, it is evidently sufficient to multiply the
whole dividend by 2, the factor of 6, which does not enter into 15.
This being done, we perform the division which gives in the
first place a remainder, whose firsti term is — 75 a^ 6. As 75 still
contains the factor 3, which enters into 6, it is sufficient to multiply
this remainder by 2, in order to continue the division wbicb^ beisg
performed, we have the first principal remainder
411 a« 6* + 274a63 — 137 6*.
Now it is easy to perceive that in* this remainder there is still a
common factor 137 6' ; and since this factor does not enter into
the second polynomial, we may suppress it as making* no part of
the common divisor, and the question is- reduced to seeking the
greatest common divisor of the two polynomials
6a3+ 19a2 6 + 8a6»'— 5 6«
and 3a»+ 2a6 — 6«.
By performing the division of these two polynomial, we find for an
Bour. Alg. 6
4il
EUmejits <f Algebra.
Axact quDtieot 2 a + ^ i ; -^ that the remaiDder da' +2ab — 6^
is the greatest common divisor sought.
38. Remark. It may be asked if the suppression of factor?
common to all the terms of one of the remainders which, we have
made in the course of the calculation, have no other object than
simplifying these calculations, or if these operations are indispensa-
ble. *Now it will be readily perceived, that these suppressions are
necessary ; for, if in the preceding example, we had not suppressed
the factor, 37 i*,it would have been necessary, in order to render
the division of the first term of the new dividend by the first term
of the divisor possible, to multiply the whole dividend by 137 (' ;
but then we should introduce into the dividend a factor which is
fouBd likewise in the divisor, whence it follows, that the greatest
common divisor sought is combined with the factor 137 6', which
ought to make no part of it.
The following example is suited to confirm what we have just
said.
39. Let it be required to find the greatest common tlivisor of the
two polynomials
ab + 2a^ — 3b^ — 4be— ac — c^
9ac + 2a' — 6a6+ 4c* +Bbc—12b\
First Operation.
and
2a^ + b
— c
a — Sb^
— 4bc
— c«
^2 a* -.6 6
+ 9c
a— 12b*
She
+ 4c*
1st. rem. ' 6 b
— 10c
or
a+ 9b'
— \2bc
— 5 c*;
(3J_ 5c;(2o + 3i + c).
Second Operation.
\
2a*— bb
— 8&
+ 8c
a_12&*)2a + 36 + c
+ Bbc
4- 4c»
a — 126'
+ She
+ 4c'
— 46
+ 4c
Greatest Common Diviior. 43
Therefore 2 a + 3 J -f- c is ihe greatest common diTisor.
After having arranged the two polynomials, we can, without anjr
preparation, perform the division, which gives for the first re-
mainder,
6b
— 10c
— 5c«
0+ 9 6^
— 12Jc
In order to continue the operation it would be necessary, by
taking the second polynomial for a dividend, and this remainder for
a divisor, to multiply the new dividend by 6 6 — 10 c, or simply
by 3 6 — 5 c, because the factor 2 belongs already to the first term
of the dividend ; but before performing the multiplication, let us
see if this factor 3 i — 5 c, will not divide the second term of the
remainder, namely, 9b* — 12 be — 5 c*. Now this division suc-
ceeds, and gives for an exact quotient 3 & — c; whence it follows,
that the remainder may be put under the form
(3 6 — 5c) (2a + 36 + c).
As the factor 3 6 — 5 c is found in this remainder, and does not
]belong to the, new dividend, (since this factor, being independ-
ent of the letter a, must (30) be found among the coefficients
of the different powers of this letter, which is not the case,) we caa,
without inconvenience, suppress this factor.
This suppression is moreover indispensable, because, without it,
. we must introduce this factor into the dividend ; and then the two
polynomials, containing a common factor, which they had not
before, the greatest common divisor would be changed j it would
involve the factor 3 6 — 5 c, which makes no part of it.
The suppression being made, we perform the new division,
which gives an exact quotient; therefore 2a-f-36-|-c is the
greatest common divisor.
40, We propose, as the last example, to find the greatest com-
mon divisor to the two polynomials
a* + 3a3 6 + 4a«6* — 6a6»Hi-26* .
and 4o^6 4-2a6» — 2 6»,
or simply, 2 a* -j- ab — 6*, since the factor 2 6 may' be suppressed
in the second.
44 J?fcw»^ <f4lgara.
First OperattQfip
4- 26 a« 6a — 38 a 63 ^ 16 6*
1st. rem. ^^1 al^^ +29 b^,
or —63(61 a — 29 6).
Second Operation.
Multiplication by 2601 ths square of 51.
6202 a' + 2601 ab — 2601 6« ) 51a — 296
— 5202 a» + 2968 ab ) 102a + 1^5
"" + 5559 a b — 2601 6«
— 5559fl6 +316l6«
5d. rem, + 560 6^.
The exponent of the letter a in the dividend exceeding by two
units that of the same letter in the divisor, we multiply the wbole
dividend by the cube pf ?, that is, by 8. With this preparation,
we perform three successive divisions, and obtain for the 6rsl
principal remainder — 51 a 6* -[- 29 6*. Suppressing the factor
6' in this remainder, we have for a new divisor — 51 a -^ 29 6,
or, changing the signs, which we are allowed to do, 51 a — 29 6 ;
the new dividend is moreover 2 a^ + o 6 — 6*.
Multiplying thisj dividend by the square of 51, or 2601, then
performing the division, we obtain, for the second principal re-
mainder, + 560 6* ; which shows that the two proposed polyno-
mials are prime to each other, that is, they have not a conHnon
fector. Indeed it results from the second principal (35), that the
greatest common divisor must be a factor in the remainder of each
operation; thus it must divide the remainder 5606*; but this
remainder is independent of the principal letter a ; therefore, if the
two polynomials have a common divisor, it must be independent of
a, and consequently (30) be found as a factor in the coefficients
of the different powers of this letter, which contain each of the two
proposed polynoinials, and Uus iis evidently not the case.
These examples wiH sofficeaoraake the learner acquainted with
the course tp be pursued, in fiudjng the great^^t ^onp^noa divisor pf
two polynomials.
41 • General Rule. We begin by suppressing in the two
polynomials the simple factors common to the two terms, (It
1
Chreaiest Common Divisor. 45
may happen that the simple factor which is found in the
dividend, and that which the divisor contaios, have a common
divisor ; in this case, we put it aside, as making a part of the
common divisor sought.) This suppression being made, ii^ pre-
pare the dividend so as to render possible the division of its first
term by thai of the divisor (35, 36) ; iheny toe perform the division,
which gives a certain remainder of a degree less than the divisor,
in which we suppress the simple or polynomial factors contained
in the coefficients of the different powers of the principal letter*
We then take this remainder for a divisor, the Heond polynomial
for a dividend, and perform the same operation tvith these as toith
the preceding, nis series of operations is to be continued until
we obtain a remainder which exactly divides the prectdin^ ; then
this remainder is the greatest common divisor ; even if we obtain a
divisor independent of the principal letter^ which indicates that
the two proposed polynomials are prime to each other, except
they have a common factor independent of the letter, which
factor would not have been discovered at the commencement of
the operation.
The following examples will serve to illustrate the method above
given,
il.) 5np» + 3nj?" J* — 2npq^ — Snj*, >
2mp^ J* — 4mp* — mp^ q + 3mj>g*. J
The greatest common divisor is p — q.
I
(2.) 36 a* — 18 a* —27 a* + 9 «%
and 27 a* 6« _ 18 a* 6« — 9 a^ 6".
The greatest common divisor is 9 a* (a — 1).
The theory of the four fundamental operations of algebra and
that of the greatest common divisor enables us to perform a great
number of questions. We shall hereafter establish new rules as we
have occasion for them. We proceed now to the resolution of
problems of the first degree.
Examples in the Reduction of Fractional Expressions to their
Simplest Terms,
1 g g -f- ^^ a-^-x
46 JElements of Algebra.
lia^ — 7ab 7a
2.
lOflc — 56c 5 c
12 fl3 x^ + 2 fl2 a;5 ^ 2a^x^
ISli 6-2 x + 3 6-2x2 ~ 3 62 •
5 a^-\-5 ax 5 a
a2 — x^ a — x'
o3 — x^ a^ ^ax-\-x^
6.
(a — z)2 • a — X
„3_2„2 „2
n2— 4« + 4 "" »--2'
a;a-f- 2 a;-^ 3 __ x — 1
2 23+3 3:2 + 3; __ 2 3; + !
2;3 — X* — 2x a; — 2*
q3 ^3 + c3 a;3 ^ a^b^^abcx-^c ^ x^
^- Sa^aZ. c2 a;2 a6— ci
2 a;3 — ( 3 c + <f + 2) x^ -f (3 c + ff) 3; __ 2x-~3c— (/
a:4-^a; "" xS+^ + i '
aa^52^c2^2a6 + 2Qc+26c _ fl+6 + c
^^' ;^_«_62 — c2 — 26c ■" a — 6 — c*
fl2— 3a5 + ac+2 62--26c _ g — 26
^^* a2^i:62 + 26c — c2 — a + 6 — c'
(fl + 6)(« + 6 + c)(« + 6-c)
2a262 + 2a2c2 + 262c2_a4_64 — c4
__ (q+6) (« + 6 + c) (fl+6— c) V
— 462c2— (a2__^2__c2j2
___ rt + 6
(c + « — 6) (6 — a + c)'
Problems of the First Degree.
Preliminary JVoiions on Equations.
42. The problems usually treated of in algebra are those whose
enunciations, translated into algebraical language, give rise to equa-
tions. On looking at ^he solution of the problem of article 3, we
PreUminary JVotions of Eqnatums. 47
shall see that it is composed of two distinct parts. In the first, the
relations which the enunciation of the question establishes between
the known and unknown qunntities, are written algebraically. In
this manner we arrive at the expression of two equal quantities,
which i^ called an equation. Such is the expression 2 x -^ b = a.
In the second part, a series of other equations is deduced from the
equation of the problem, of which the last gives the value of the
unknown by means of the known quantities. Such is the result
_ a — b
at which we arrived. This is what is called solving the equation.
* As the lules to be followed, in putting a problem into an equa-
tion, are rather vague, we shall begin with the second part, in
which they are fixed and invariable.
From the definition of an equation, it is composed of two parts,
separated from one another by the sign =. The part on the left
is called the Jirst member ^ and that on the right the second member.
There are several kinds of equalities.
(1.) The equalities which exist between known numbers repre-
sented by letters, such as
a — b = c — d, ^ = j
which may be verified immediately by putting, instead of a, b, c,
and d, the particular numbers for which we suppose these equalities
to stand.
(2.) Such as are evidently true of themselves, and which can be
verified as they actually stand, as
25=12 + 13; Sa—bb=za — b + 2a — 4b.
These are called identical or verified equaliiies,
(3.) Such as cannot be verified except by substituting in place
of one or more of the letters, designating unknown quantities, cer-
tain numbers whose values depend on the known and given num-
bers which enter into the equality.
To distinguish this from other equalities, we give it the name of
equation. It is of this that we now propose to treat.
Equations are divided into different classes ; those in which there
is only the first power of the unknown quantity are said to be of
the^r^^ degree^ such are
3« + 6=:17 — 6 a?, ax + b =z ex -\' d.
48 Elements of Algebra.
The equation
is of the second degree.
The equation
4x^— 6afl + x=i2a^+ 11
is of the third degree.
Generally the degree of an equation is the same as the greatest
exponent of the^unknown quantity in the equation.
A distinction is made between numerical and literal equations.
The first are those which contain only particular numbers, with the
exception of the unknown quantity, which is always designated by
a letter. Thus,
4a? — 3 = 2a? + 6, 3a? — 0? = 8,.
are numerical equations. They are the algebraical translation of
problems in which the grren quantities are particular numbers.
The equations
ax -^ b = ex + d^ aa^ -j- b X = Cy
are literal equations. The given quantities of the problem are here
represented by letters. It is customary, in order to distinguish the
known and unknown quantities, to denote the latter by the last let-
ters of the alphabet, as x, y, anU z.
We. proceed to show bow to solve an equatioa of the first de*
gree with one unknown quantity, that is, to find a number, whieb^
being substituted for the unknown quantity in the equation, shall sat'
isfy it; that is, make the first member identically equal to the-
second.
Of Equatifms of the First Degree wi^ One Unknown Quan;tity,
43w* We must regard it ais an axiom, that we may, without altei>>
ing the truth of any equation, add the same number to both. it«
SQiembers, or subtract it from them ; and also, that both members
may be multiplied or divided by the same number ; that is to say^
that if the two members were equal before, they are still equal when
changed in the manner expressed above.
The following transformations are of continual use in\tfae soltttioa
of equations.
Equations of the First Degree with One Unknoum Quantity. 49
(1.) When the tvro members of an equation are simple polyno-
mials, it is generally necessary to transpose some of the terms from
one member to the other.
Take the equation 5x — 6=8-|-2a?. To disengage x from
this equation, we must endeavour to get it by itself in the first
member. Subtract 3 x from both members, and the equation is
still true ; thus we have 5a? — 6 — 2a? = 8. Here we see that
the term 2 x, which had the positive sign in the second member,
has a negative one in the first.
Secondly, add 6 to both members, and the equation becomes
6jj — 6 — 2a? + 6 = 8 + 6,
or, since the terms ^— 6, -|- 6, destroy each other,
6a? — 2af=:8 + 6.
Then the term which was negative in the first member, takes a posi-
tive sign when it passes to the second member.
Take the equation a x -\'b =i d — ex. Add c « to and take b
from both members, and it becomes
aa?-j-6 -f-ca? — 6 = £l — ex -{- ex — 5,
which, reduced, is
ax -j- ex :=z d — b.
Generally, if a term is to be removed from one member to the
other, it must have its sign changed.
44. (2.) When the terms of an equation are fractional, it must
be reduced to another whose terms are whole numbers. Take the
equation
Reducing all these fractions to a common denominator by the
ordinary method, we have
40x_45 _ ,, , 12x
and because (43) the two members of an equation may be malti-
plied by the same number, let us multiply this by 60, which is
equivalent to suppressing the denominator 60 in the fractional
terms, and multiplying the whole term by 60. The result is
40jc — 46 = 660 + 12 a?.
Bour. Alg. 7
50 Elements ofMgtkra.
In performing this operation, we may arrive at the reduced
equation directly, without writing down the common denominator,
if we take care to roukiply every whole term by that denominator.
Take the equation
5x 4x 7 13
The denominators have evidently common factors, and the least
common multiple of the denonfiinators is 24. It is then to this
denominator that the fractions must be reduced.
Make this reduction, and omit the common denominator 24, and
the result is
10 a? — 32 a? — 312 = 21 — 52 a?.
(The whole term, — 13, being multiplied by 24.)
Hence this general rule may be deduced ; To clear an equation
of fractions^ begin by finding the least common multiple of all the dc"
nominators, (This number is the product of all the denominators, if
they have no common factor.) Then multiply every whole term by
this common multiple^ and every fractional term by the quotient of
this common multiple divided by tJie denominator of that term^ and
omit that denominator.
To apply this rule, take the equation
ax 2d^x . , Abc^x Scfl . 2c^
a
b ah +4a = -^--p- + ^_36.
The least common multiple of all the denominators is c?b^. Mul-
tiply each whole term by c?b^^ and the numerator of each frac-
tional term by the quotient of a^ 6^, divided by the denominator of
that term, and omit the denominator. The result is
a^bx — 2a^bc^x + 4a^6» = Abh^x — 5a® + 2a'b^t^ — Za^b^.
45. To solve the equation 4a? — 3 = 2a?4-5. It becomes
by transposition 4a? — 2a? = 5 + 3, which reduced, is 2 a? = 8.
8
Divide both members by 2, and we have a? = ^ = 4.
If 4 be substituted instead of a? in the equation, it becomes
4x4 — 3 = 2x4 + 5, or 13 = 13.
Take the equation of article 44
5z 4x ,Q_7 13«
12"^ "3 ^"^-g 6~-
This, when cleared of fractions, is
Equations of the First Degree imth One Unknoum Quantitff. 61
10a?— 32 a? — 312 =21 — b2x.
By traDsposition, it becomes
10a? — 32 a? + 62a? = 21 + 312;
and by reduction,
on ooo 333 111
30 a? = 333, or a? = -gg- = ^,
which may be verified by substituting for x this value in the origi-
nal equation.
Let (3a — x) {a — b) + 2 ax = 4b {x + a).
The multiplications here indicated must be performed, in order to
reduce the two members to simple polynomials, so as to disengage
the unknown quantity x. It then becomes
3a3 — ax — 3a6 + 6a? + 2aa? = 4 6a? + 4a65
which, when transposed and reduced, becomes
ax — 2bx =: 7 ab — 3a».
Now, a a? — 3 6 a? is the same thing as (a — 3 b) a?. Therefore,
(a — 3 6) a? == 7 a 6 ~ 3 a« .
Divide both members by 3 a i,
J , 7a6 — 3a«
and we have a? = oi— •
a — 3o
To solve an equation of the first degree, however complicated it
may be ;
(1.) Begin by clearing the equation of fractions, if there ar$
any, and performing all the algebraical operations which present
themselves on both members of the equation ; thus an equation is
obtained^ the members of which are whole polynomials,
(2.) Transpose to one member {the first is generally chosen) aU
the terms into which the unknown quantity enters, and to the other
member all the known terms.
(3.) Reduce to one all the terms which contain x, if the equation
is numerical ; and if it is algebraical, one product composed of two
factors must be formed of all these terms, one of which is x, and the
other the aggregate of the quantities by which it is multiplied, vnth
their respective signs,
(4.) Divide both sides by the number or polynomal by which x
is multiplied, and perform the division, if possible.
This rule can be applied, in all its parts, to the following ex-
ample;
53 ElemeniM of Algebra.
(i±i)i!Lrii)_3a = l^^-3. + ^^.
a — 6 a + b ' o
This, cleared of factions, is
b{a + by(x — b)—3ab(a' — b')
=zb{a — b){4ab — b^) — 2b{a^ — b^)x-^{a^ — b^){a^ — hx);
when the multiplications are performed,
a'bx + 2ab^x + Px — a» 6^ — 2ab^ — 6* — SaH + 3a4»
which becomes, by transposition and reduction,
4a«&r + 2ab^x — Si^a? = 4a«6a — 6a6» + 26* + 3tf% + a* j ,
collect into one all the terms which contain x, and
J (4 a « + 2 a 6 — 2 6 « ) a; = 4 a a 6 « — 6 a 6» + 2 6* + 3 a% + a*,
_ g* + Sa^b + 4 gg^g — 6 gfts -^- 2 M
^"" 6(4g» + 2g6 — 26») '
which cannot be reduced to a simple polynomial.
46. In an equation such as 3 a? — 2 = 4 a? — 7, the transposi-
tion of the unknown terms to the first member, aad the knowo ones
to the second, gives
2x — 4a? = 2 — 7, or — a?=: — 6.
To explain this result, observe, that the order of the transposi-
tion may be inverted, that is, the unknown terms may be put in
the second member of the equation, which will give
7 — 2 = 4a? — Sx, or 5 = ir, ora? = 6;
that is, whenever the result is such as — ^ = **~ 5, the signs of the
two members must be changed. This is evidently the same thing
^s transposing the unknown terms into the second member, and the
known ones into the first member, and then writing the second first.
We now proceed to the resolution of problems.
.47. We have already said, that the first part of the solution of
an algebraical problem cannot be reduced to any fixed rule. Some-
times the enunciation of the problem furnishes an equation imme-
diately ; sometimes it is necessary to discover from the enunciation
conditions from which an equation may be formed ; and frequently
it is not the conditions of the enunciation which must be translated
into algebraical language, but certain conditions derived from them.
The first of these are called explidi conditions^ and those which
Equations of the First Degree mth One Unknoton Quantity, 53
are derived from the enunciation implicit conditions. Nevertheless,
the extended application of the following rule will almost always lead
to an equation. Consider the problem as solved^ and having repre-
sented the known quantities either by letters or numbers, and the
unknown quantity by a letter, ivriie down in algebraical language
all the reasonings and operations which would have been necessary
to verify the value oj the unknown quantity had it been given.
Problem I . To find a number, the half, third, and fourth of
which, added to 25, give 448.
Let X be the number sought ; then 5, x, and j, are the half,
third, and fourth of that number. From the enunciation, these
three, added to 45, gives 448, that is,
2
Transpose 45, and
I + I + I + 45 = 448.
5 + I + ^ = 403.
Cleared of fractions, this becomes
6a? + 4a? + 3a? = 4836, or IScc = 4836;
whence
4836 ^^^
X = -YT- = 372.
372 372 372
and ^ + ^ + 2if + 45=186 + 124 + 93 + 45=448.
This question is one of that species which, in arithmetic, is solved
by the rule of false position ; we see with itat ease it is solved
by algebra.
Problem 2. A workman is engaged for 48 days, and receives
for each day that he works 24 cents and his food. Each day that
he does not work, his food costs him 12 cents ; and at the end of the
48 days he receives 504 cents. How many days did he work, ahd
how many was he idle ?
If we knew these two numbers, and multiplied them respectively
by 24 and 12, and subtracted the second product from the first, the
remainder would be 504. This must be expressed algebraically.
Let X be the number of days during which he works ; 48 — x
is tbe^ number of those during which he is idle, 24 a? is the number
of cents which he gains, and 12 (48 — x) that which he loses.
54 Elements of Algebra.
Consequently
24 a?
— 12 (48 ~ a?) = 604,
or
24 a?
— 676 + 12 a? = 604;
therefore
36 a? =1080,
and
^-36 -^^5
whence
48 — a? = 18.
Therefore, he works 30 days, and rests 1 8. For 30 days of work
he receives 30 X 24, or 720 cents 5 in 18 days of rest he loses
18 X 12, or 216 cents. Now, 720 — 216 = 604.
This problem may be generalized by calling n the whole number
of days of labor and idleness^ a the sum he receives for each work-
ing day, b that which he loses for each day of idleness, and c his
total gain. Let a? be the number of working days, and then n — x
is the number of days of idleness. Then ax and b{n — x)
represent respectively the sums gained and lost.
The equation of the problem is
ax — 6 (n — x) = c,
whence ax — 6»-f- b x =z Cj
(a -}- b) X =z c -j- b n^
J c -^ bn
and a? =
Therefore
n — a? = n
a -f-'A
c -j- 6 n an-^-bn — c — bn an — c
a-\- b a + 6 a -\- b'
Problem 3. A fox is pursued by a greyhound^ and is 60 of her
oum leaps before him. The fox makes 9 leaps while the greyhound
makes 6, but the latter in 3 leaps goes as far as the former in 7,
How many leaps does the greyhound take before he catches the fox?
It is evident that the distance the dog has to run, consists of the
60 leaps which the fox is in advance of him, added to the dis-
tance the latter runs, from the moment when the former begins the
chase. If then these two distances can be expressed in terms of
one unknown quantity, the equation of the problem is easily stated.
Let X be the number of leaps, made by the dog. Since the fox
makes 9 leaps, while the dog makes 6, it follows that the fox makes
^, or ^ leaps, while the dog makes one, and therefore -5- ;s the
number of leaps made by the fox while the dog makes a?. It might
Equations of the First Degree vnth One Unknown ^antiiy. 55
be imagined, that to obtain tbe equation it would be sufficient to
put X and 60 -f- -^ equal to each other ; but, by so doing, we
should commit a manifest error ; for the leaps of the dog exceed
tiiose of the fox ; and we should thus put equal to each other two
heterogeneous numbers, that is, two numbers referred to different
units. We must then express the leaps of the fox in terms of those
of the dog, or the converse. Now 3 leaps of the dog being equi^
alent to 7 of the fox, one leap of the dog is equivalent to | leaps
of the fox, and therefore x leaps of the dog are equivalent to
7 X
-g- leaps of the fox.
7x 3
Therefore, -5- = 60 + 5 a? ;
or 14 a? = 360 + 9 a?,
or 5 a: = 360, and a\ = 72.
Tjierefore the dog makes *72 leaps, and the fox makes 72 X •§'
or 108 leaps.
J^erijication.
The 72 leaps of the dog are equivalent to 72 X h ^^ 168 of
the fox, and 168 = 60 + 108.
The two following problems are well worthy the attention of
learners, as an exercise in algebraic calculation.
48. Problem 4.* A father, who has three children, leaves his
property to them by will, in the follomng way ; to the first he
bequeaths a sum a, together with the nth part of what remains ; to
the second a sum 2 a, with the nth part of what remains after the
first part and 2 a has been deducted ; to the third, a sum 3 a, with
the nth part of what remains after the two first parts and 3 a has
been deducted. The property is thus entirely divided; what is its
amount ?
Call this property x. V by the aid of this quantity, we can form
the algebraical expressions of the three parts, we must subtract their
sum from x, and the remainder, put equal to nothing, gives the
equation of the problem.
Since x denotes the property of tbe father, and x — a the re-
mainder after a has been taken away, the share of the first is
, X — a an -|- x — a
a + , or • .
' n ' It
56 Elements i^ Algebra.
To form the second part, we must subtract from x this first pan
and 2 a, which gives
^ an 4- X — a
CO — 2 a ;
or, when reduced,
nx — San — a;-f-a .
»
Now, the second part is 2 a added to the nth part of this remain-
der ; in^'olher words, it is
nx — 3aR-— z-f-a
2a +
2a n* •4"'*^ — ^^^ — X -^-a
If we take from x the two first parts and 3 a, the remainder is
an + X — a 2an34-nz — San — x 4- a
X — 3 a ' -L !—
n n2 '
which, reduced to the same denominator and simplified, is
n^x — 6a n^ — 2nx + 4an-{~^ — *
■ Q •
n^
The third part therefore is,
^ n^x — 6a n* — 2nz-f-4an-j-a; — a
o a -4- q 9
• n* '
or
San^-f-n^x — 6a»3 — 2nx-{-4aii-^x— •«
But according to the enunciation of the question, the property of
die father is thus entirely used up. Therefore the difference be*-
tween x and the sum of the three parts must be equal to asero ;
that is, '
an4-x — a 2aii^+nx — 3an — x + a
X ■ 5 !— f
(=0:
San^-j-**^^— '^^'•^ — 2»x-(-4a» + * — «
: ^
which, the denominators being cancelled and the whole reduced,
becomes
n^a?-— Can' — 3n*a?-|- lOan^ + ^na? — 6an-— « + ^=^5
whence
_ 6an'-- 10an'4-5an — g a(6n'— 10n'+5n— 1)
*— „3_3^«^3^_l - n» — 3n»+3n — L
L
Equations of the First Degree with One Unknown Quantity. 57
We might obtain an equation and a result more simply by ob-
serving that, since the portion of the third child is 3 a, added to the
nth part of the remainder, and the property is then all disposed of,
the portion of the third is 3 a, and the remainder spoken of is
nothing.
The expression for this remainder is
tflx — 6ona — 2nx-^^an -}- X — a
-^ ,
which, put equal to nothing, gives
n^x — 6an^ — 2na? + 4an+a? — a = 0,
_ 6ana — 4<»« + a _ a(6iia — 4>i-{-l)
^^ ^— ffi — 2n + l "" ng — 2n + l'
To prove the numerical identity of this expression with the
former one, it is sufficient to show that the second may be derived
from the first, by suppressing a factor common to both its terms.
Now if to the two polynomials
a (6 n3 — 10 n» + 6 n— 1),
and (na_3n=» +3n— 1),
be applied the rule for finding the greatest common divisor (41),
we shall see that n — 1 is their greatest common divisor, and by
dividing both terms of the first expression by n — 1, we obtain the
second.
This problem will show the learner the importance of finding
out, in the enunciation of a question, all the circumstances which
facilitate the formation of an equation, the neglect of which may
lead to solutions much more complicated than the question actually
requires.
The conditions from which the expressions of the three parts
were formed, are the explicit conditions of the proposed problem ;
the condition from which the* most simple equation was derived is
an implicit condition, which a little attention shows to be contained
in the enunciation of the question.
To obtain the values of the three parts, it is sufficient to put for
07, its value in the expressions obtained above.
Let a = 10000, n = 6.
Then
_ 10000(6x^*^X5+1) _ 10000X131 _ 1310000 _ g
^'^ 25—2x5+1 "" 16 "^ 16 ""
Bour. Alg. 8
58 EkfMnti of Algebra,
The first child must have 10000 + Q^^Q^Sj 10000 ^^ ^^^^^^
There remains then for the other two 81875 — 24375, or
57500.
mu J U r./^/^/^.^ . 57500 — 20000 ^„^^^
The second has 20000 + = , or 27500.
o
There remains then 57500 — 27500, or 300(50 for the third
share.
Now 30000 is triple of 10000, which verifies the solution.
A less direct, but more simple and elegant solution of this pro-
blem, may be given. It is founded on the condition, that the sub-
traction of 3 a and of the two first parts from the whole, leaves no
remainder.
Let r, r', r'', be the three remainders mentioned in the enuncia-
tion ; the expression for the three parts will be
r r' r/'
Now (1.) by the question r'' = 0. Thus the third part is 3 a.
(2.) What is left, after the share of the second and first have been
deducted, is r' , or ^ ^— . And this remainder is the share
of the third ; therefore
(n — 1)1^ ^ , - San
^ i— = 3 a : whence r' = 7.
n n — 1
Then the share of the second is
^ - ,1 3'a» ^ , 3 a ,1 . 2a» + a
• 2 a + - . r or 2 a H -, that is V-' *
\ n n — 1 * n — 1' n — 1
The remainder, after the sharp of the first has been taken away,
r (n — 1) r
IS r or ^ ^~.
n n
Moreover this must be the sum of the second and third shares ;
hence
n — 1 .dan-fa 5an — 2 a
n ' n — 1 n — 1
rrii r 5an — 2a n 5an^ — 2a»
Therefore r = — - x 7 = —7 ttq- .
n — 1 n — 1 (n — 1)'*
Aj j-ii/* , \ 5an^ — 2 an
And accordingly the first part is a H — . — 7 tts —
° "^ ^ n (n — 1 )'*
j^ 6an — 2a 5an — 2a an^ + San'-^a
Equations of the First Degree with One Unknown ^antity, 59
The whole property is, therefore,
. 2 an 4- a , an^ + San— a
^<' + -;r:rr+ (I-ij* — »
3 a («2 — 2« + l) + (2fln4rq)(w — l)+fln'4-3an — g
or „• _2 w + i '
6an» — 4o» + a a{6n^ — 4n^l)
which Is the result obtained above.
This solution is more complete than the preceding, since the
whole property and the three shares are found at the same time.
^^ 49. Problem 5. A father leaves in his wUl^ to his eldest childy
a sum a, and also the nth part of the remainder of his property ;
to the second^ a sum 2 a, with the nth part ofwhfd remains^ when
the first share and 2 a have been deducted ; to the thirds a sum
38, with the nth part of the new remainder; and so on. . Mortr
over J the shares of dU the sons are equal. 1 fFhat is the property
of the father, the share of each of the children^ and the number oj
children ?
This problem is remarkable, since it contains more conditions
than are necessary to determine the value of the unknown quanti-
ties.
Let X be the property of the father ; a? — a is what remains when
a is taken away. Therefore the share of the eldest is
, X — a na + X'-^a
a -\ or ' .
' n ' n
The subtraction of this part and of 2 a from the whole, leaves
^ an-4-a; — a nx'-^San — x+a
so —— » a — ^ 4 or — ^_^__. —
n n
of which the nth part is § — .
n*
Then the share of the second is
nx — San — x -^ a 2an^ + nx-^3an — x'+a
2 a -| -2 , or
a ifc«
In the same way the other shares may be found ; but as they
must all be equal, the equation of the problem may be formed by
simply putting the two first shares equal to each other, which gives
an+x — a 2aw^ -f-nx — 3a» — xl-^-a
whence x =^ an' — 2an + a.
60 EUmenii ofAlgAra.
Substitute this value of x in the first share, which then becomes
an + €LT^ — 2an-\-a — a aifi — an
— : — — ^ , or , or i» n — a,
or a (w — 1) ;
and as all the shares must be equal, dividing the whole property
by the share of the first we shall have the number of children ; this
, . an^ — 2 an A- a .
number is — , or n — 1.
an — a
The whole property is, therefore, a{n — 1)*.
The share of each son is a{n — 1).
The number of children is n — 1.
' It remains for us to determine whether the other conditions of
the problem are satisfied ; that is, whether, if the second have 3 a
with the nth part of the remainder, the third 3 a, with the nth part
of the remainder, the share of each is a (n — !)•
The difierence between the whole property and the first part is
a{n — 1)^ — a(n — 1), and the share of the second is
fl(CT_l)8 — a(n— l)_2g
2a +
n
or
2 a (n— 1) + a ( n— 1) ^ — a (n — 1),
n
a(«-l)-)-«^(»-l)» ^^ a(n -l)(l+n-l)
n n
ora(n — 1).
In like manner, the difference between the whole and the two
first shares is a{n — 1)* — 2a{n — 1); and the share of the
.!.• J • o I tf(n— 1)* — 2a{n — 1) — 3a .... ., ,
third IS 3 a -) ^ ^ -^ '-^ , which is evidently
fl(n — l) + fl(w— !)• . ^ .
n
The fourth share is
4 a + a( »-l>-3«(^-I)-4a
"^ n '
ain — i)+ain — rf
or — ^^ ^-^ — ^ ;
n '
and so on. Therefore all the conditions contained in the enuncia-
tion are fulfilled.
EquatioM and Problems tuith 2W Unhunvn Quhntities. 61
Cf Equations and Problems of the First Degree with Two or
more Unknown Quantities,
«
50. Although several of the problems which we have solved
contained more than one unknown quantity, we have employed
only one symbol of an unknown quantity in their solution. This
was because the conditions of the enunciation enabled us easily to
express the other unknown quantities^ by means of one of them ;
but this cannot be done in all problems which contain more than
one unknown quantity.
To know bow such problems are to be solved, take first one of
those which have been already solved by meansi of one unknown
quantity.
To find two numbers, whose sum is a, and whose difference is
J (4).
Call these numbers x and y ; then
x + y=za^
X — y = b.
Now, it is an axiom, that if to two equal numbers A and B be
added two other equal numbers C and D, the results .^ -{- C and
B+D are equal ; that is, if w3 = B and C =: Dj A -{- B
= C + D.
Also, if from two equal numbers be subtracted two equal num-
bers, the remainders are equal ; that is, if w9 = j5 and C =z D^
then^— C = J5 — D.
Apply this principle to the equations of the proposed problem ;
we shall have,
by adding 2 a? = a + 6,
and by subtracting 2y = a — b.
We obtain from the first, x = T 5
and from the second, y = — ^ — -
A ja + 6 , a — 6 2a ■> a + b a — b
And -i- + -^— = ^ = a and —^ ^-
2
Take the problem of the labourer (47), considering only the gen-
eral enunciation.
63 Ekments of Alg^a,
Let X be the number of working days, and y those of idleness ;
a X 'and b y are the sums respectively which he receives for his
work, and loses by his idleness.
The equations are ,
x + y =n,
ax — 6 y z= c.
Now we have already seen that two equal quantities, multiplied
hy the same quantity, are still equal ; if therefore we multiply both
members of the first equation by 5, the coefficient of y in the
second, we shall have
6 a? -|- 6 y = 6 rj,
which, added to
gives
ax — b y=:Ci
bx'i-ax=Lbn-\-c,
- bn4-c
and X = — r^.
Multiply the first by a^ the coefficient of x in the second, and we
have
Subtract the second from this, and there remains
I JL an — c
ay + by = an—c, or y = ^q-y.
The introduction of a symbol to represent each of the unknown
quantities, has the advantage of finding either of the numbers
sought independently of the other.
Eiiminaiion.
51. Given the two equations,
5 a? + 7 y = 43,
lla? + 9y = 69,
which may be regarded as the algebraical translation of the enun-
ciation of a problem containing two unknown quantities.
If one of the unknown quantities had the same coefficient in both
equations, simple subtraction would give another equation contain-
ing only one unknown quantity, from which the value of that un-
known quantity might be found.
Eliminatiotk. ^
Mukiplj both members of the first equation by 9, the coefficient
of y in the second, and both members of the second equation by 7,
the coefficient of y in the first. This gives
45 a? + 63 y = 387,
77 a; + 63 y =; 483,
which may be substituted for the two first equations, and in which
y has the same coefficient.
Subtract the first from the second, and we have
32 0? = 96, or a? z= 3.
In like manner, multiply the first equation by 11, the coefficient
of X in the second, and the second by 5, the coefficient of x in the
first. This gives
65 a? + 77 y = 473,
55 a? + 45y = 345.
Subtract the second of these from the first, and we obtain
32 y = 128, or y = 4.
Then a? = 3, and y = 4, are the two values of x and y, which
verify the enunciation of the question. For
5.3+7.4=15-1-28 = 43,
and 11.3 + 9.4 = 33 + 36 = 69.
The operation by which the values of the unknown quantities have
been found, is known by the name of elimination^ and consists in
making one of the unknown quantities disappear by means of cer-
tain transformations of the proposed equation.
The preceding method has a close analogy with that for the
reduction of fractions to a common denominator ; and, like this last
operation, is susceptible of some simplifications.
Take the equations
8 a? — 21 y = 33,
6 a? + 35 y = 177.
To render the coefficients of x and y equal, observe that 21 and
35 have a common factor 7 ; it is sufficient, therefore, to multiply
the first equation by 5, and the second by 3, which gives
40 a? — 105 y = 165,
18 a? + 105 y = 531.
Add these equations, and we obtain
58 a: = 696, or a? = 12.
64 Elements of Algebra.
In like manner, the two coefficients of x contain a common fac-
tor 2 ; it is sufficient, therefore, to muhiply the first equation by 3
and the second by 4, which gives
24 X— 63 y = 99,
24 X + 140 y = 708.
Subtract the first from the second, and we have
203 y = 609, or y = 3.
N. B. It is important to find out whether the coefficients have any
common factors, as the calculations are then much more simple.
As a third example, take the equations
1-2 + 2 = 5-2* + ^'
They must first be cleared of fractions by the rule (44), and
thus the two following equations are obtained,
8a:_48 + 6y + 12 a? = 96 — 9y + 1,
y—2x+ 12 = 1 — 12 a? + 36,
which, reduced, become
20a? + 16y = 146, ' 4a? + 3y = 29,
9 0? ^- y = 25, ^^ 9 0? 4- y = 25.
Muhiply the second equation by 3, and subtract the first from the
result. This gives
23 a; = 46, or a? = 2.
Now, y = 25 — 9a? = 25 — 9 . 2 = 7.
52. We now proceed to the solution of three equations contain-*
ing three unknown quantities.
Let the equations be
5a? — 6y + 4z z=: Ibj
7x + 4y — 3z =z 19,
2a? + y + 6;? = 46,
To eliminate z by means of the two first equations, multiply the
first by 3 and the second by 4, and add the two results ; as the
coefficients of z have contrary signs, we obtain
43a? — 2y= 121.
Multiply the second equation by 2, one of the factors of the
coefficient of 2: in the third, and add the result to the tliird, which
gives
16a? + 9y = 84.
V
JEliminaiion. 65
The qaestion is then reduced to finding the values of x and y,
which satisfy these new equations. Multiply the first by 9 and the
second by 2 ; adding the results, we have
419 a? = 1257, or a? = 3.
We might now determine y as we have determined x ; but it
may be done more simply by putting for x its value in the second
equation, which gives
48 + 9 y = 84, or y = ^^~^^ se 4.
Put for X and y their values in the first of the three proposed equa-
tions, and it becomes
l6 — 24+4z=ii5, or « = ?^ = 6.
Let there be m equations, with m unknown quantities. To find
the value of the unknown quantities, combine successively one of
the -equations with each of the m — 1 others, so as to eliminate the
same unknown quantity from each ; thus are obtained m — 1
equations, containing m — 1 unknown quantities, on which perform
the process which has already been performed on the first equa-
tions ; that is, eliminate another unknown quantity, by fcombin-
ing one of the new equations with the m — 2 others, which leaves
m — 2 equations, containing m — 2 unknown quantities. Con-
tinue this series of operations until at last there remains one equa-
tion, containing only one unknown quantity, from which find the
value of that unknown quantity. Then, by going back through the
series of equations which have been obtained, the value of the
other unknown quantities may be successively determined.
63. The mode of elimination, here given, is known by the name
of the method by addition and subtraction ; because the unknown
quantities disappear by addition and subtraction, when the equa-
tions have been so transformed that one unknown quantity has the
same coefficient in two of them.
There are two other methods of elimination commonly used. The
first, called the method by substitution, consists in finding from one
equation the value of one unknown quantity, as if the others had
been determined, and substituting this value in the other equations,
which produces new equations, containing one unknown quantity
less than the original ones, on which the same operation is repeated.
The second, .called the method by comparison, coasists in finding
Bour. Alg. 9
66 Elements of Algebra.
the v^lue of the same unknown quantity from all the equations, and
putting these Talues equal to one another. This gives rise to new
equations, containing one unknown quantity less than the original
ones, on which tiie process is repealed.
But these two methods have an inconvenience from which the
method by addition and subtraction is free. The new equations
contain denominators which must be made to disappear. The
method by substitution may be employed with advantage when-
ever the coefficient of an unknown quantity is unity in one of the
equations, because then this inconvenience does not exist. Some-
tim'es it will be necessary to employ it. But generally, the method
by addition and subtraction is preferable ; it has, besides, this ad-
vantage, that if the coefficients be not too great, the addition or
subtraction may be performed at the same time with the multiplica-
tion, which makes the coefficients equal.
54. It often happens that the given equations do not each con-
tain all the unknown quantities. In this case, with a little skill, the
eliminations may be shortened.
Take the equations ' '
2x — Sy + 2z =: IS (I),
4m — 2a? z= 30 (2),
4y + 2z = 14 (3),
5y -I- 3 m = 32 (4).
It will be perceived immediately that the elimination of z be-
tween equations (1) and (3) gives an equation in a;andy; also,
that if u be eliminated between (2) and (4), a second equation id
X and y is obtained ; these two unknown quantities can then be
easily determined.
First, the elimination of z between (1) and (3) gives
7 y — 2a? = 1 ;
that of u between (2) and (4) gives 20 y + 6 a? = 38.
Muhiply the first of these by 3, and add ; which gives
41 y = 41, or y = 1.
Substitute this value in the equation 7y — 2^=1; we find
0? = 3.
Substitute the value of a? in (2), which becomes
4 M — 6 = 30, or M = 9.
The substitution of the value of y in (3) gives ;r.= 6.
Elimination. 67
The student may take as aa exercise the following equations ;
7a. — 2^+3w= 17]
4y — 2z'\-t =11 which give
6y_3a;_2tt= 8> afi=2, y = 4,2rz=3,tt = 3, ^= 1.
3 2r-|-8w = 33^
55. In all that has gone before, we have supposed the number
of equations equal to the number of unknown qgantities. This
must be the case in every problem where there are unknown quan-
tities, in order that it may be determinate, that is, that it may not
admit an infinite number of solutions.
Suppose, for example, that a problem, with two unknown quanti-
ties X and y, leads to tlie single equation 5a? — 3y= 12;
whence a? =
12
+ 3y
5 •
^
If for y be put successively
1, 2, 3, 4,
> 5,
6 . •
K • • • ^
the values of x are
^ 18 21 24
27
6 . .
A • A #
* 5 • y 5 '
5'
^^ • V
•
and each of the systems of values
a? = 3
y =
1,
18
^=5
y =
2,
21
* 5
y =
3,
will satisfy the equation.
If there are two equations with three unknown quantities, one ot
the unknown quantities may be first eliminated by means of the
given equations ; and thus an equation is obtained, which, as it con*
tains two unknown quantities, may be satisfied by an infinite num-
ber of values of these unknown quantities, whence an infinite num-
ber of values of the third unknown quantity may be obtained^
Then, in all cases, in order that a problem may be determinate, it
is necessary that its enunciation contain at least as many different
conditions as there are unknown quantities, and that each condition
be expressible by an equation.
56. We go on to the resolution of problems containing two or
more unknown quantities.
68* Elements of Alg^ra.
Problem 6. A person possesses a capUcX of $90000, on which
he gains a certain rate of interest ; hut he owes $20000, ^br which
he pays interest at another rate. The interest which he receives ik
greater than that which he pays by $800. A second person has
$35000, on which he gains the second rate of interest ; but he
owes $24000, /or which he pays the first rate of interest. The sum
which he receives is greater than that which he pays by $310.
tVhat are the two rates of interest ?
Let X and y be the two rates of interest. The interest of $30000
at the rate of x per cent, is x taken as many times as there are
hundreds in 30000 ; that is 300 x. Also, the interest ot $20000
at y per cent, is 200 y. And, from the question, the difference of
these two is $800.
Then the first equation of the problem is
300 a? — 200 y = 800.
The second condition in the question, translated into algebraical
language in the same way, is
350 y — 240 a? = 310.
As both numbers of the first equation are divisible by 100 and of
the second by 10, they are equivalent to
3 a? — 2 y =8,
35y— 24a? = 31.
To eliminate a:, multiply the first equation by 8, an^ add the
result to the second, which gives 19 y = 95, or y s: 6, Substi-
tute this in the first equation, which becomes
3 a; — 10 =: 8, or a? = 6.
Therefore the first rate of interest is 6 and the second 5 per
cent.
30000 at 6 per cent, gives yearly 300 X 6, or 1800; 20000
at 5 per cent, gives 200 X 5, or 1000 ; and
1 800 — 1000 = 800.
iti the same way the second condition might be verified.
Problem 7. There are three ingots of different metals mixed
together. A pound of the first contains 7 ounces of silver y 3 ownccM
of copper y and 6 of pewter ; a pound of the second contains Id
ounces of silver ^ 3 of copper ^ and 1 of pewter ; that of the thirds
4 diifices of silver, 7 of copper , and 5 cf pewter. How wMch of
each of the three ingots must he taken to form af&urthy which shall
Miminaiion. 69^
cofitiotfi in a pounds 8 ounces of silver, 3| of copper, and A\ of
pewter ?
Let a?, y, and z be the number of ounces which must be taken
from the three ingots to form a pound of the fourth. Since in the
first there are 7 ounces of silver in a pound of 16 ounces, h follows
that in one ounce there is yV of an ounce of silver^ and in x ounces
there must be -=^ of silver. In the same way we might show that
"Y^ and -^ are the quantities of silver taken from the second an^
third ingots to form a pound of the fourth ; but, by the question,
this contains 8 ounces of silver ;
or 7a?+12y + 42f= 128.
By reasoning in the same way on the quantities of copper and
pewter, codtained in the fourth, we have
3 a? + 3y + Tjt = 60,
and 6a?+ V +62: = 68.
As in these equations the coefficients of y are the most simple, it
will be convenient to eliminate y first.
Subtract the first equation from the second, multiplied by 4, and
we obtain
6 a? + 24^ = 112.
Subtract the second equation from the third, multiplied ^y 3,
and the result is
16 a? 4. 8 a? = 144.
Subtract the first of these two from the second, multiplied by S^
and we liave
40 a? = 320, *or a? = 8.
iSubstitute this value of a? fn the second of the abote, and it
becomes
120 + 8z = 144, or z = 3.
The values of z and of a?, substituted in the third of the original
equations, give
48 + y + 16 = 68, or y = 6.
Thus, to form a pound of the fourth ingot, we must take 8
ounces of the first, 6 of the second, and 3 of the third. If in 16
70 Elements of Algebra.
ounces of the first, there are 7 of silver, in 8 ounces there must be
7X8-. T Ti 12 X 5 , 4 X 3
—rrr- of ihe same. In like manner, — - — and [l are the
ID 10 16
quantities of silver contained in 5 ounces of the second ingot and 3
of the third. Now,
7X8 12 X 5 4x3 _ 128 _
. 16 "^ 16 "^ 16 "" 16 "~ '
and the fourth ingot contains 8 ounces in the pound of silver, as the
question requires. In the same way the conditions arising from the
quantity of copper and pewter may be verified.
67. The following problems are intended as an exercise for the
student.
(I.) One workmam can finish a things in b days ; a second ca%
finish c ihinz^s in d day^ ; and a third j e things in f days. How
long wiU they ic, ad working together, in finishing g things ?
Application.
XA.LIi9t
-■' adf + bcf+bd€
a = 27
i = 4,
c = 35
rf = 6,
e = 40
/=12,
g=zm.
'
days.
Answer, 12 days.
(2.) If 32 pounds of sea water contain 1 pound of salt, how
much fresh water must be added to these 32 pounds, in order thai
in 32 pounds of the new mixture the quantity of salt may be reduced
to 2 ounces, or \ of a pound?
Answer, 224 pounds.
(3.) How many times do the hands of a clock meet between noon
and midnight, and at what hour» ?
Ans. 11 times; at 1*-5'JV, 2*- 10'|f, 3*-16Vtj &«•
(4.) A number is composed of three digits whose sum is ii, the
units digit is double of that of the hundreds, and 2^1 added to this
number gives the number inverted. What is the number ?
Answer, 326.
(5.) A person who has ^100000 puts part to interest at 5 and
part at 4 per cent, ; his income is ^4640. What are the two parts ?
Answer, $64000 and $36000.
(6.) A person has a certain capital, which brings a certain inter-
est. A second person, who has $10000 more than the first and
gets 1 per cent, more, has a greater income by $800. A third
Eliminaiion. 71
who has ^15000 more than the first, and gets 2 per cent, moref has
a greater income by f 1500. Wliat are the three capitals, and the
rates of interest^
Ans. The capitals are |30000, $40000, and $45000.
The rates are 4, 5, and 6 per cent.
Problems which give rise to JVegative Results. Theory ofJVegative
Quantities.
, 68. The use of algebraical signs in the resolution of problems
often leads to results which at first view are embarrassing ; on reflec-
tion, however, it will appear, not only that they are capable of
explanation, but that by their means the language of algebra may
be still further generalized.
Let us take the following problem. To find a number which,
added to a number b, gives for their sum a number a.
Let X be the number required, then evidently
b -\- X = a, or a? = a — b.
This expression or formula gives the value of x, in all particular
cases of the problem.
For example, let
a = 47, b = 29;
then a? = 47 — 29 = 18.
Let now a = 24, J = 31 ; then a; = 24 — 31. Since
31 = 24 + 7,
this expression may be put under the form a? = 24 — 24 — - 7, or,
by reduction, a? = — 7. This value of a? is what is called a negative
solution. What is the meaning of it ?
If we return to the statement of the problem, we see that it is
impossible that 31 added to any number can give 24, a number less
than 31. Therefore no number can solve the question in this case.
Nevertheless, if, in the equation of the problem, 31 -|- a? == 24, we
put, instead of the term + ^j l^c negative value — 7, it becomes
31 — 7 = 24, a true equation, which amounts to saying that 31
diminished by 7 is 24.
The negative solution a? = — 7 indicates the impossibility of
solving the problem in the sense in which it was proposed ; but if
we consider the solution independently of its sign, that is, x =z 7y
we may see that it is the solution of the following problem, To find
73 Elements €f Algebra,
a number which, subtracted from 31, gives 24; which only differs
from the first, viz. To find a number which, added to 31, gives 24,
in this, that the words added to are supplied by the words subtracted
from.
The new question, when solved directly, gives the equation
31— cc = 24;
whence 3J — 24 = vC or a? = 7.
Let us take the following problem. A father is a years, his son
b years old. In how many years will the son^s age be one--
fourth of his father* s 9
Let X be the number of years, then a -f- ^ and b -{• x represent
tbe ages of the father and son at the end of this number of years;
then the equation is,
, , a + X a — 46
^+^ = •"4 — » ora? = — g — .
Let a = 54, and J = 9 ; then
54 — 36 18 ^
The father being 54 years old, and the son 9, in six years the
father will be 60, and the son 15 ; now, 15 is the fourth of 60;
therefore 6 is the answer to the problem.
Let us suppose a = 45, 6 = 15;
45 — 60
then 0D=> — g — .
This expression may be reduced to a: = — ^^ 5 by the ordinary
ru}e3. How is this negative result a? = — 5 to be explained ?
If we return to the equation of the problem, we shall find that in
this case it becomes
This contains a manifest contradiction ; for the second member is
-J- + 2 5 and each of these two parts is less than the correspond-
. ing part of the other member. But if we substitute — -5 for -(- «» it
45 — 5 40
becomes 15 — 5 = — j — > or 10 = -j-;
an exact equation, which indicates that if, instead of adding to the
ages of the two, we take away 5 years, the age of the son will be
one-fourth of that of the father. Thus the solution which has been
Problems with JVegoHve Re$ulis. 73
found, considered independently of its sign, is the solution of the
following problem. A father is 45 years old^ and his son 15 ;
when WAS the age of the son one-fourth of that of the father 9
The equation of this new problem is
16 — a = — j~;
whence v 60 — 4 a? = 45 — a?, and a? = 5.
The least consideration of the problem will show that as the
15 1
ratio of the ages of the two is ^, or »» the age of the son cannot
become one-fourth of that of the father, but has been so already ;
because, as has been pro\red (6), by adding to both terms of a
fraction the same number, the fraction is increased in value. On
the contrary, it is diminished in value by the subtraction of the same
quantity from both terms.
59. We are led by these analogies to the establishment of the
following principle.
(1.) fVhen the unknovm quantity is found to have a negative
valuey it is indicative of some incorrectness in the manner of stating
the question, or, at least, in the equation which is the algebraical
translation of it. (See the Remark at the end of this article.)
(2.) This value, independently of its sign, may be regarded as
the solution of a problem, which differs from the proposed problem
in this, that certain quantities, which were additive in the first, are
subtractive in the second, and the reverse.
Demonstration. The first part of this principle may be easily
demonstrated. The finding a negative value for x must arise from
our being led, by the nature of the equation, to subtract the greater
of two numbers from the less, which is impossible.
Thus the values a? = — 7, a? = — 5, (58) arose from the equations
« = 24-31, x=ii=l^.
Now, if no absolute number,* when substituted for x, can verify
the equation of the problem, after the transformations (43, 45)
have been made, it follows that the original equation itself cannot
be verified by the substitution of any absolute number for x ; for
the correctness of the transformations has been shown for all equa-
tions which are capable of being verified.
<—>.■.«— w^«—.— I II. I i»ii » II, ■ 11 I I I I I - I. H .. i.'ii. III... I I wm
* An absolate number is one considered withoat reference to its
sigB, as in arithmetic.
Bout. Alg. 10
74 Elementi of Algebra.
Sometimes the imposdibilitjr of solving the problem in the way
{n which it has been put, is evident on the mere inspection either
of the enunciation or the equation; the two preceding problems
are examples of this. Sometimes it is difficult to discover this
impossibility ; but it is always made evident in the course of the
solution.
We BOW go on to the second part of the principle.
Observe, first, that if, in the equation, — a? be substituted for
-)- Xy all the terms containing x^ which were additive, are now sub*
tractive, and the reverse. If there be, for example, the term -|- ^ ^i
when — a? is put instead of a?, it becomes + a X —a?, or — ax*
In like manner, if we have the term — & a?, it becomes — b X — ^
or -j- & ^- If this new equation be translated into ordinary language,
a new enunciation is obtained, which differs from the first only ia
having some quantities subtractive which before were additive, and
the reverse.
It remains to show that the substitution of — a? in the place of
X gives £ = p, the former result being x = — jp» (p is here con-
sidered as an absolute number).
Now, whatever the original equation of the problem may be,
we can always reduce it by known transformations to the form
a a? = 4* ^ (^ ^"^ ^ being absolute numbers).
From this equation,
— h h
X = « or a? =: — -, or a? = ■*— »,
a ' a ^^
\ip be put for the absolute number -. But if — a; be put instead
of X in the original equation, the same transformations will reduce
the new equation to the form
— aa?=: — ft,
from which »==* = *=»,
— a a '^^
which was to be demonstrated.
It will hence be seen how we are to interpret negative results*
When the sign is taken away, they may be regarded as the solu-
tions, not of the questions proposed, but of questions of the same
nature, certain conditions of which have been modified ; and the
surest method of obtaining the new question is to change x into
•*^x in the equation of the problem, and translate dM resttk into
ordinary language.
Problenu wiA ff^i^e RuuUt. 76
60* Remark^ The principle which has been established is
rigorously true for the equations only, and not always for the
enunciadona of problems ; that i^, the enunciation of a problem
may be correct, although the resolution of the equatiou gives
a negative value. The cause of this is, that the algebraisti in
the application of his methods to the resolution of a problem, is apt
frequently to interpret certain conditions in a sense exactly opposite
to that in which they ought to be taken ; in this case, the negative,
solution corrects the effect^ of the wrong view which he has taken
of these conditions. Thus, the equation is false, although the
problem is capable of being resolved ; and it is only when the
equation is a faithful translation of the enunciation and of the mean-
ing of all its conditions, that the principle is applicable to the enun-
ciation. We shall see examples of this in what follows ; but it is
mostly in the application of Algebra to Geometry that the principle
iff applicable, not to the enunciations, but to the equations.
Ql. In the preceding demonstration, we have been led to mul-
tiply -4- a by — a?, to divide •'—6 by + «j and — 6 by — a, Qnd
l^e results were obtained by applying to simple quantities the rule of
ik^ signs established for the multiplication and division of polynomials.
It may iippear at first view necessary to demonstrate these rules with
reference to insulated simple quantities, and this is what most authors
have attempted to do. But the demonstrations which they have given
have only the appearance of rigour, and leave much uncertainty in
tbe tpiqd* . ^6 S9y then, that the rule for the signs, established for
polynomial quantities, is extended to simple quantities^ in order fo
interpret the peculiar results to which algebraical operations lead.
Those who do not admit this extension deprive themselves of one of
the principal advantages of algebraical language, which consists ill
comprehending, under one formula, the solutions of several questions
of the same nature, whose enunciations differ only in the way of
stating certain conditions, that is, in certain quantities which are addi-
tive in the first, being subtr^ctive in the second, and the reverse.
The es^tension to simple quantities of the rules established for
polynomials, may appear desirable from the following considera<«
tipns.
The demonstration of article 17 for the multiplication of the bi-
nomials a — h and c — d^ evidently supposes a]>i ajad c^d*
For, if the contrary be the case, the course of re^^oning loses all
meaning; nevertheless, having once established the rule for the
76 Elmmiis ofAlg$bra.
signs, we need not call it in question, whatever be the magnitudes
of a, i, c, and d.
If this be granted, the product of o — ft by c, being ac — ftc,
it follows that the product of a negative quantity a — 6 (a being < ft)
by a positive quantity c, is negative.
Also, the product of ft by c — d, being ftc — ft rf, it follow that
the product of a positive quantity ft by a negative expression c — d
(c being <] d), is negative.
Lastly, the product of a — ft by c — d being
a c — ft c — a d -f" ft d,
which may be put under the form *
bd — ad — bc-^-aCf or d(ft — a)'-^c{b — a),
if we suppose d^c and ft ^ a, or ft -— a positive, it follows that
d[b — a)>c(6 — a), or d{b — a) — c(ft — a)
is positive. The product therefore of a negative expression a-— ft
by a negative expression c — d {a being <[ ft and c<^d) is positive.
It is this which constitutes one of the distinguishing character-
istics of Algebra. In arithmetic and geometry, the things reasoned
on are real, and such that the mind can form a distinct conception
of them ; while in algebra, the subjects of the reasoning and of the
operations are often imaginary, or contain symbols of operations
which cannot be performed ; but the exactness of the results which
are obtained by these means, and which may always be arrived
at by processes more rigorous, but much longer, is a sufficient
sanction of the methods which we have followed.
62. As the rule for the signs applied to simple quantities is of
constant use in algebra, we shall here present it at one view. We
shall also see methods of deriving new expressions peculiar to alge-
braical language.
We begin by addition and subtraction.
To add + ft or — 6 to a quantity expressed by a, we must
write the result thus, a -|- ft or a — ft ; that is, we must write the
two simple quantities one after the other, with their respective
signs. (13.)
The subtraction of -f- ft or — ft from a, must be written thus,
a <— ft or a -}- & ; that is, change the sign of the simple quantities
to be subtracted, and write it with its new sign, after that from
which it is to be taken. (14.)
ProbUnii toUh Negative Results. 77
Id multiplication and division,
+ aX +b ov — ax — h gives the product 4" ^ ^ 7 fi 7 \
— flX -^h or -{-ax — h gives the product — ab^^ *'
4-0 — a . , . , a
XT O"" jTa S*^®s the quotient + r
q^ or ^-r gives the quotient — ▼
These rules give rise to the following important remarks*
(1.) In algebra, the words add and sum do not always^ as in
srithmetic, convey the idea of increase ; for the result a-— 5, which
arises from the addition of — b to a, is, properly speakingi the
difference between the number of units in a and that in b ; whence
the result is less than a. To distinguish this kind of sum from an
arithmetical sum, call it the algebraical sum. Thus the polynomial
is an algebraical sum^ if we regard it as the result of the union
of the simple quantities
2a3, _3 o«6, 4. 3i* c, and — 2 a«c,
with their respective signs ; its reed meaning is the arithmetical
difference between the sum of the units contained in the positive
and negative terms.
It follows, tiiat an algebraical sum may, if particular values be
substituted for the letters which it contains, be a negative -quantityy
or one which is preceded by the sign — .
(2.) The words stAiract and difference do not always convey
the idea of diminution ; for the difference between a and — 6,
being a 4* &> is greater than a ; it is called an cdgebraical differencCf
because it can be put under the form a — ( — b).
By means of these denominations, negative values may be re-
garded as solutions of equations. For example, in the equation
SI 4*^ == 24 ; the result, x=i — 7, indicates that we must add
— 7 to 31 to obtain 24 ; and, in fact, 31 4- ( — 7), or 31 — 7,
is 24.
Id like manner, in the equatioD
ic I 454-»
15 4-« = — j~i
the resuh, x = — 5, shows that we must add — 5 to the two
ages, in order that the age of the son may be one*fi>arth of that of
the father.
78 Elemmti qf A^Ara.
For 16 + ( — 5), or 16 — 6 = 10,
46-1- (_6), or 45 — 6=40.
63. The necessity of employing negative expressions in alge-
braical calculations, and of proceeding with them as with absolute
quantities, leads to two other propositions, which are of frequent
use in algebra. Every negative quantity — a, is less than 0, and
of two negative quantities, the least is that whose numerical value is
the greatest.
Thus — tf < 0, and — a < — 6, if a be numerically greater
than &.
' Demonstration. To explain these two propositions, deserve,
that if from the same number we subtract a series of continuaHy
increasing numbers, the remainders will continnally diminish. Take
any whole number, as 6 for example, and from it subtract succes*
sively 1, 2, 3, 4, 5, 6, 7, 8, 9^ be. and we shall have, writing the
differences in the same line,
6—1, 6—2, 6—3, 6—4, 6—6, 6—6, 6—7, 6—8, 6—9, &c.
which, reduced, become
6, 4, 3, 2, 1, 0, —1, —2, — 3, &DC.
Whence it will be seen, that — 1 must be regarded as less than
nothing, because the last expresses the difference between 6 and
itself, whereas — 1 expresses the difference between 6 and a
greater number.
For the same reason, — 1 is greater than — 2, — 3 is gireater
than — 12, although the numerical values of the first expressions
are less than those of the last.
AnoAer Demonstration. Since, in order to interpret the peculiar
results which the algebraical solution of a problem leads to, we
have agreed to consider negative expressions as quantities, we
ought, in performing with them the same operations as with absolute
numbers, to arrive at true results. Now, we may regard it a« an
ipciom, that if a number a is greater than another number 6, and
the same number d be added to both, the first result a -}- <2 is greater
than the second 6 + d.
Then, if we admit that > r— a, and — a > — (a -f- m),
(a and m being absolute numbers), if we add a + m to the two
members of each, we find a -{' m"^ m, and m ]> 0, which is true.
On the coptrary, if we supposed < — a» and — a < —
(a -)- ^)9 the same reasoning would give us a -}-%<! "* &tid.
ffi <[ 0, which is absurd.
Discuiiion ofPrchlemt of the First Degree. 19
The two preceding propositions must be admitted, if we ar6 to
treat negative expressions as absolute quantities. These propo-
sitions are an algebraical method of speaking, analogous to that
which we often make use of in common language. We say
often that a man is worse than nothings to express that his debts
exceed his property ; and, of two persons of equal fortune, each of
whom owes more than he possesses, that the richest is he who owes
least.
Discussion of Problems of the First Degree^ containing One or
more Unknown Quantities.
64. After a problem has been solved generally, that is, after
«
the given quantities are expressed by letters, we can determine
what the values of the unknown quantities are, when particular hy-
potheses are made respecting the given quantities. The determi-
nation of their different values, and the interpretation of the peculiar
results at which we arrive, form what is called the discussion cfthe
problem.
The discussion of the following question will present nearly all
the circumstances which are usual in problems of the first degree.
Rf A B B
Problem 14. Two couriers set off at the same time from two
different points^ A and B, of the same straight line AR, and in the
same direction AB. The one wlio starts from A goes m, and the
other n miles an hour. At what distance from the points A and B
wiU the two couriers meet ?
SoltUion. Let 71 be the point at which they meet; let the
unknown distances AR and BR, expressed in miles, be called x
and y, and let a be the distance AB between them at the time
when they start. Then
X — y =z a. . • . . . (1).
Since m and n are the number of miles per hour (or the respee^
tive velocities of the two couriers), it follows, that the times of trav*
elling over the spaces x and y, are expressed by ~ and ^ ; and
e99 ew
these times are equal ; therefore
| = |,ornx — »y = (3).
80 Elements ofAlg^a.
The combination of equations (1) and (2), by the known methods
of elimination, gives
am an
m — n ^ fit — n
which values are easily verified.
As long as we suppose m ^ n, or m — n ^ 0, or positive,
these values are positive, and the problem, as the enunciation stands,
admits of a real solution. And it is plain, that if the courier A
goes faster than j5, he gains on him every instant ; the distance
which separated them at first diminishes more and more, till at last
it is entirely destroyed, and the two are at the same point of the
line which they are travelling.
But if we suppose m <^nf or m — n <^0^ or negative, the
values of x and y are negative, and are
am an
n — m'^ n —
m
In order to interpret these results we observe, that it is impossible
for the two couriers to meet in the direction AB ; for as B goes
faster than A^ the distance which separates them increases every
moment. But if, instead of supposing that they proceed in the
direction AB^ we suppose, on the contrary, that they go in the
direction BA, the circumstances of the problem become the same
as those of the case where m'^ n'y and it is plain that the two
couriers will meet in the point Rf of BA produced. This is shown
also by the principle established in article 59. Change the signs of
X and y, and the equations become
— X -j-y = a^
— - = — ^ C
m n )
or
which give
am » an
n — m ^ n — m
These values verify the new enunciation, in which the couriers were
supposed to start in the direction BA.
Let m = n, or fl> — - n = 0, the general values become
^_ am an
How are these results to be interpreted ?
Discussion of Problems of the First Degree. 81
The enunciation shows, that the solution of this case of the prob-
lem is absolutely impossible ; that is, in whatever direction the two
couriers start, they can never come together ; because, being at a
certain distance from one another at starting, and going equally fast,
they ought to preserve the same distance.
The result -jr- may be regarded as a new sign of impossibility.
The equations of the problem become, when m = n,
X — y z=: a '^ Cx — y =- a
which are evidently inconsistent.
Nevertheless, it is customary to regard such results as
atn an
a? — -Q-, y — "0"'
as a species of value, to which is given the name of infinite. The
reason is as follows ;
When the difference m — n, without being absolutely nothing,
is supposed to be very small, the two results
am , an
and
m — n m — n
are very great.
For example, let
m — n = 0,01, wi z= 3, whence n = 3 — 0,01 = 2,99.
= 300 a, = 299 a.
m — n 0,01 m — n
Let m — n = 0,0001, m = 3, whence n = 2,9999 ; then
"*"* z= 30000 a, -^- = 29999 a.
m — n m — n
In a word, if the difference of the velocities of the couriers be
nothing, the two couriers will meet ; but the distance between their
point of meeting and that of starting, will be greater and greater,
the more that difference is diminished. If, then, we suppose that
difference less than any given quantity, the distances
am . an
and
m — n m — n
will be greater than any given quantity, or infinite. We say then,
Bour. Alg. 11
82 Elements ofAlgAra,
for brevity sake, that when m — n = 0, the results become
am an • /• *.
X = -g-, y = -g-, or inanite.
Since is less than any absolute magnitude, we can take tUs
symbol to represent the last state of any magnitude which may be
diminished as much as we please. And since a fraction is greater,
according as its numerator is greater compared with its denominator,
A .
an expression such as jr (where A is an absolute magnitude) is well
suited to represent an infinite quantity, that is, a quantity which is
greater than any assignable quantity.
An infinite quantity is also expressed thus, gd ; and a quantity
less dian any given magnitude, or 0, can be expressed by ^ ; for a
fraction is the smaller, according as its denominator is greater com-
pared with its numerator. Thus, and ^ are synonymous sym-
bols ; so also are -^ and cs.
We have insisted particularly on these last notions, because there
are questions of such a nature that infinity may be regarded as a
real answer to the question. Frequent examples of this are seen
in the application of algebra to geometry.
If to the hypothesis m = n, be added this, a :^ 0, the two values
become
,
a: = g and y = g.
What meaning is to be attached to this result ?
Returning to the enunciation, it will appear that, as the two cou-
riers go equally fast, and set out from the same point, they will be
always together, and consequently they meet at every point of the
line which they pass over. Indeed, by this double hypothesis,
m — n, a = 0, the equations become,
x — y =0^ C a? — y =
which result in the same. Therefore, the question is entirely inde-
terminate (66), because there is in reality only one equation be-
tween two unknown quantities.
Discussion of Problems of the First Degree. 83
The expression ^, in this case, is a symbol that the question thus
enunciated is indeterminate (55).
If the two couriers do not go equally fast, that is, if m ]> or
<^ n, while we suppose a = 0, the resuh is a? s 0, and y = 0.
As the two set out from the same point with different velocities)
they cannot evidently be together except at the point from which
they set out.
The preceding hypotheses are the only ones which lead to
remarkable results. They serve also to show to beginners in what
manner algebra answers all the circumstances concained in the
enunciation of a problem.
65. When a problem has been solved generally, we may find,
by simple changes of sign in the formulas obtained for the unknown
^quantities, the solutions of other general problems, the enunciations
of which do not differ from those of the proposed problem, except-
ing that certain quantities which were additive become subtractive,
and the reverse.
Let us take, for example, the problem of the workman, which
was solved in article 47. If we call c the sum he has to receive,
the equations are,
a? + y=:n, ax — 6y = c,
bn 4- c an — c
or X = — r-r , y = — r-ir-
But, if we suppose that at the day of settling, the workman owes a
sum c, instead of having to receive it, the equations are,
a?+yz=n)^ C x + y =: n,
6y — aa?=c ) {ax — 6y= — c,
(in which the signs of the second equation are changed).
It is plain that without solving these equations again, we may
obtain immediately the corresponding values of x and y by simply
changing the sign of c in the preceding values, which gives
bn — c an + c
X = — r-T' ; y = - — rnr.
To prove this rigorously, let us designate -^ c by d ; the equa-
tions become
a? + y = n,
ax — fr y = i{,
which differ from those of the first enunciation only in having d
84 Elemenii of Algebra.
instead of — c. Then we must have
bn-{-d an — d
^"^ a + h ' y-^ a + 6 •
If we put instead of d its value — c, the equations become
6n-f-f — c) an — ( — c)
a -f- 6 ^ ^ a -f- '
or, by applying the rules established in article 62,
hn — c an -\- c
^ - T+b ' y ~ a + 6-
The results which correspond to the two enunciations may be
comprehended in the same formula,
bndt-c anz:pc
The double sign d= is called plus or minus ; the upper signs refer
to the case where the workman has to receive a sum c, and the
lower to the case where he has t^pay it.
These formulas, moreover, contain the case in which, upon the
settlement, the workman is supposed to have nothing either to
receive or to pay. It is only necessary to suppose c = 0, which
gives
hn an
^ "" ^r+^' y - T+l'
Let there be the two general equations, supposed to arise from
the translation of the conditions of a problem into algebraical lan-
guage,
a a? -|- J y = c,
dx +fy =zg.
By multiplying the first by/, the second by 6, and subtracting the
second result from the first, we have
(af—bd) X = cf— bg, OTX =z ^E^.
In like manner, we shall find
ag — cd
y " af—bd'
This being premised, to pass from these formulas,
(1.) To those corresponding to the equations
a X — b y =z Ci
dx+fy=ig,
Inveiiigation of Equations and Problems of the First Degree. 85
it is sufficient to change b into — 6, which giires
^_ g/+^g ag—cd ^
^^ af+bd" y "" a/+ 6rf'
(2.) To those corresponding to the equations
ax — J y = c,
it is sufficient to change h into —6, and /into — f which gives
.^_ — c/+^g _ ^8 — cf .,^ <^g—cd
^^ —af+bd "" bd—af y^bd—af
The demonstration is exactly the same as that of the preceding
example.
General Investigation of Equations and Problems of the First
Degree.
66. In order to generalize the investigation of problems of the
first degree with one or more unicnown quantities, we propose to
establish formulas which will represent the values of the unknown
quantities for any set of equations, containing an equal number of
unknown quantities.
In the first place, every equation of the first degree with one
unknown quantity may, by means of the usual transformations, be
reduced to the form ax z=zb^ a designating the algei)raic sura of
the quantities by which the unknown quantity is multiplied, and b
the algebraic sum of all the known terms.
We may evidently deduce from this equation a? = -.
«• —
In the second place, we observe that every equation of the first
degree with two unknown quantities, may be represented by
a 0? + J y = c.
For, if the proposed equation contains denominators, we may make
them disappear (44) ; by uniting all the terms containing x and all
the terms containing y in the first member, then transposing all the
known terms to the second member, we designate the algebraic
sum of the first by a x, the algebraic sum of the second by 6 y,
and the algebraic sum of the last by c ; a^ b^ c are then whole
quantities affected with either sign.
Let the two proposed equations be < ,^ X &' --5
86 Elements of Algebra,
We obtain by multiplying the first equation by h'^ the second by
hf and subtracting one from the other,
{ab' — ba')x=zcl/ — bc',
ch' — bd
whence x = ^3^, ;
we find, in like manner, y = —17 tz,'
' ' '' ab' — ba
Let there now be the three equations
ax +Jy •\-cz = d! (1),
a' X +b'y +&Z —d' (2),
a'^x + V'y + d' z = d/' (3).
In order to eliminate z^ we multiply the first equation by (/, the
second by c, and subtract the second from the first.
It becomes
[a€f—ca')x+ {b& — cb')y = d& — cd'. . . (4).
Combining, in like manner, the second equation with the third, we
find
(^a'&' — &a'')x+{b'&' — &b'')y = d'&' — &d'' . . (5).
In order to eliminate y, we must muhiply equation (4) by
1/ &' — & b^^y and equation (5) by b cf — c i', and then subtract,
which gives
[{a& — ca') {b'&'—&b'')—{a'c'' — &a'') {b& — cb'y\x=z
{d& — cd') i^b' d* — dy') — {d d" — d' d') (6C — qiOj
by performing the operations indicated, reducing and dividing the
two members by c', we have
(a y d' — ad b'' -^^ ca'V — ba' d' ^bd a" — c V a") x =
dy d^ — ddb" + cd'y' — b d'd' J^bd d" — cV d".
Then finally,
_ db' d' — dd h" + cd' b"—bd' d' -^-b d d" — c b' d"
^"^ ab' d' — ad b" -{-ca' b" —b a' d' \^bda" — c b' a"*
By performing analogous operations for eliminating x and 2:, and
afterwards x and y, we shall find for y and z,
_ ad' d'—ad d" + c a'd" — da' d' + dd a" — cd'a!'
y '^ ab' d' — ad b" ^ ca'b" — b a' d' + bda" — e b* a" '
z
_^ ab'd"-^a d' y'-^-da' b" — b a' d" -^-b d' a" — db' a"
"" a b' d' — adb" + e a' b" — 6 a' c" -f 6 d a" — cb' a"'
Investigation of Equations and Prohleiks of the First Degree. 87
As beginners may not be much used to making all possible ab-
breviations in the calculation, we will here give a method of passing
from the value of x to that of y and z, without being obliged to go
over all the preceding operations from the beginning.
We observe that the set of equations (1), (2), and (3), will
remain the same if we substitute for x, a, a^, a^^, the quantities
y, &, b', Vf and the reverse ; then, if in the expression which gives
the value of a?, we change x into y, then a, a\ d'^ which are the
coefficients of a?, into 6, h\ h"^ which are the coefficients of y, and
the reverse, we shall obtain a result which will be no other than the
value of y.
Performing this change, we have
_ da' c" — dc' a" -^cd'a^ — a d' d' J^ad d"—c a' d"
^ '^ ba' c" —be' a" -{-c b' a" — ab'c" + a d b" — ca' 6"*
or, changing the signs of the numerator and denominator, and
writing in each, the last three terms first, and the first three terms
last,
_ ad'd'—add"-{'ca'd"—da'd'-\'dda" — cd'a"
^'^ ab' d' — ad b" J^ca'b" — b a' d' \-bc a" — c b' a"'
In like manner, we should obtain the value of z by changing
a?, a, a\ a'* into ar, c, c', (/', and the reverse.
We can easily see the method which must be pursued, if we
had four equations with four unknown quantities, he.
N. B. By reflecting a little on the manner in which these formu-
las have been obtained, we shall easily perceive that for any
number whatever of equations, containing an equal number of
unknown quantities, a?, y, 2; . • . , there can exist, in general, but one
set of values of j?, y, z . . . , which will verify the equations.
First, the proposition is evident for an equation with one un-
known quantity a a? = J. There is only the value - which can
satisfy it.
Let us consider two equations with two unknown quantities.'
After we have multiplied the first equation by the coefficient of y
in the second, and the reverse, the result which we obtain, by
subtracting one from the other, may be substituted in one of the
two proposed equations. Now this result, containing only one
unknown quantity, admits for that unknown quantity but one value,
which, carried back into one of the proposed equations, will likewise
88 Elemmts of Algebra.
give but one value for y. The same reasoning will apply to three
equa^tions with three unknown quantities.
67. The use of accents, in the notation of coefficients, has led
to the observation of a law, according to which we can easily find
the preceding formulas, without being obliged to perform the elim-
ination*
Let us consider, first, the example of two equations with two un-
known quantities. We have found, for the values,
cb' — b& ac
/
ca'
^''ah'—haf' ^ " ab' — ba''
(1.) To obtain the common denominator to these two values^ form
with the letters a and b, which designate the coefficients of x and of
y, in the first equation^ the two arrangements a b and b a, then
interpose the sign — , which gives a b — b a ; finally, accent in
each term, the last letter ;
it become ah' — b a'.
(2.) In order to obtain the numerator relative to each unknown
quantity, replace, in the denominator, the letter which desis^nates
the coefficient of that unknown quantity, by the letter which desig-
nates the known quantity, still leaving the accents in the same place.
Thus, a b' — b a' w changed into ch' — b d, for the value of x,
and into a c' — c ^',for the value ofy.
Let us consider the case of three equations with three unknown
quantities, a, b, c, designating the coefficients of ic, y, z, and d the
known quantity.
(1.) In order to have a common denominator, take the denomi-
nator a b — b a, which corresponds to the case of two unknoum
quantities, (the accents being omitted) ; introduce the letter c into
each of the terms a b and b a, at the right, in the middle, and at the
left ; then place between them alternately the signs plus and minus ;
there results
abc — acb-|-cab — bac-|-bca — cba.
Afterwards place, in each term, the accent ' over the second letter,
and the accent " over the third letter ; and we have for the denomi-
nator,
ab'c'' — ac^b'' + ca'b'^ — ba^c'' + be' a'' — cb'a''.
(2.) In order to form the numerator of each unknoum quantity,
replace in the denominator, the letter which designates the coefficient
of that unknoum quantity, by the letter which designates Jhe knovm
Investigation of Equations and Problems of the First Degree, 89
quantityy having the accents in the same place. Thus for x tJiai^t
a into d ; for y, b into d ; and for z, c into d.
This law, which may be regarded as the result of obsenratioo for
two or three equations, is capable of being extended to any nomber
of equations ; but the demonstration is very complicated, and does
not belong to the elements of algebra. The learner may consult,
for this purpose, the second part of the Algebra of Gamier, which
refers to one of M. Laplace. This demonstration is taken from
the Memoirs of the Academy of Sciences, 1772.
68. Let us attend to the use which may be made of these for-
mulas, in particular applications.
Let there be the two equations
6jn — 7yz=34, 3a? — 13 y = — 6.
By comparing them with the general equations,
a 0? -f" ^ y = Cj a' a? + 6' y = (/,
we have
a = 6, 6 = — 7, c = 34, a' = 3, i' = — 13, (/ = — 6.
Let us substitute in the formulas
*
ch' — he' ac' — ca'
"^""aft' — 6a" y ~" ab' — ba"
in the place of a, 6, c, a', 6', (/, these values ; and we have
X _ 34X — 13 — (— 7)X— 6 _ — 34X 13 — 7x 6
\^')^— 5X— 13 — (-.7)X3 "■— 5X13 + 7X3
_ _ 442 ^ 42 _ —484 _
— _ 65 + 21 "" — 44 ~ " '
. . _ 5X— 6— 34x3 _ —30 — 102 _ — 132 _
^ ^^ ^~5X — 13 — (— 7)X3"" —65 + 21 ""—44 """*'
and a? = 11, y = 3, are the values which will satisfy the two pro-
posed equations.
We can immediately assure ourselves of this, by substituting
them in the equations. But, in order that the demonstration may
be independent of every particular example, we remark, that in
order to pass from the formulas relative to the equations
ax -{-hyznc
and a^ a? + 6' y = (/, to those which belong to the equations .
a a: — 6 y = c, and a' a? — i' y = — d^
it is sufficient (65) to change h into — h^ V into — iy, and V into
— </, which gives
Bour. Alg. 12
90 Ekmenti af A^ebra.
— g X — ft^ — (— ft) X — c^ _ gX — g^ — c X a' .
*- aX—b' — i—b) Xfl' ' ^■" ax— ft' — (— 6)Xa '
und, in order to deduce from these new general formulas, the values
whicb belong to the particular equations, we must make
a =: 6, 6 = 7, c = 34, a' = 3, ft' = 13, (/ = 6.
Then, finally, to obtain the values relative to the proposed equa-
tions, it is sufficient {o make, in the general formulas before obtained,
a = 5, 6 = — 7, c = 34, a' = 3, ft' = — 13, (/ = —6,
then to perform the calculations, according to the rules laid down
for simple quantities.
The rule consists, in general, tn substituting, in the place of the
coefficients, a, b, a', b' . • • , their particular values considered with
the signs by which they are affected in the particular equations, and
in performing all the operations indicated according to the establish'
ei rules.
These applications justify again the necessity of extending to
simple quantities the rules of the signs established for polynomial
quantities, since it is the means of rendering general formulas of
^e first degree, applicable to every particular example.
We proceed to the discussion of these formulas.
j69. It results from inspection, that in particular applications, we
can obtain four kinds of values for answers to problems of the
first degree, that is, positive values, negative values,- values of
the form ^, in fine, values of the fom % The probleo. of
<
the couriers has given rise to these four results, which we now pro-
pose to explain in a general manner.
First, the positive values are usually answers to questions in
the sense of their enunciation. However, we observe that for cer-
tain problems all the positive values do not satisfy the enunciation.
If, for example, the nature of the problem requires that the numbers
sought should be whole numbers, and that we should find fractional
lumbers, the problem cannot be resolved. Sometimes, also, the
nature of the problem does not permit that the unknown numbers
should exceed the numbers known and given a priori, or that they
should be less than other numbers. If the values obtained,
although positive, do not satisfy that condition which the enunciation
requires, but which cannot be expressed by an equation, the prob-
lem cannot be resolved. Thus the positive values of unknown
Investigation of Equations and Problems of the First Degree. 01
^antities are^ properly speakings direct answers to the equations ;
and they are solutions of the question j only so far as their nature
agrees with that which the enunciation requires. In order to con-
ceive how a number can verify an equation, without verifying the
problem of which it is the algebraic translation, it is sufficient to
remark, that the same equation is the algebraic translation of an
infinity of problems ^ of which some admit all possible absolute
numbers for a solution, and others admit only numbers of a certain
nature.
70. We already know how to regard negative values in problems
of one unknown quantity. To complete the view already taken, we
proceed to consider the principle of article 59, for a problem
having several unknown quantities.
It is evident, in the first place, that if we obtain negative values
for some of the unknown quantities, the equations of the problem
cannot be satisfied in the sense in which they have been established ;
for, if a set of absolute numbers, put for a?, y, z, could verify them,
the equations which have been deduced by the method of elimina-
tion, would themselves exist for that set. So that the equation which
contains only one of the unknown quantities, for which we have
obtained a negative result, would be verified by an absolute num-
ber, which would be contrary to the hypothesis. We must then
rectify the enunciation of the problem, or at leastj the equations
which are the algebraic translation of it.
If, in the equations, we change the signs of the unknown quan-
tities, for which we have obtained negative results, the terms afTectr
ed by these unknown quantities will necessarily change their signs,
and the enunciation of the problem will be generally modified
so that certain quantities, which were additive, will become sub-
tractive, and the reverse.
Finally, these modifications once made, the new enunciation is
verified by the values first obtained for the unknown quantities,
their signs being left out of consideration. Let us take, for exam-
ple, three equations with three unknown quantities,
ax -{-by -{- cz=: d, a'x + b'y -j- (/;2r = d\ a"x + bf'y -f- d'z = d'\
and suppose that these equations have given x = ^i, y = — * g,
2r = — r ; let us change, in these equations, y and z\ nto -— y and
— z^ or rather into \f and z' (designating for the moment, — y and
— 2f by y' and 2/).
&i Elements of Algebra.
. We bave
a^'x + h^'y' + &' zf = d''.
Now, as these equations differ from the preceding, only by
having substituted for y and z^ y' and zf^ they will necessarily
give X =z p, y' = — q, z^ =z — r; whence, putting — y and
— z in the place of y' and 2/...,a?=p, — y=: — j,
— 2r = — r, or rather, x =z p^ y =zq, z =^r; which was to be
demonstrated.
Thus the principle of article 59 is true with regard to problems
of the first degree having several unknown quantities.
We shall conclude with this observation, that sometimes the enun-
ciation of a problem is not, by its nature, susceptible of any modifi'
cation ; in this case, the negative values are only solutions of modified
equations, which may moreover be regarded as tlie algebraic
translation of other problems susceptible of modification.
71. It DOW remains to interpret such expressions as ^, jr.
Let there be, in the first place^ the equation with one unknown
•quantity, ax^zh; whence « == -.
(1.) If, for a particular hypothesis, made with respect to the given
quantities of the question, we have a = 0, there results x =z r:.
Now the equation becomes, in this same case, X a? = J, and
can evidently be satisfied by no determinate number. Let us
however remark, that as the equation may also take the form
— = 0, if we put in the place of x numbers continually
increasing, - will continually diminish, and the equation will ap-
proach aearer and nearer to being ei^act ; so that we can take
for X a value sufficiently great to make - less than any assign-
X
able quantity.
It is for this reason that Algebraists are accustomed to say, that in
this case Infinity satisfies the equation ; and there are some ques-
tions for which this kind of result forms a true solution ; at any rate
it is certain that the equation cannot admit of a solution in a finite
number, and this is all which we wish to prove.
InvestigcUion of Equations and Problems of the First Degree. 93
(2.) If we have, at the same time, a = 0, i = 0, the value of x
takes the form a? = tj.
Now, the, equation becomes, in this case, X a? = 0, and every
finite number, positive or negative, will satisfy this equation.
So that the equation (or the problem of which it is the algebraic
translation) is indeterminate.
72. It is important here to remark upon the expressioa
jc, that it does not always announce an indeterminateness ; but rath-
er the existence of a factor common to the two terms of the frac-
tion, which factor becomes zero by the effect of a particular
hypothesis.
Let us suppose, for e&le, that we have found for the result
of the solution of a problem,
a3— 63
X =
a2 — 69'
If, in this formula, we make a = 6, there results a; = ^r .
But we remark, that a^ — 6^ can (31) be put under the form
(a — 6) (a* + a 6 + 6^), and a® — 6^ is equal to (a — 6) (« + 4) ;
so that the value of x is reduced to
^^ {a-h){a^ + ah + l^)
^~ (a-b){a + h) •
Now, if before making the hypothesis a = 6, we begin by sup-
pressing the common factor a — 6, the value of x becomes
a^ 4. a 6 + 62
an expression which is reduced to
3a9 3a
x^^,orx=^^
5
on the hypothesis of a = 6.
Let there be, for a second example, the expression
gg — 6a _ (a + 6)(a — 6)
(a— 6)2 — (a — 6) (a — 6)'
By making a = 6, we find, for the value of a, a? = 5?, on account of
the existence of the common factor a — J ; but if we first sup-
n -i- h
press this factor, it becomes x =: . , an expression which
2 a
is reduced to a; = -g-, when we make a = 6.
D4 Elements of Algebra.
We hence conclude, that th^ sifmbol ^ is sometimes^ in Algebra^
characteristic of the existence of a factor common to the two
terms of the fr<iction which is reduced to this form. So that,
before pronouncing upon the true value of the fraction, we must
ascertain if these two terms do not contain a common factor. If
there is not one, we conclude, that the equation is really indetermi^.
note. If there is one, we suppress it, then make again the particular
hypothesis, which gives the true value of the fraction, which may still
A A Q ,
be presented under three forms, -g, -g-j rjj in which case, the
equation is determinate, impossible in a finite number, or indeter-
minate.
This observation is very useful in the discussion of problems.
73. Let us return to our subject, and consider now two equations
with two unknown quantities,
We have found (66), for the values of j? and y,
ch' — b d ac' — ca'
^ — aU — ba" y "" 'oF^Ta/'
Let us suppose that we have ab' — 6a' = 0, the numera-
tors bd — 6 c', ad — ca\ being of different values from ; the
values are reduced to
In order to interpret these results, let us remark, that from the
ab'
equation aV — 6 a' = 0, we obtain a' = -r-, whence, substituting
this value in the equation a^ a? + 6' y = (/, we have *
or, making the denominator to disappear, and dividing by 6^
aa? + 6^ = — ,
an equation whose first member is identical with that of the first
ax '{'by z=c^
while the second member is essentially different ; for, from the
inequality cl/ '^b d we de'duce c ^ -rr' i
We see then that the two proposed equations cannot be satisfied
simultaneously by any set of finite values ofx and y.
Investigation of Equations and Problems of the First Degree. 95^
If we have, at the same time,
a6/_6a' = 0, cJ' — 6(^ = 0,
the value of a? is reduced to a? = r, a value which needs to be inter-
preted.
The two proposed equations may, in consequence of the relation
ab^ — 6 a^ = 0, be put under the form
5 c^
aw + by=zc, aa? + 6y=-^,
equations which necessarily return into each other ; for from the
be'
relation c i' — - 6 (/ = 0, we deduce c t= -yj-.
So that in order to resolve the problem^ we have but one equa-
tion with two unknown quantities. Then, the question is indeter-
minate.
ha'
Since the relation, ab' — 6 o' = 0, gives b' z= — , whence,
substituting in the relation
bd — b& = 0, ii^'_6(/ = 0,
' a
or reducing, ca' — a c' = 0, we may conclude that if the value of
X is of the form jt, the value ofyis generally of the same form^
and the reverse.
We say generally^ for, if we had at the same time 6 = 0, i' = 0,
the two expressions, aV — b a'j cb' — b&, would be necessarily
zero, without there hence resulting any determinate value for
acf — ca\
In this particular case, the two values of x and y are reduced to
J ac' — c a' A
a? = g, and y = ^ , or ^.
Reciprocally, if we had at the same time a = 0, a^ == 0, there would
hence result x = ^ or -jr-, and y = />•
But this particular case is hardly admissible, since the proposed
equations would be reduced to two equations with one unknown
quantity,
namely, ^/^ Z ^ ] supposing 6 = 0, 6' = ;
^ ^^ 6^ y = ^ C supposing a = 0, a' = j
96 Elements of Algebra.
while we are considering here the case of two equations with two
unknown quantities.
74. It would be difficult to apply the preceding method to the
case where we had more than two equations j but we can supply
the deBciency by the following reasonings.
In order to present the subject clearly, let us consider four equa-
tions (1), (2), (3), (4), containing the four unknown qoantities x^y^
A
Let us designate by jz the value of a?, at which we have arrived
by the assistance of elimination, and suppose that, for a certain
hypothesis made upon the given terms, we have D = 0, .4 being
" any quantity whatever or equal to ; in the first case, the pro^
posed equation cannot be satisfied by finite values ; and in the
second, they are indeterminate^ or susceptible of being verified by
an infinity of sets of values of x^ y, z^ u.
In fact, it results from the method of elimination, that the set of
equations (1), (2), (3), and (4), may be replaced by four other
equations, of which one is D x =z A; the second is an equation in
terms of x and y ; the third, an equation in terms of a?, y, z, and
the fourth, one of the proposed equations, the equation (1), for
example.
This being premised, (1.) if the value of x is reduced to the
A
form ^, since the equation in x becomes then X ^ = wSy and as
k is besides a consequence of the simultaneous existence of the
proposed equations, these equations must be impossible in finite
numbers, since the equatioh X -a? = -^ cannot be satisfied by any
finite number.
(2.) If the value of a? is reduced to the form ^ (without the ex-
istence of any common factor between the two terms of its expres-
sion), the equation D x sn A becomes X 'J? = 0, and may be
satisfied by an infinity of values of x. Substituting each of these
values in the equation in x and y, of which we have spoken above,
we shall obtain an infinity of values corresponding to y ; substitut-
ing all these sets of values of a? and y, in the equation in a?, y, z, we
shall find an infinity of values for z ; finally, let us substitute all
these sets of values of a?, y, and z, in the equation (1), and there
will hence result an infinity of values for u ; and all these sets, thus
obtained, will necessarily satisfy the four proposed equations.
Investigation of Equations and Problems of the First Degree. 97
76. The first part of this proposition is subject to no^restriction ;
whenever we find for the value of one of the unknown quantities, a
result of the form -jr, it is a certain sign that the equations are impos-
sible m finite numbers, at least for that unknown quantity.
As to the second part, it admits of numerous rpodifications ; that
is, we may obtain for one or several of the unknown quantities,
results of the forn) g, without for this reason concluding that the
equations are indeterminate. Sometimes even, one of the unknown
quantities having a value of the form -^ we obtain the others under
the form -^.
The following sets furnish a proof of this ;
C ax +8y+ cz z=i dj '
(h) set of equations < ax -H^y+ ez =i d'y
( ax -|-6y-|" cz:=:dP\
C ax +6y+ cz zrz dy
(2.) set of equations < a' a? -(- A y -j- cz =i d\
{ a''x 4-*y+ c« = d'^
(3.) set of equations < a?+ y+^^^^J*
(. X'\'y-^nz=:r.
If we apply the general formulas of article 66, to the first set, we find
_0 _0 _0
and still, by considering this set, we easily discover that it can*
not exist in finite numbers (the first members remaining \he
same), except we have d = d'' = d'^. It is true, that from the mo-
ment when this relation exists, the set becomes indeterminate^ since
it is reduced then to a single equation with three wnJcnown quanti-
ties ; but it is not less certain that, in its actual state, there is an
incompatibility between the equations.
The first of the formulas of article 66, applied to the second set,
gives J? = ^, a result that reqidres to be explained.
In order that the second set may Qxist, (the first members re-
maining the same,) we must have
d — ax z=.d' — a' X, and d — ax:=,d" — d' x.
Bour. Alg. 13
98 Elements of A^ebra.
These equafioDS give successively
d' — d , d" — d
X = , and X = —. ;
a' — a a" — a ' *
but these two values of x ought to be equal, so that we have the
equality of condition
d'-^d d" — d
a' — a a" — a '
'
So long as this relation among the quantities a, J, a\ d\ a!'^ i!\ is
not satisfied, the second set will be impossible, although we have
obtained values of the form ^r for each of the unknown quantities.
If this relation is satisfied, the value of x becomes determinate
and equal to
d' — d d" — d ,
a! — a a" — a
But the values of y and z are indeterminate, since we have only
one equation between two unknown quantities.
The formulas of article 66 applied to the third set, give
(ni — n)(p — q) (n-~m)(p — g) ^_0^
that is, two values are of the form -g-, and the other of the
form g.
In this example, in order that the two first equations may
be possible simultaneously, (the first members remaining the same,)
we must have p = g ; in which case, the values of x and y are
reduced also to the form jr.
By admitting this condition we say, the value of ar, for which
^
we have found ^r, becomes determinate, and the two others are in-
determinate.
For, the set- is then reduced to the two equations
which, subtracted one from the other, give
(m — n) 2: SHE » — r ; whence z = ^ .
Inveitigatum ofEquationt and Problems of the Fir$t Degree, 99
Carrying back this value into the two equations, we obtain the sim-
ple equation
+ mr '^pn
y = -^^— •
^ m — n
The cases which we have just examined are sufficient to con-
vince us that in the applications of general formulas to particular
sets,
•
(1.) The values of unknown quantities may be presented, some
A
under the form jr^ others under the form jt.
(2.) That the symbol g, obtained for each of them, is not neces-
sarily characteristic of an indeterminateness in the equations.
A .
(3.) That the symbol -g- is always a character of impossibility ;
but the symbol g is sometimes a character of indeterminateness^ and
sometimes a character of impossibility. It sometimes also an-
nounces the presence of a common factor (72).
That we may know what to hold as to the true import of g,
we cannot do better than return, in each particular set, to the equa-
tions of that set, and seek again directly the values of the unknown
quantities, with the help of these equations.
76. Let there be, for the first example, the set of equations,
a? + 9y + 6z=:16,
2x + 3y + 2z =: 7,
3a? + 6y + 4z =: 13.
By applying the general formulas, we find for the three unknown
quantities
_0 _ _0
^ — 0' y — 0' ^""0'
but if we operate directly upon the equations, we find, by multiply-
ing the second by 3, and subtracting the first from the second,
5a? = 21 — 16 = 6;
whence x =s 1.
100 Elements of Algebra*
Substitutiog this value in the three equations, we obtaia
for the 1st, 9y + 6;? = 16, or 3y -|- 2z = 6;
for the 2d, , . . . 3 y -f 2 z = 6 ^
and for the 3d, 6 y •+• 4 2r = 10, or 3 y -}* 2 ;? = 6.
, The value of x is then determinate and equal to 1 ; as to the
values of y and z, they are indeterminate^ since we have only one
equation with these two unknown quantities.
The proposed set enters into the second of the preceding article ;
since, if we divide the first equation by 3, and the third by 2, the
coefficients of y and z become the same.
It is, moreover, formed in this manner. After having placed
arbitrarily the first members of these three equations, so that the
coefficients of y and z, multiplied crosswise in the equations, con-
sidered two and two, may always form equal products, we have
also taken arbitrarily the second members of the first and second
equation, which gave
a? + 9y-|-65r=:16 ^ Tgit-fSy + aarzz^,
2a?4.3y + 2z= 7 >whence < 2» + 3y + 2ar = 7,
i J 3 j^
3«?+6y4-42r=s k j f sO? + 8y + 2« = g.
k
Then we determined ^ or'd'', from the relation
d" — d d' — d
a" — a a' — a
of article 75, by putting
a = g, d = ^, o' = 2, d' = 7, a" =|,
13
which gave d'' = -^ ; whence 2 d'' or /? =;= 13.
Let there be the new set
llo?— 8y-|- 62: = 49,
6 a? — 12 y + 9z = 16,
4a?— 20y -f^ 15 2r = 15,
which enters also into the second of article 75, but does not satisfy
the relation
d^' — d _ d — d
a" — a «' — a'
By applying the formulas, we should find
Inve$iigai%on ofEqucUions and Problenu of the First Degree. 101
_0 __A _B
But let ul operate directly on the equations.
Multiplying the first by 3, and the second by 2, then subtracting,
we obtain 23 a; =: 115 ; whence d? = 5. Substituting this value
in the three equations, we find
— 6z = 6 )
— 9z = 9 }
— 16;r = 6 )
8y — 6z = 6
12 y — 9 z =z 9 ^or rather
20 y
The last two equations are evidently impossible simultaneously ;
and, if we apply to them the formulas relative to two unknown
A JB
quantities, we shall obtain y = -g-, z = -rr-.
So that of the three values jt^ obtained above, for x, y, z^ the
first has a determinate signification a? = 6, and the two others are
infinite.
77. In fine, while we operate directly upon particular examples,
we have other characteristics of impossibility or indeterminateness.
In the first set, treated of in article 76, if we consider the two
equations
3y + 25? = 5, 3y + 2z = 5,
at which we arrived by the elimination of x ; and, in order to
obtain y or z subtract these two equations, the one from the other,
we have = 0.
In the same manner, in the second set, if we subtract one from
the other, the two equations
4y — 32r=:3, 4y — Sz ^ I,
we have 0=2, which is absurd.
The results, = 0, and = .fl, are the true characterics of
indeterminatenesSf or simultaneous impossibility of the equations.
78, We shall terminate the discussion of equations of the first
degree by the examination of a particular case. It is that in which,
in the general equations, all the known quantities which are in the
second member, are simultaneously zero.
In this case, it evidently follows from the law of the formation of
the numerators, for the general values of unknown quantities^(67),
that these numerators are all destroyed at once, that is, that we
102 ElmmU of Algebra.
have .^ = 0, JB = 0. Since, moreover, there exists do particular
relation between the coefficients a, 6, c, a'^ 6^ (/, of the unknown
quantities, D, which results from a certain combination of these
coefficients, is generally different from 0. So that we have, for the
values of the unknown quantities, a? = 0, y = 0, 2r =:: 0. These
values evidently verify the proposed equations.
If, however, besides the hypothesis that the known quantities of
the second member, are simultaneously zero, we have still between
the coefficients of the unknown quantities, the relation X) == 0, the
general values are reduced to the form a? = j:, y = jr, &c.
Now we say, that in this case, the equations are indeterminate,
but the ratios of the unknown quantities are constant numbers,
which may be obtained by the help of the proposed equations.
Let there be the three equations
oa? + 6y + cir = 0, afX'\' Vy + i/z = 0, af'x -|- l/'y -f- ^'z = 0,
in which we suppose (67) that we have D or
They may be put under the form
fl * J- 6 y J- c = 0, a' - + 6' ^ + (/ = 0, a'' - -f 6'' ? + (/' = 0.
Jtfow, we draw from the first two, by treating - and - as two un-
known quantities.
I he' — ch' y ca! — ad
Whence we see that hy giving to z values entirely arbitrary^ the
values ofx and y wiU be obtained by the help of these two propor-
tions whose second results are constant and equal to known quanr
tities.
But it remains to know if these values satisfy the third equation.
We find by substituting them in this equation,
ah' — 6a'' ab' — o a' '
or reducing and writing the terms in a proper order,
aV d' — adV + ca'V — ba'd' + J(/a'' — ci'o'' = 0,
a condition which, by hypothesis, is satisfied^
Invutigation of Eqwxtions and Problems of the First Degree. 109
■
79. This leads naturally to the examination of a circumstance
of which the second problem of the will, resolved (49), has given
an example ; it is that where the enunciation of the question leads
to a number of really different equations, greater than that of the
unknown quantities to be determined.
Let us suppose,, for more generality, that the question* contains n
unknown quantities, and gives rise to m different equations, m being
>. n. We must first combine with them a number n of the propose
ed equations^ in order to obtain the values in the n unknovm jiuin-
tities ; then substitute these values m — n rematntng' equations^
which give rise to as many relations between the given terms;
and these last relations ought to be verified, in order that the prob-
lem may be possible in the form in which it has been enunciated.
The m — n relations thus obtained, are called equations of con-'
dition,
80. Recapitulation of the preceding discussion. It results from
this discussion, (1.) that a set of equations of the first degree with
an equal number of unknowji quantities, can in general be satisfied
but in one way (66) ;
(2.) That every positive value, found for an unknown quantity,
answers directly to the equations of the problem, without corres-
ponding always to the enunciation (69) ;
(3.) That every negative value answers only indirectly to the
enunciation or to the equations which are its algebraic translation,
but always corresponds to the equations considered in a sense purely
algebraic (59 and 70) ;
A
(4.) That every expression of the form jr, found for one or sev-
eral unknown quantities, indicates an incompatibility in the pro-
posed i^t of equations, at least, in finite numbers, for all unknown
quantities (71, 73, and 74) ;
(5.) That every expression of the form n indicates, either an
indeterminateness, or an incompatibility (71, 73, 74, and 75) 5
but the value of an unknown quantity may be reduced to jz,
by means of the presence of a common factor in the two terms of
the fraction. This must be carefully examined (72) ;
(6.) That if all the second members of the proposed set of equa-
tions are zero, the values become also ; that if to this 'hypothesis,
104 Mements of Algebra.
we add that in which the common denominator of th^ values of
unknown quantities is 0, the number of the sets of values is infinite ;
but that these values are subject to have a constant relation between
them (78) ;
(7.) That, whei^ the number of equations is greater than that of
the unknown quantities, the problem is possible only, inasmuch as
the values of the unknown quantities, determined by a number of
equations equal to that of the unknown quantities, satisfy the other
equations (79).
81. Enunciations of new problems susceptible of discussion, or
whose resolution presents some interest.
Problem 15. A banker has two kinds of money ; it requires a
pieces of the first to make a crown ; and b pieces of the second to
make the same sum» Some one demands c pieces for a crown.
How many pieces of each kind must the banker give to satisfy
(ist kind, ^±^f ; 2d kind *-^^).
Problem 16. To find the two adjacent sides of a rectangle^ by
supposing, (1.) that the two sides are to each other in a given ratio
m : n; (2.) that if we alter the sides of this rectangle (by addition
or subtraction) of the given quantities a and b, the surface U)iU be
altered by the quantity p.
Supposing the sides altered by addition, we find
^^ m{p — ab) n(p — ab)
na-^-mb' " na-^mb'
Problem 17. It is required to find the property of three persons
A, B, C, knowing, ( 1 .) that the sum of the estates of A and of 1
tifnes thejproperty of 3 and C is equal to p; (2.) that the'^um of
the estate of B and of m times the property of A and C, is equal to
q; (3.) that the sum of the estates ofC and ofn times the property
of A and B, is equal to r.
(This question may be very simply solved by introducing an
unknown auxiliary quantity in the course of the calculation ; and
this unknown quantity is the sum of the three estates.)
Problem 18. To find the property of 6 persons, A, B, C, D, E,
Yjjrom the following conditions ; (1.) the sum of the estates of A
and B isa; that of the estates of C and Dish; the sum of the
Problems and Equations of the Second Degree. lOS
estates E and F is c; (3.) the estate of A is worth m timei.th^
estate ofC; the estate of TH is worth n times the estate of IE; the
estate of F is worth p times the estate ofB9
(This problem ma^ be solved by means of a single equatfon
witfr one unknown quantity.)
These problems are extracted from the Algebra of M. LbuiHieri
of Geneva, a work distiTiguisfaed for its selection of questions pro-
posed as exercises.
Resolution of Problems and Equations of the Second
Degree.
83. Ii^TRODUCTioN. When the enunciation of a problem leads
!0 as equation, such as a ot* = fi, in which the ucrknown quantity
is multiplied by itself, the equation is said to be of the second
degree ; and the principles established in the preceding articles are
insufficient for it solution ; but since, by die divistoni of both its mem-
bers by a, the equation becomes a^ =: -, it will be seen that the
question is reduced to finding a number^ tohich, when multiplied by
* . . • -
itself shaU produce the number expressed by - ; this is what ispro^
posed in the extraction of the square root.
Formation of the Square and Extraction of the Square Root of
Algebraical Quantities.
83. Let us first take the case of a simple quantity, and in order
to discover the process we must see how we have formed the
square of a simple quantity. By the rules given (16) for the niol-
tiplication of simple, quantities, we have
therefore, to raise a simple quantity to* the square we must square
its toefficient and double each of the exponents cfit^ d^ffhrent letters.
Then to return from the square to the square root of a siqpqple
quantity, we must, (1.) extract the square root of its coefficient;
(2.) take the half of each of its exponents.
ThtJS„ \/§r^B = 8 a' J' ;
for {Sa^by =i8a^b* XScfib^;=z64i^b\ .^
JBour. Alg. 14
106 EhmenU of Algebra.
In like 'manner,
for . (25 a 6^ c')' z= 625 a* t^ c«.
From this rule it follows, that in order that a simple quantitjr
xp^y be the square of another simple quantity, its coefficient must
he a perfect square, and all its exponents must he even numbers.
Thus, 98 ab^ is not a perfect square, because 98 is not a perfect
square and the exponent of a is not even.
In this case the quantity is placed under the sign \/ , and
is written thus, V9t^a^- Expressions of this kind are espied
radical or irrational quaniities, or, simply, radicals of the second
degree.
84. These expressions may be often simplified by the use of the
ibllpwing principle. The square root of the product of two or mor€
factors is equal to the product of the square roots of these factortf
Pk:, in s^lgebrmcal language,
••'.. \/ahcd • • • • = \/a • \/b • ^c • \/d • • • .
To demonstrate this principle, observe that, from the definition
of a square root,
(\/abed f =z abed ,
On the other hand,
(Va V^ Vc. W— 0* = (V«)*-(v'6)*'(Vc)*-W)' =zabcd .....
Then, si^ce the squares of \^abcd . . . and ^a . v^J . \/c . \/d, . •
are equal, these quantities themselves are equal.
This being premised, the above expression \/9dab^ may be put
ynder the form \/49fr* X 2»j which is V49fe* X \^2a*
. Now V49P (83) is reduced to 7 6^.
Then
V98a64 =.7i*. V2^.
In like manner,
V45a263c2rf = V9a962c2 x 5bd == Sabc.\/5bdx
\/864oa65cU z= y/lUc^b^eio x 66c = 13 o6^C* . \/Stc.
In general, to simplify an irrational simple quantity, separate aU
its factors which are perfect squares, and extract their roots (83) ;
place the product of all these roots before the radical sign, and
all the factors under the radical sigUj which are not perfect
squares. . : r.» . ' . . ,, , . I
Prohltmt arid Equatumi of the Second Degree* lOT
In the expressions
the quantities 1 1^^ 3 a i c, 12 a &' c^, are called the coefficients of
the radical*
85. We have hitherto paid no regard to the sign of the simple
quantity. But since, in the solution of questions, we are led to ttte
consideration of simple quantities preceded either by the sign -{- or
— , it is necessary to know how to proceed with quantities of this
*
kind. Now, as the square of a simple quantity is the product of
that simple quantity, muhiplied by itself, it follows (62), that what^
ever he its sign^ its square is positive ; thus the square of -{- 5 a* 6'
and the square of — 5 a^ 6^, are each + 25 a* 6^
Hence we may conclude, that if a simple quantity he positive^ its
square root may he regarded either as positive or negative ; thus,
V9a4 = d= 3 a* ; for 3 a* and — 3 a* raised to the second power,
are each 9 a*.'
If the proposed simple quantity be hegative, the extraction of its
square ro$t is impossible^ because we have just seen that the square
of every quantity, positive or negative, is positive. Thus, v'*- 9^
y/ — 4 £j2, y^-^5i are algebraical symbols representing operations
which are impossible to be performed. They are known by the
name of imaginary expressions^ and they are symbols of absurdity,
which are often met with in the resolution of problems of the second
degree.
We may always, by extending the rules, give to these symbols
the same simplifications as to irrational expressions upon which the
necessary operations can be performed. ^ Thus v^Tg is reduced to
Also, V— 4a3 = vJfl^ • V— 1 = 2 a \/^T
y— 8 a2b = V4aax— 26 = 2 a vll26 = 2 a v25 \/^^.
86. We now proceed to make known a law of formation for the
square of any polynomial, from which we can deduce a rule ibr the
extraction of the square root.
We have already seen (19) that the square of a binomial a-^-bf
is equal to a* -^ 2 ah + h*.' Let it be required to form the
sqyare of a trinomial a -{- b + c. And for tt^e present de^igpaia
a -f- 5 by the single letter s, it becomes
(a+6 + c)»=(# + c)*«#* + 2tc + c*. .
Now we hare
«s> = (a + 6)* = a» + 2 a 5 + 6*,
Stc = 2(a + i)c=:2ac + 26c.
Therefore,
(a + 6 + c)* = a« 4- 2a J +J»4-2ac +2ftc + c*.
That is, the square of a trinomial is composed of the sum of the
squares of the three terms ^ and of twice the product of the. three
terms multiplied together^ two and two*
This law of composition is true for any polynomial. For let us
suppose it to be true for a polyaornial containing a certain number
of terras, it is true for a polynomial which has one term more.
For this purpose leta + 6 + c + rf+.... + i + A;bea poly-
nomial of m -f- 1 terms, and designate the sum of the m first terms
by s ; then js -^ h\^ the given polynomial, and we have
(# 4- A;)2 = ** + 25A4-**;
or by substituting for s its value,
(44.4)«=(a+6+c+d+....+i)'+2(rt+i4^+d4-.,,.+*)A+*».
Now, the first part of this expression is composed, by hypothesis,
oftha squares of all th» terms of the first polynomial and twice the
product of these terms multiplied two and two ; the second part
contains twice the product of each of the terms of the first midtiplied
by the new term k, and the third part is the square of that term^
Therefore the law of composition, enunciated above, is true for the
polynomial of m + 1 terms, if it be true for that of m terras. But
we have seen that it is true for the trinomial ; therefore it is also
true for the polynomial of four terms. Being true for that of four,
it is also true for that of five terns, and so oo. Therefore, it is
general.
We may also enunciate this law in the iblbwing maniier* TT^
^square of a polynomial contains the square of its first term plus
twice the product of its first term multiplied by the second, plus the
' square of the second ; plus tvnce the product of each of the first
two terms multiplied by the third, plus the square of the third ; and
'.also iwiee the product of each of the fint three terms multiplied by
the fourth, plus the square of the fourth ; and so on. This enuti-
'^tion, which is evidently comprehentled in the first, wffl lead ti$
more easily to the rule for the extraction of the equate. rdot of a
polynomial .'- ' -^
Problems and E^juaihnM of ih^ Second Degree. IM
Actordifig to this la^ we shaft find
(5 fl3 — 4 a by = 25 a« — 40 a*V + 1 6 w'6*, '
{^^—2ab+4by=i9a*— 1 2a%+4a^b^+24a*b'—\ 6alfl+ 1 6M,
:^9a^—l2a^b + 2S(fb^—l6ab^+l6b\
(5a95 _ 4a6c + 6ic» — 3a*c)* = 25a*6* — iOa^b^c + IGa^b'^
— 48ab^(? + 366*c*— 30a*6c + 24a%c' — BGa'ic^ +:9a*c<
■ We wiirnow proceed to the extraction of the square root.
87* Let tis designate by JV the polynomial whose root is lo hf
jfbund, and R the root, wijich we suppose determined , let these
|wo polynomials be arranged with reference to the powers of ooe
of the letters which they contain, as of a, for example.
The first two terms of JV, thus arranged, will immediately give
the first and second term of R ; for it evidently follows from th?
law of the formation of the square (86), (1.) That the square 43f
the first term. ofK contains the highest exponent of the letter
a, which enters into the square of R. (2.) Thaf ttaice the
product of the first term of ^ by the second^ contains also an expo^
nent higher than that of the succeeding terms. Thus the two parts
which we have just mentioned, not being capable of reduction with
any other terms, are necessiarily the two terms of JV which have
the high^t power of a, and the highest but one. Hence it follow^;,
that if JV be a perfect square. ( I .) Its first term must be a perfect
square^ and its root^ extracted by the process of article 83, is the
first term o/*R. {2.) Its second term must be divisible by twice the
first term ofR; and by performing this division, the quotient wUl
be the second term ofR.
To obtain the following terms, form the square of the binomial
already found, and subtract it from JV; the remainder, which we
call JSTj contains still twice the product of the first term of R by
the third, plus the product of the second by the third, together wilb
a series of other parts. But double the product of the first term ef
R by t/ie third must have a greater exponent of a than any of the
following terms, and therefore cannot be reduced vviih any of these
tei*m$« Then this double product must be the first term cf-N^'; and
the first term of this remainder must be divisible by tmce the first
term ofK; and if we perform this division, the quotient is the third
term ofR.
To obtain the next term, we must form ivnce the product of the
first and second terms by the thirds plus the square of the thirds and
I !• EkmnU of 4^«6ra.
tvhiract their sum from the remainder N' ; this will give « new
remainder N"^ which contains twice the product of the iSrst term
of R by the fourth, plus a series of other terms. But we may
prove, as before, that the first term of N'*' is twice the product of
the first term ofR multiplied by the fourth ; therefore the quotient
of the frst^ term of N^' divided by twice the first term of K is the
fourth term cfR; and so on.
N. B. It is indispensable, after having obtained the two first
terms of the root, to subtract the square of this binomial from the
polynomial JV*; for, commonly, the square of the second term of
jR contains the same power of a as twice the product of the first
term by the third, and therefore these two terms may be reduced
Co one. It is not, therefore, till after this square has been sub-
tracted, that we may be sure that the first term of the remainder h
equal to twice the product of the first term of R by the third.
The same remark is applicable to the other terms.
It is left for the learner to deduce from the preceding reasonings
the general process for the extraction of the square root of a poly-t
noniial. It is only necessary to bring together all the portions in
italics.
Wfe now proceed to apply this rule to a particular example. Let
it bcf required to extract the square root of the polynomial,
49a^b^ — 24ab^ + 25a* — 30a^b + I6b\
2/Sa* — 20a% + 49a%^ — 2Aab^ + J 66* ) 6a« — 2ab + 46^
— 25a* + 30Q^i — 9a%^ ^Jo^
Isl rem. 40aV — 24aP + 1 66^
— 40a»5«+24ai3— 166*
— — ^— ^^^- -- - ^
3d rem.
After having arranged the expression with reference to the pow-
ers of a, we extract the square root of 25 a\ which gives 6 a', and
1!?rite this on the right hand of the given polynomial ; then divide
the second term — 30 a^ 4^ by 10 a*, or twice 6 a* (which is writ-
ten underneath 6 a*) ; the quotient is — 3 a 6, which is placed at
the right of 5 a*. The two first terms of the root are, therefore,
6 o" — 3 a i. The square of this binomial is
25 a* — SOa^fi + 9aV,
which, subtracted from the given polynomial, gives a remaiipder
whose first terms is 40 a' b*. Dividing this first term by ipa%.Of
Problems and Equaiiont of ike Second Degree. Ill
twice 5 a^ we obtain the- quotient + 4ft% this is the third term of
the rooti and is written on the right of the two first terms*' Form
twice the product of .
6 a' — 3a 6 by 4 6%
and the square of 4 ft*. Their sum is
40 a* b* — 24 ah^ + 16 i*,
which, when subtracted from the first remainder, leaves nothing*
Tlierefore, 5a* — Sab + 4 6* is the root sought.
Beginners may apply this rule to the squares developed in
article 86.
88. If the given polynomial contain several terms which have
the same power of the principal letter, it must be arranged accord-
ing to the directions given in division (29), and the above process
must be applied by considering, as one part, the algebraic sum of
the terms which have the same power of the principal letter, and
substituting in the enunciation of the process, instead of the words
first term of the polynomial, ^rs^ term of the remainder, first term
of the root, the expression ^rs^porfion of the polynomial or portion
affected with the highest power, first portion of the remainder, ^r5^
portion of the root. Examples of this nature very rarely p%sent
themselves.
89. We will terminate this subject by the following remarks.
(1.) A binomial can never be a perfect square, because the
square of the most simple polynomial, that is, of a binomial, con-
tains three distinct parts, which do not admit of any reduction.
Thus the expression a* -|~ ^' i^ "^^ ^ perfect square ; it wants the
term d= 2 a ft to make it the square of a db i.
(2.) In order that a trinomial may be a perfect square, it must,
when arranged, have its extreme terms squares, and the middle
term > equal to twice the product of the square roots of the two
others. The square root of such a tnnomial may be obtained im-
mediately ; extract the square roots of the extreme terms, and give
to these roots the same or contrary s^igns, according as the middle
term is positive or negative. See then if twice the product of these
roots will give the middle term of the trinomial. Thus
9a^ — 4SaH^ + 64a«6*
has for its square root
V955 — ^64 a° M, that is, 3 o' — 8 ai*,
113 JE^enenlff 9^ J%efira%
4 o? -j. 13 (I fr — 9 J* is not a perfect square ; for although 4 o*
and 9 h^ are the squares of 2 a and 3 6, and 12ai=:2«X6i,
yet — 9 6^ is not a perfect square.
(3.) Whenever, in the series of operations which the general
process contains, the first term of one of the remainders is not
exactly divisible by twice the first term of the root, we may con-
elude that the given polynomial is not a perfect square* This is
an evident consequence of the reasoning by which we- arrived at
fbe process.
(4.) The simplifications of article 84 may often be applied to the
sqi^aro WKDts of polynomials which are not periect squares.
' For example, let there be the expression
,"'''■ V«^6j+4a2 62+4a63 ;
the quantity under the radical siga is not a perfect square, hut it
may be. put under the form a6(a^ + 4a6 + 4 5*). Now the
idctos within the parenthesb is evidently the square of a + 2^fr;
whence we conclude (84)
Va3&+4a?&9^4a69 = (a + 26) y/iSJb^
>
Calculus of Radicals of the Second Degree.
90b Since the extraction of the square root gives rise to new
atgjebraical expressions, such as \/a, 3 x^by 7 \/ 2, which are called
irra^ionai. quantities^ or radical quaniiiies of the second degree, it
iniiii^^fisary to lay dowa the rules for performing on these expresh
sions the four fundamental operations.
^ Two radicals of the second degree are said to be simUar^ when
the quantity under the radical sign is the same in both. Thus,
3 oi V^ and 5 c s/b^ are similar radicals ; so also are 9 \/2 and 7 >/2.
. Addition and Subtraction. To add or subtract two similar radr
Ifi^ls, a4^d or subtract the two coefficients, and apply the commofi
inadicol sign to their sum or difference. Thus,
Sav/i + 5c\/b =: (3 a + 5c)\/b;
Sax/b — 5cx/b :=: (3 a — 5 c) \/b.
In like manner,
7v5a + 3 V5a= 10^2^5 7 ^2^ — 3 v2a = 4 v^,
Problems and Equations of the Second Degree. 1 13
Two radicals, which do not at first sight appear similar, may
become so by the simplifications of article 84.
For example,
2 v'46 — 3 V^ = 6\/5 — S\/5 « 3^/^.
If the radicals are not similar, the addition or subtraction can
only be indicated. Thus in order to add S\/b to 5 x/a, we write
simply 5 \/a -}- 3 ^/b.
Multiplication,- To multiply two radicals by each other, multi"
ply together the two quantities under the radical sign, and place
the radical sign over their product. Thus,
i/a X V^ == V^b.
This is the converse of the principle of article 84.
If there are coefficients, multiply them together, and write their
product before the radical sign*
For example,
3 V5i6 X 4 V20S = 12 v^IoOflSl = l^Oax/b,
2 a \/bc X 3 a \/bc = 6 a" \/W^ = 6 a^6 c,
2av^4r63 X —2ax/W+^=z — 6a^{a^ + b^).
liivision. To divide one radical by another, divide the two
quantities which are under the radical sign by one another^ and
place the quotient under the radical sign. Thus,
-v/a aa
;75
a
For the square of each of these expressions is t, therefore they are
equal. If there are coefficients, we write their quotient as the
coefficient of the radical*
For example,
5a\/b _ 6^ \b
26v<^ "" 2b ^7^
I2ac\/6bc
= 3«^=3«^/3c.
4c\/2b
91. There are two transformations of frequent use in finding the
numerical values of radicals.
The first consists in placing the coefficient under the radical
sign. For example, let there be the expression 3 a \^Sbi
Bour. Alg. 15
Ii4 ElmenU i^4]^^a.
we see ^t this is reduced to
V9o» X \/5l, or V9a8.5& == \/45an,
by applying the rule for multiplication to the two radicals. Thus
the square of the coefficient may be placed under the radical sign.
The principal use of this transformation is the following. Suppose
it were required to find, within a unit, the value of 6 y 13. Since
13 is not a perfect square, only an approximate value of this expres-
sion can be obtained. This root is equal to 3 plus a certain fraction »
but by multiplying this by 6, we have 18 plus the product of this
fraction by 6 ; so that the total result may give a whole number
greater than 18. The only method of determining this whole paft
exactly, is by putting 6^13 under the form
v^>ri3 = \/wy<ri^ = V468.
The square root of 468 is 21 plus a fraction ; therefore, 6 \/13 is
21 plus a fraction.
In the same manner, 12 y7 = 81 plus a fraction.
The object of the second transformation is to make the denomi-
nators qf sucji expressions ^s
j' + vg' p—W
rational, a and p being any entire numbers whatever, and q any
number which is not a perfect square. Expressions of this kind
often occur in the resolution of problems of the second degree. This
purpose is effected by multiplying both terms of the fraction by
P — Vffj if the denominator is |? -j- \/q, and by p 4" V?* if ^^^
denominator is p — yj. For, since the sum of two quantities
multiplied by their difference is equal to the difference of their
squares, we have, by this multiplication,
a_ a{p — \/q __ ap — Oj^q
p + v/j "" (p + y?) (p — V?) """ p^ — q '
«___ _ a{p + \/q) _ ap + as/q
p—vq {p—yq){p + vq) f—q '
expressions which have rations^l denominators*
To show the utility of this transformation^; suppose it required to
find the approxiniate value of the expression
7
s\.
Problems and Equations of the Second Degree. 1 1 5
This id reduced to ^
7 (3 + v/5) 21+7V/5
9 — 6 ' °' 4 •
Now 7 v^5 is the same as
%
V7* X 5 or V245,
the value of which is 15 to within 1. Therefore,
7 _ 21 + 15 + a fraction _ 36 _
3 — ^5 "" 4 — 4 — y
to within a fraction indicated by i, so that 9 is the value within |.
If a more exact value of this expression is required, it is sufficient
to calculate v^245 to the required degree of approximation, to add
21 to the root obtained, and then divide the sum by 4.
Required the value of the expression
7^/5
v'll+\/3
to within 0,01, We have,
7^/5 _ 7^/5(^11 — v3) _ 7 V55 — 7 ylS
V11+V3"" 11—3 "" 8 '
DOW
7 v65 = ^55x49 = v2695 =» 51,91 to within 0,01.
7 v^l6 = V15 X 49 = v/735 =27,11....;
SO tbat
7^5 51,91—27,1 1 _ 24,80 _ ^
Vll +V3T 8 """ 8 ' "^ ' '
the result required i; '3,1 0» which is exact to within jl^, as may
easily be seen.
By a process analogous to the preceding, we shall find
r-y^^^-s - 3,1 59 to within 0,001 .
5v/12 — 6^6
N. B. We might have calculated the value of this kind of
expressions by finding an approximate value of each of the radicals
which enter into the numerator as well as into the denominator.
But as this would not give an exact value of the denominator, no
very precise idea could be formed of the degree of approximation
which would be obtained ; while, by the method just described,
the denominator is made rational, and we know upon what to de-
pend for* the required degree of approximation.
116 EhmenU ofAlgibra.
V.
Having established the principles on which the extraction of the
square roots of numbers and algebraical quantities depend, we can
now proceed to the solution of problems of thB second degree.
. Problems and Equations of the Second Degree.
92. We distinguish two kinds of equations of the second degree ;
equations of two termsy or incomplete^ and equations of three ierms^
or complete.
The first are those which contain only terms affected with the
square of the unknown quantity and terms altogether known. Such
are the equations
•g-ar — ^ 'T Tj^ — i4 — ar -f* ^^y ,
they are called equations of two terms, because, by means of the
general transformations of articles 43 and 44, they may always be
reduced to the form aar^ = b. For let us consider the last of
these equations and the most complicated of the three; we havCi in
the first place, by freeing it from denominators,
Sar^ — 72 + lOa^ = 7 — 24a?» + 299,
or by transposing and reducing,
42a?» = 378.
Equations of three terms, or complete equations, are those which,
besides the square of the unknown quantity, contain the first power
of this quantity. Such are the equations •
6j:a_7ic = 34,
ja:9_|^+ 3 _,8 — |x — a7»+ V/;
which can always be reduced to the form aaP -{- bx :=2 c hy
means of the transformations already mentioned.
Equation of Two Terms. The solution of the equation aa^=^b
presents no difficulty. From it we deduce, in the first place,
a^ ziz "J whence x = v-.
^ a ^ a
. If - be some number, whole or fractional, its square root may be
cz
obtainqd, either exactly or by approximation. If - be an algebraic
quantity, we apply to it the principles established for algebraic
quantities.
Problems and Equaiiant of the Second Degree. 117
Nevertheless we observe, that as the squares ot + m and — - m
ar« equ^ly + m^y so, in like manner,
9
(WD'
and
fftV
(- -Jiy
give equally for a result -. So that the equation is in reality sus-
ceptible of two solutions,
For if we substitute each of these values in the equation a a^ := &,
it becomes
a X C+ ^^ =6, or a X - = i, or 6 = i,
and
a X ( — .^-) = 6, or a X - = i, or 6 = 4.
For the first example let there be the equation
We find, by transposition,
a^ =3 16, whence a? = db -v/l^ = ± 4.
For a second example let us take the equation
ioja — 3 + /y r^ = tV — oj^ + Vr*-
We have already seen (92) that this equation may be reduced to
42 «^ = 378, whence a^ = VV = 9, and a? = zfc 3.
Again, let there be the equation 3 o;^ = 5, we deduce from it
As 15 is not a perfect square, we can only determine the values
of X by approximation.
93. Complete Equation of the Second Degree. To resolve the
equation ax^ -^ b x = c^ we begin by dividing the two members
by the coefficient of a^ ; it becomes
^"'"a^'^S' ora» + pa? = j,
by putting, for the sake of greater simplicity,
5 = pand '- = ?.'
a '^ a ^
\
Thlsl beiitg preiloi&(ted9 Mft remark, that }f vft can alter the fkst
member so as to make it the sqnttre 6f a bhioarial, a simple exti^actiOD
of the square root will reduce the equation to an equation of the
first degree. Now by compafing the firift member with the square
of the binomial X'\' a^ov with a^ + 2 a a? + «S we see that a?* -j- P *
is composed of the square of the first term ; it follows that x plus
twice the product of this firsf term or by a second term which is
then ^ (for we have px :=.2 X 5 a?) ; whence, if to a?^ + jj a? we
add the square of ^ or —-, the first member of the equation be-
c6iBes: the square of £ 4~ »• ^"^ >° order that we may not destroy
P^
the equality of the two members, we must likewise add ^ to the
second member.
By this transformation it becomes
whence, by extracting the square root,
(The doubk sign d= is put here, because the sqeare of
+ Jj + 2 and — J^ + g are each| + q.)
Deducing ibe value of a? from tlie last equation, we obtain
Whence results this genferal rule for the solution of a complete equa-
tion of tlie secoad^ diegree. After having reduced the equations to
the form x* -f- P x = q ; add to the two members the square of
half the coefficient of x in the second ierm^ extract the square root
of both members, and place the double sign =h before the square
root of the second member, and, lastly, deduce the values of x from
the equation thus obtained.
The double value of x, which we have obtained by this method,
may be thus expressed in ordinary language ; half the coefficient of
X taken with the contrary sign plus or minus the square root of the
known term, increased by the square of half the toeffiaent ofx.
Problems an4 EfputHons ^ the Skcond Degree. |19
94. As a first example, le^ there be the ^uatiqn
By freeing it from denonvnators it hecomeff ^
10 a* — 9« + 9 = 96 — 8a?— 12a^ + 273,
or by transposing and reducing
22 a;« + 2 a? = 860 ;
dividing both menabers by 22,
Now by adding to each member (7V)^ ^^^ equation becomes
«" + A^ + u\r = vv + (A)».
Whence, by extracting the square root,
Then
4-
2
a result which is conformable to the expression, given above for the
double value of a?.
It remains to perform the numerical calculationsu First, we must
reduce ^y -j- (^j)^ to a common denominator.
Now,
aeo / 1 Y _ 360 X 22 + 1 _ 7921
22 + V22y •" (22)2 "" (22)3-
By extracting the square root of 7921 we find exactly 89 ; so that
J
Therefore a? = — ^V =*= If-
By separating the two valuer it becomes
« U T" ¥^ — t i[j — %
* — ITT Tt — 'iH — TT»
So that of the two values of x wfaic^ will satisfy the proposed equa-
tion one is a positive whole number, ^e other a negative fractional
one. '
For a second example, let there be the equation
6aj8 — 370? = — 5T,
which is reduced to
«• — y « = ^^y .
1
120 £S*m*nU of AlgAra.
If we add to both members {\jf, it becomes
^- V^c + (*})' = - V + (H)»J
whence, by extracting the square root,
and lasttyi
J
In order to reduce (f |)^ — y to a whole number, we see that
(12)^ = 12 X 12 = 6 X 24 ; so that it is sufficient to muhiply
67 by 24, then 37 by itself, and then divide the difference between
the two products by (12)^.
Now 37 X 37 == !369, 67 X 24 = 1368,
then
/37\2_67 _ J[_^
V12/ 6 ""122'
an expression, the square root of which is j^^.
Therefore « = f J =i= Ta>
QP ) * — Ty T^ TJ — T^ — F >
\X yy y^ yy O,
This example is remarkable, inasmuch as each of the values of
X are positive, and are direct answers to the question, of which the
proposed equation is the algebraical translation.
Let there be the literal equation
4 a^ — 2 a?2 + 2 a 0? = 18 a6 — 18 62.
By transposing and changing its signs, and dividing by 2, it becomes
ar^ — ao? = 2a2_9aJ 4. 9J2;
by completing the square,
extracting the root.
Now ^*_9o6 + 9i«
has evidently for its root . .
V2
X
~3i)j
Problem and Equations of the Second Degree. 121
.hen , = |±(»^°_06)
These values will both be positive if
2a>3ft and 36>a,
that is, if 5 > I and < ^.
The following equations are proposed as an exercise for the stu-
dent; a^ — 7 0?+ 10 = 0;
values a? = 2, a? = 6 ;
values { ^ ^ _ ll\l } to within 00, 1 ;
a^ + b^-2bx + c^='!^,
n
gives X = p^^-a (*»=*= v/aamS + ftSmS — asy).
95. The equation aa^ + bxz=:c may be resolved without divid-
ing by the coefficient of x^, but the transformations are more com-
plicated.
The term a a^ may be put under the form (x ^a)% and the term
b X under the form
whence aa^-i-bx 2iTe the two first terms of the square of
277^^ ^r J- 1 ihe first mem-
ber becomes a perfect square.
Performing this transformation the equation becomes
a aj* + J a? 4- -— = c -I- -p- t
extracting the root,
transposing,
X \/a =z —
Bour. Alg. 16
%s/a
j:
\
\U2 Eiemtnts cfAlgAra.
Dividing both members by \/a, and observing
6_
h h
(1) that
divided hy \/a
divided by v^a
We obtain
= - lc + ii= k+^ (90.)
V/a^^T-4^ ^a^4£^ ^ ^
or ^ = -^^V4a c+58,
the same result which was obtained more easily by putting the
equation under the form
a^A — a? =1 -.
' a a
96. We proceed to apply the preceding principles to the solution
of some problems.
Problem 1. To find a number suchj that twice its square
added to three times the number^ may be 65.
Let X be the unknown number. The equation of the problem is
2a? + 3a?=:66.
Whence
^_ S . |65 , 9 3^23
*"-~4=*= J"5■ + T6--~4=*=T•
Therefo^e
and « = — I — y = _ Xy3.
The first value is the one which answers the question in the
sense in which it was proposed. For
2 X (6)* + 3 X 5 = 50 + 15 = 65.
To interpret the second, observe, that if we change x into -— Xj
in the equation
i«i*-f 9a; = 65,
Problems and JEquaHans iffih^ Second Degr$: 1S3
the term 3 m ooly changies its sign, siace ( — «)^ = 9^. laytead of
0? = — |db V
we have a? = | dt »/ ;
that is, J? = y , or a? z= — 5,
which values differ from the former only in their signs. We may
therefore say that the negative solution — y , considered independ-
ently of its sign, is the answer to the following question. To find
a number such that twice its square diminished by three times the
number m 65 ? For we have
Problem 2. A man bought a certain number of ells of doth
for 240 francs. ]ff\ with the same sum^ he had bought 3 ells lessf
it would have cost him 4 francs an ell more. How many ells did
he buy ?
Let 0? be the number of ells ; then — is the price of each. If
for 240 francs he had bought x — 3 ells, each ell would have cost
^ . But according to the enunciation the first price is greater
than the second by 4 ; we have then the equation
240 _2^ _ . .
X— 3 X
whence we deduce
ar^ — Sx=i 180,
*=^j
. ^^ 3 ± 27
+ 180 = — ^— ;
2
then 0? = 16, and x = — 12.
The first value 0? =: 15 satisGes the enunciation; since 15 ells
for 240 francs gives V/ or 16 francs, for the price of 1 ell ; and
12 ells for 240 francs gives for the price of an ell W or 20 francs,
the second of which is greater th^n the first by 4.
As to the second solution, we can form' a new enunciation to
which it will correspond. For by going back to the original equa-
tion, and changing x into -^ ^, it becomes
240 240 ^ 240 240 .
z = 4, or — rs = 4,
«^x— 3 — »"" ' X x + S
which may be considered as the algebraical translation of the follow-
ing problem.
1
134 EkmenU of Algebra.
A person buys a certain number of eUs of cloth for 240 francs.
Jfhe had paid the same sum for 3 dk more, each ell would have cost
4 francs less. Required the number of ells which he bought.
By resolving the equation of this^ new problem we shall evidently
find X = 12, and a? = — 15, for the equation becomes
a?3 + 3.'c = 180,
instead of
a? — 3 a? = 180.
Problem 3. A merchant discounts two bills, one of 8776
francs, payable in nine months, the other of 7488 francs, payable
in eight months ; he pays for the first 1200 francs more than for
the second. What is the rate of interest at which he discounts it ?
To make the calculation more simple, let x be the interest of
100 francs for a month, and 12 x that for a year ; 9 x and 8 x are
the interest of it for nine and eight months; therefore 100 + 9 a?,
and 100 + 8 a?, are what 100 francs amount to at the end of nine
and eight months. So that in order to determine the actual value
of the two notes of 8776 francs, 7488 francs, we must make the
proportions
100 + 9a? : 100 : : 8776
100 + 8a?: 100 : : 7488
877600
100 + 9 x'
748800
lOO + Sx'
and the fourth terms express what the merchant has paid for each
note.
By the enunciation, we have the equation
877600 748800 _
100 + 9 a; 100 + 8 z "" ^^^" '
which, being freed from denominators and reduced, becomes
216 oj^ + 4396 X = 2200,
whence ^
^ = _^± [2200 /2198Y,
216 >|216^V216/
Reducing the two terms under the radical to the same denominator,
_ — 2198d=v/5306 403
« 216 '
or multiplying by 12
19 ^ — —2198 ± s/^mi
18
Problems and Equations ofih^ Degrte, 125
To obtain the value of 12 a: to within b,Ol it is sufficient to extract
the square root of 6306404 to within 0,1, since it must afterwards
be divided by 18. This root is 2303,5 ; then
-^^_— 2198±2m5 105,5 ^ . .
and
12 x = :^*^ = _ 250,08.
The positive value 5,86 is the rate of interest required.
As to the negative solution it can only be considered as united
with the first by the same equation of the second degree. For if
we go back to the enunciation, and change x into — a?, we shall
find it difficult to translate the new equation into an enunciation
analogous to that of the proposed problem.
Problem 4. A man buys a horse^ which he sells some time after
for 24 guineas. By this bargain he loses as many guineas in the
hundred as his horse cost him at first. Required the price of the
horse ?
Let X be the number of guineas which the horse cost, a? — 24 is a
first expression for the loss.
But since he loses as many guineas in the hundred as there are
units in a?, on every guinea he loses ^^, and on x guineas he loses
-rgg. We have then the equation,
{QQ = a? — 24,
whence a^ — 1 00 a? = — 2400,
a? = 50 db V2500 — 2400 = 50 d= 10.
Then x = 60, and x = 40.
Both values equally satisfy the question. For suppose the price
of the horse to be 60 ; since 24 is what he sells it for, his loss is
36. But he loses 60 per cent, on the sum which the horse costi
60
that is, 1^ X 60, which is also 36.
Again, let 40 be the price. Then the loss is 16, and moreover
he loses 40 per cent, on 40 which is 40 X {Kq> which equals 16,
UQ EkmnU of4fgehrm.
General Discussion of the Equation of the Second Degree.
Hitherto we have only golved problems of the second degree, in
which the given quantities were expressed by particular numbers.
But in order to solve general problems, and interpret all the results
which can be obtained, by assigning particular values to the given
quantities, we must resume the most general equation of the second
degree, and examine the circumstances which arise out of every
hypothesis which can be made as to the value of the coefficients.
This is the object of the discussion of equations of the second degree.
97. Before entering on the discussion, we shall point out another
method of solving equations of the second degree, which will lead
to some important properties which belong to the values of the
unknown quantity.
We have already seen (93) that every equation of the second
degree can be reduced to the form
c^+px=Lq (1),
p and q being either numerical or algebraical quantities, whole or
fractional, positive or negative.
This being premised, if, in order to make the first member a
perfect square, we add ~ to both meinbers, the equation becon^es
or
(^+1)'=.+^.
p«
Whatever maj be the vsdue of 9 -{- ^ , we may always designate
its square root by m ; then the equation becomes
(a; + ly = < or (x + j)' — m» = 0.
The first member of tlua equation being the difference of two squares,
may (19) be put under the form
(aj + ?_m) (* + ! + «) = (2), .
the first member of which is the product of two factors, and the
second is 0. Now the product may be made equal to 0, and the
equation (2) satisfied, either by making
a?+^ — m = 0, whence « = '— % + m.
Discussion of the SquaHon of the SecQnd I)egrtt. 127
or by making a? + ^ 4" •'^ =^ ^>
P
whence a? = — 5 — m;
or substituting for m its value
■1
'=-i+]i+i
'=-3-4? +1
It is moreover evident that the first member of equation (2) can-
not be made equal to 0, except by giving x a value, which destroys
one of its factors.
Then, since equation (2) is derived from (1), and the reversei
every equation of the second degree admits of two values of the
unknown quantity^ and cannot have more.
This method of solution, which, perhaps, is a little longer than
the first, has the advantage of showing more clearly that there are
two values of the unknown quantity and only two.
98. These values have some very remarkable properties.
(1) since the equation
«* + /> a? = g, or 0^ -{-p X — 2 = 0,
is reduced, by a series of transformations, to the form
where
it follows that the first member
» !x^'\-px — q
of every equation of the second degree^ whose second member is 0,
is composed of two binomial factors of the first degree in terms ofxj
X being a term common to both, and their second terms being the
two values ofx, taken with contrary signs.
From this property, by which the equation itself may be found^
when the values of the unknown quantity are given, these values
have been called the roots of tiie equation.
Let there be the equation ^
ai2-f3a? — 28 = 0,
which being resolved gives
X = 4, and « = — 7,
128 ElemerUs of Algebra.
the first member results from the product of
(x_4) {x+^)^,
for this product is
a^ — 4x + 7x — 2S=zar^ + Sx — 2S.
(2.) If we designate the two roots by a/ and a?^, according to the
preceding property,
ar^ +p X — g = («r — a/) {x — xf'\
or, by performing the operations,
^ + JP -2? + 9 = a?2 — (x' + ir'O a? + ^ a?"'-
By comparing the analogous terras in the two members we find
xf '\-x'^ =. — p, xf xf^ z=. — gr.
Therefore, (1) the algebraic sum of the two roots is equal to the
coefficient of the second term, taken toith the contrary sign. (2) the
product of the two roots is equal to the last term, of the equation or
the known term, when transposed to the first member of the equation.
These two properties may be verified from the general expres-
sion of the two roots.
If we add
we obtain
Now by mulliplying the same equations, we obtain
N. B. The preceding properties suppose that the equation has
been reduced to the form
a^ -\-p X — ^ = 0;
that is, (1), that it has been divided by the coefficient of aj^j (2),
that all the terms have been transposed and arranged in the first
member.
99. Discussion. Let us take the general equation
x^ -^-px =z q;
which, being resolved, gives
4
s+?-
Discussion of Equmons of the Sebond Degre: 199
- In opder that the falufe of this expression, which contains a radi-
cal, may be found either exactly or by approximation, the quantity
under the radical sign, that is, ? + t;> must be positive (8^). Now
Y being necessarily positive, whatever be the sign of p, the sign of
9 -j-'T d^pci^ds principally on that of j, or on that of the known
term.
Ill the first place let q be positive, in which case the equation is
(distinguishing the signs of the coefBcienls), whence
4
Now it is manifest that the two values of x are real and can be
.determined exactly, if gf -|- j be a perfect square | or^ if.o|iiej:wis€t
to any degree of approximation we please. -
The first of these two values is always positive, and i;s a direct
answer to the equation, or to the problem of which the equation ^is
the algebraical translation ; for the radical
J
P
being numerically greater than ^^y the expression
has necessarily the same sign as the radical.
The other value is, for the same reason, negative, because it
must have the same sign as that of the radical. Considered in-
dependently of its sign, it answers no longer to the equation which
has been established, but to one arising from the substitution of
— 0? for + a? in that equation ; that is, to
x^ =Fp 0? = J.
Fpr the solution of this last equation gives
'^i- - ■
inUtehdifiWs from the former only in its sign*
Bout. JiJg. - * • '• . "17 "
2
lao Elments of 4ig€kra.
A
It is also remarkable that the same equatbo conneots two. ques-
tions w]siQ$Q eauQoiati0n» difier only in the nwaning of ceErtataic^iH
ditions. (See die two problera^s of article 96.)
100. Let q be negative, in which case the equation is of the forrii
and the values of a? are
1-2
In order that the extraction of the root may be performedj k is
necessary that 9 <Ct"' '^^^^ condition being satisfied, the two
▼alues will be real.
Moreover, since
sf
— ?
J8 nnmerrcalfy leto fhtin-^, the tWo values are both negatrre,' whea'j»
is positive, or when the et^uation is of the form
and both positive when pis negative, or when the equation is of the
form ar^ — p x =z — q.
The same consequences may bjg (^^duced from the two properties
demonstrated above, that in overy equation of the second degree^
a^+p*-?=o, . ;
the algebraic sum of th$ two r^ots u< equal to the coefficient of the
second term taken with the tontrai*y ^g^y and that their product
18 equal to the last, or known term^ when, traj(isj^osed. to <^c fir^f
member c^ the equation.
For, let q be positive in the second term, and therefore negatly^
in the first } the prpduct of the two ropts is theijefore. negative^
and they have contrary signs. Also^ thei^ sjurn is posj^jye or neg-
ative, according as p is negative or positive ; that is^ tije ro^t which
is numerically the greater of the two will always have the contrary
sign to the coefficient p.
But if q be negative in the isecond member, attidbbhseqae^ly
positive in the first, the protjiuct of the iwq. roots is positive; they
have therefore the same sign, that is; are both negative if p be posi-
tive, or both positive if jp be negative, aince in .tlie.&rrt case tknir
algebraic sum is negative, and in that second positive* ': .^ .^b.
Discussion of Equations of die Second Degree. ISI
We can also show, a priori, that whenever y is begative in tfie
second member, and /> negative in the first, the problem admits of two
direct solutions, provided there exist between p and q the relation
The equation
aj* — p X 2= — ' },
by changing the sighs, may be put under the form
p X — 0?* 2att g, or 0? (p -^ a?) = y,
virhich is evidently the algebraical translation of the following pr6b«
lem : To divide a number p into two parts, of which the product is
a given number q ; for if x be one of these parts, p — a? is the otheri
and their product is x {p — x).
To prove that this problem admits of two direct solutions, we
remark that the equation is the same whether x be supposed to.
represent the greater or the less of the two parts ; the equation
cannot therefore give one more than the other j it ought therefore
to give them both at the same time.
Also, the two parts sought should be such that their sum may be
equaftoj?, and their product to J. Now these same relations exist be-
tween the roots of the equation
ajSJ — pa? 3= — q, or . a?*-^!?* + y =
and its eoeffici^ts ; hence the parts required are equal to the two
roots of this equation.
In the second place, in order that the solution of this problem m^y^
be possible, we must have f <i^«
For whatever be the two parts sought, we may always designate
the difference of the two {»arts\by d; since tbfir sqm.is p,we have,
by the theorem of article 4, . ,
fpf the ©rea^r part | + g, ,
and for the less g — 5.
The product of these two is ^ — -j, which is necessarily Wss
than ^, unless we suppose the two part^ equal, in which case d=zOj
and the product is reduced to-^.
133. Elementi of Algebra.
h i^ therefore absurd to require thiit the product, which we have
denoted by 5, should be greater than -^, Whence we may con-
elude, thai whenever the known quantity in the second member is
negative^ but numerically greater than the square of half the coeffir
dent of the second term^ the proposed question is imppssible*
Remark. It follows that the greatest product which can be ob-
tained by decomposing a number into (wo parts, and muhiplfinf^'
these parts together^ is the square of half the number. For this
product may be expressed by j- •— j, which is less than ^, but
which becomes equal to it if we suppose that d = 0, or that the two
parts are equal.
Examination of some particular cases.
101. (1.) If, when q is negative, that is, when the equation is of
the form a?+pa?=:; — q
(p being of either sign), we suppose qz=z^, the radical
J
9
4-?
becomes nothing, and the values are both reduced to^r = — 5; in
this case we gay..tbat.the two roots are equal.
' For if we go back to the original equation, and substitute V for
g, it becomes ^
whence
s
a^
+ P^ + j-0, or (^+1) =0.
The first member of this equation ,is the product of two equal fac-
tors. We may say, therefore, that the two roots of the equation
are equal, because both factors, when made equal to 0, give the
same value for x.
(2.) If, in the equation (x? -{- px = j, we suppose j = 0, the
values of a? are reduced to
' ■'>--f>|,orx = 0,
IHicussion of Equatiom <3f the Second Degree. 13S
The equation is then a?^ -[- pa? = 0, or ap(a? +P)|=^5 Xvhicb
may be verified either by making a? = 0, ora?*|-p=0; whence
X = '— p*
(3;)r Ifiin the general equation o?^ -f-jaia? =^5 we suppose j> ==.0,
there results the equation a^ =: q, or a? = rfc \/q ; that is, in this
case, the two values of x are equal, and of contrar}' signs ; real if q
be positive, and imaginary, if it be negative. The equation belongs
to the class of equations of two terms treated of in article 92.
(4.) Let us suppose at the same time p z= 0, and q = 0; the
equation is reduced to x^ == 0, which gives two valves of x, each
equal to nothing,
102. It remains to examine a peculiar case which often occurs
in the solution of equations of the second degree.
For this purpose let us resume the original equation
a x,^ + b X = c, which being resolved gives
— 6 d= s/^'+Tac
X
2a
Suppose that, according to some particular hypothesis made iii
the given quantities of the problem, a = ; the expression for x
becomes
_ — bdob
wbence
•■ ' •■ 26 •• - '
. . x=:^r ora?=;^5 .■
the second value takes the form of an infinite quantity,, and may be
regarded a^ an answer, if the question admit of infinite solutions
(71). ,
The first result jr remains to be interpreted.
The hypothesis a = reduces the original equation tobx =z cy
b
whence a? ss? -f a finite and determinate expression, which must be
> ■
regarded a;5:th|B true yalue;of j^ in this cai^e.
J To leave no doubt oti this subject, let as take the driginar
equation J ' .
\< ''j ''^ I
aa^ + b a?''=± 1c, a|^(J p^t at == -.
Tbeii the equation [becomes
-54--ZZC, ox c^ — hy — a=i:0.
Let tt = ; this last equation becomes c y* — t y = 0, which
gives y = 0, and y = -.
Substitute these values in the eqiialibn a? = - ; we decluce from it
y
u
If we suppose 5 = 0, as well as a =: 0, the value -t itself be*
* •
comes n, or infinite*
For the equation c y^ — J y — o = 0, is reduced, by thlsr hypo-
thesis, to cy^ = 0, the two roots of which are 6adh equal -to 6.
Therefore the corresponding values of x are both infinite.
If, at the same time, a = 0, 6 = 0, arid c = 0, the proposed
equation is altogether indeterm-inate. This is the. only case in
which the equation of the second degree is so.
We proceed to apply the general discussion to a problem con-
taining all the circumstances which are ordinarily met with in the
solution of problems of the second degree.
103. Two luminaries of different degrees of brightness are.^
placed at the points A and B. .fft whai point of the straight line
AB vnil the quantity of light hceived from the two bodies be equal.
'^(W^'take for granted this princifle of natural philosophy, ttat
the brightness of the same light, as seen from two different distances, '
varies in the inverse ratio' of ^tbe squares of the distdnees.)
Let a be the distance AB of the two lights, h the brightness of
the light A at the distance of* a unit, and c that of B at the same
dbiaiM^e; Let O be the required point. Let AC 'Sl t; then
BC=;a—x.
Since the brightness of the light A^^t the distance unity, is 5, b^
tl^pi^Dciple of nati^a) philosO|d3y Stated ab<>ve9 its brigbtDess. at
the distances 3, 3, 4, &c. must be , ,
b b b .
Discussion of Equatiot^ af tk» Sscond Degree. ISfi
and at thq distance a?, i^ must be -5. For similar reasons, the bright-
ness .of £ at the ilistQoce a -*-^ d» i9
{a — x)9 '
but, from the enunciatioo, the first degree of hrlgbtness is the same
as the second ; therefore
x« ■" (a — xf
which, when developed and reduced, is
(6 — c)o^ — 2 a i ^ =: .^ o* 6.
This equation gives
a (6 db \/f7)
a6 . I a968 o^ft
« = r db -- — —-- — ,orir=:
b — c ^(b'-'c)^ ft— c'
»»^-— »^i ■ ■ » I ■ I
The expression may be simplified, since b db \/ffc may be put
under the ibrm
\/h » \/b dz ^b . \/Cf or \/b {\/b dz \/c),
and
6 — c z= (v^6)9 — (yc)8 = (y6 — x/c) . {s/b — x/c).
The root which has the positive sign before the radical gives
_^ a \/b {\/b -{- \/c) a\/ b
In like manner, the second value of x is
_ a\/b (\/b — s/c) _ a\/b
. ^"" (v6+vc)(v6— yc)"" yft + vc' ..
These simplified values may be obtained immediately from.. the
proposed equations. For, from, the equation
extract the square root of both members, and we have
a — X
which, cleared of fractions, becomes.
o Vfi' — x\^b=z zkx \/c, or a^ = • ,. - ^ - .
Z'
W6 ^ ^ EkmiMtof.Mffbm.
(• I
N;. B. The first values were tie most complicated, becapse the
equalion of the second degree was solved by the general method,
which is not so simple as the one which we have just follawed.
We have therefore
/, V a\/b >, r as/c
(l.) . 07= 1 ' . — T— k \a. — a: =: — -7--] --,
r. //. / whence \ .... ^ . ^
(2.) a? = —7 \ 9 a — x =.
'htt b ^ c.
The first value of a?, . ; > *s pQsitive, and less than a, be-
cause , . is a fraction less than unity; this value,. therefore,
4^b + s/c
gives, for the point required, a point C, between A and Bk Also,
this point is nearer to B than A ; for, since 6 > c,
V6 + V6, or 2 V* > Vi + V^^i or ^f, ^ ^^ > §»
therefore
a\/b ^ a
yb -{- \/c 2
This tmist be the case, as the brightness of A has been supposed
greater than that of i?.
The corresponding value of o — a?, namely,
a \/c
is positive, and less than g, which may be easily verified.
Ttie second value of x,
a\/b
\/b— \/c'
is positive, and greater than a ; for
This second value gives a point C^ situated in the prolongation of
AB to the right of the two lights. We may see that, since' the
light spreads its^f equally m all directions,^ the£e_n^st be another
point in the prolongation olAB^ equally illuminated by eacfa light ;
! - - . ^ .V,
DUcusrian of SSquatioHf of4h& Skond Degree. (97
and that this point must be nearer to the wesk^ light Aan lo'the
other. .^ \
We may see why these two values ^r^ coo^ected by the same
^uatjon. If we take the unknown quantity x to represent w2C^
instead otAC^ we have JBC = a? — a; and the eqqation of tbp
problem is
and since {x -^ of and (a — xf are identically the sanie, this is
the same, equation as the one already established, which must
therefore give AC as well hs AC.
The second value of a — a?, namely, '
— a\/c
\/b — vc'
• * •
is negative, and necessarily so^ since x'^ a; but, if the signs of
the equation
V* — %/c
be changed, it becomes
a\/c
•end thisTalue oSx*^ a represents the absolute value ofBC^n '
Leib <:'€•
TlljS' MBST vAiiDE of a?, namely, i
Is'.positive, but leSs than ^, since
The corresponding value of a — jp, naaiely, .. .
a\/c
is positive, and greater than s*
On this hypothesis the .point C is situated betweep A wadJB,
.but nearer to A tbao io J3.
Bwr. Alg. 18
136 iESfemcnlt ^4lgAr<i.
The SXCONO TALUK 6f «|
a\/b a\/b
or —
\/b — \/f^ \/c — V*'
is negative. To explain this, we return to the equation, which,
when — « is substituted for x^ becomes,
b _ c
Since, before this change, a — x represented the distance of
the required point from j5, that distance must now be expressed
by a 4" ^9 ^^^ ^he required point is to the left of A ; as, for ex-
ample, at C"* Since the brightness of the light B is, by hypothesis,
greater than that of Aj the second of the equally illuminated points
must be nearer to A than B.
The corresponding value of a — cr,
a — v^c a\ /c
\/b — vc' V^ — \/b^
18 positive ; which it must be, since, when x is negative, a — - «
reaUy is an arithmetical sum.
Let b ^=: c.
Each of the two first values of x and a — x becomes ^ » so that
the point which i$ equally enlightened by both is the middle of
AB. This also follows from the hypothesis.
a /h
The two oth^ values are reduced to —^9 or become infinite;
that is, the second point, which is equally illuminated by both, is at
a distance greater than any assignable quantity. This result is
perfectly conformable to the hypothesis ; for, if we suppose the
difference b — c, instead of being equal to nothing, to be extremely
small, there is a Second point, but at a very great distance ; which
is indicated by the expression for x,
a \/b
x^b -^ v'<^'
whose denominator is very small when compared with its numera*
tor. If then 6 = c, or \/b — \/c=: 0, the required point can no
longer exist, or is at an infinite distance.
While on this subject, we may observe, that if, when 6 =: c, we
lake the values of x, which are not simplified, viz.
Diicitiiion ofEquationt cfihe See(md Dtgru. 1S9
a(h + \/be) a(b — s/bt)
b — c ' b — c
The first, which is the same as
X = — ; • becomes ■ >
V* — Vc
and the second, which corresponds to
a\/b ,
X = — T-r — 7i becomes -.
\/b + \/c
But j^ is obtained from the common factor \/b -*- \/c, which is in
both terms of this value of x. (See art. 72.)
Both terms of the first value also contain the common factor
^5 -(. ^c\ the suppression of which gives
a^b
X =
^b — ^c'
which becomes
X =
= — ~- when 6 = c.
iLet i = c, and a = 0.
The first values of x and a — x become 0, and the second g.
This last symbol is here really that of an indeterminate quantity ;
for the original equation,
(i — c) a? — 2abx z=i — a* 6,
becomes, on these hypotheses,
0.a? — a? = 5
which may be satisfied by placing any number instead of x. Since
the two lights are of the same brightness, and placed at the same
point, they ought to illuminate every point of the line AB equally.
The solution a; = 0, given by the first value of a?, is one of the
infinite number of solutions of which we have spoken.
het a = 0, and b and c be unequal.
Each of the two values of x and a — x becomes ; which
proves, that in this case, the only point equally illuminated by both
is the point where the two lights are placed.
The equation is then reduced to (6 — c) «* =: 0, and gives the
two equal values « :ts 0, « =: 0.
14P JSMif te ^«%c6fa.
The precedkig discussion is a new instancir of the precision with
which algebra answers to all the circumstances of the enunciation
of a problem.
104. Problem 6. To find two numbers such that the difference
of the products of these numbers by the numbers a and b respectively
shall be equal to a given number s, and the difference of their
squares shall be equal to a given number q.
Solution. Let x and y be the numbers sought j we have evi-
dently the two equations
ax — by =^Sj
a^ — y^ = 5.
Fiom the first, we ol^tain x =:= ^ "*" » a value whicbi substituted
Cv
in the second, gives
(^a^ — i^)f — ibsy=s^ — a^q (I);
then
__bs ± a ^«2 — «(a2 — 62)
y ^^rgs '•
Carrying back this value into the expression for x in terms of y,
^e find
X — ' ■■ ■ ■ ' ■.
a
whence
^^ asdzb^^^ — qda^r^b^) ^
(It is necessary to observe that in these valuer oty and x the two
upper signs correspond, as well as the lower.)
Discussion, We. shall suppose, in all that follows, that a^b^q,
^jj are absolute numbers; if it were otherwise,. certain terms in the
values of x and of y would change their sigos^ and it would be
necessary to make these changes before the discussion*
Let a > b, whence a^ — h^ positive.
First, ID Older that the two values of x and y may be real, we
must have
•; -
q (o« — 5a < a*, whence q < ^^^ .
Duetisnon ofEquaiumi of the Second Degree. 141
Let us suppose this hsi condition fulfilled, and let us detern[iin&
the signs by which the two sets of values are effected.
, * ^ ^-ITJa '
The first set is < , . , u^
The two values of this set are necessarily positive, and conse-
quently form a direct solution of the problenii such as it bad beea
established.
_as — b ^^—5(08 — 62)
The second set is < , ,
8 — a i^a3 — 9(a9 — &9)
The value of x is essentially positive, for from a ]> i we obtaiii
a f > J *, and, a fortiori^
noce the radical quantity is smaller than s.
As to that of y, it may be positive or negative,
la order that it may be positive, we must have
whence raising to the square,
6a«3>o««» — a3g(a» — 6«);
or, adding to the two members, c^qijfi — 6^), and subtracting J*j^,
whence, dividing by c? {c? — 6^),
Thus, in order that the second set may be still a real and direct
solutianf we must have
■Q A
that is, J is comprehended between the two numbers
143 Elements of Algebra.
»a
(Let us remark, that the conditioD 9 }> -^ may be obtained more
easily, by going back to the equation for y.
This equation being
(a8_J9)y2_2J*y=5«— a«j,
we see, that in the hypothesis of a }> i, it is of the form
«^ — p 0? = — y,
if we have
s
s
and we know (100) that the two roots are then at ihe same tim^
positive.)
If we had, on the contrary, q <^ —g* in which case we should
s^
have for a still stronger reason,
the value of y of the second set would be negative ; and this set
(the sign of y being left out of consideration) would be no longer a
solution of the problem, as it has been established, but rather of
that in which the equations would be
ax -{- by =z Sj
and which would differ from the problem only inasmuch as s would
express a sum, instead of a difference.
Thus, in the case of a^bj the problem admits of two real and
direct solutions, while we have
s^ s^
and it admits of one only^ if we have 9 <C -3*
Taking for a, i, ;, any absolute numbers whatever, provided
always that a ^ i, and then choosing for ;, a number compre-
hended between the two limits
we diall be certain of obtaining two direct solutions.
Dtwiifftofi ofEquatioru of the Second Degree. 148
liOt there be, for. exanipley a « 6, i =3 4, t = 15, whence we
deduce
? "" 36 "" 4' ^F^:^ "" 20 "" 4'
^ We may suppose j = 10, for example, and it will become
6X l6d=4V225 — aox 10 90 ±20 11 ,7
.20 20 2 2'
_ 4 X 15±6v'225— aox 10 _ 60 =fc 30 _ 9 ,3
y— 20 ~ 20 ""2 2*
The solutions
11 9t _7 _8
*-5' y-2'
form evidently ttoo direct solutions of the equations
6a? — 4y = 16,
ix^ — f =10.
But if we suppose a = 6, i = 4, « = 1 5, j = 5, it would be
easy to discover that of the two sfets, the first only would give a
direct solution.
Particular cases which relate to the Hypothesis <)/*a ]> b»
Let q = ^ ,^ , whence y (a* — 6') = A
The two sets of values of a? and y are reduced to
as bs
So that on this hypothesis, there is only one solution of the prob-
lem, and it is direct*
Further, let g = -5 ; whence 5* =: o*g, and » = a y'gr.
as^b_s/b^ g _ flS + 6a
the first set becomes < , . _ ** '^
__6s + a^63gr 2a6
« 5 — 6 v^ja g
* — — ^ZTaq ^ Vj>
the second . . ^
_65 — a^«g _
144 MemmtpofAlgtiim*
Atid indeed let us sgppbse /^ t^ a^q in th» leqoatioo in tslms
of y ; it is reduced to
whence we deduce
26s 2ab .
Let lis carry back each of these values into
by 4- s
a ^
there resuhs
jLc^ ihtrt now Jc a <^ b, whence we have a^ — b^ negative*
The expressions of x and y may be put under the form
— as =pb ^ «2 +J, (69 _ flS)
6^ — a^
These values are all real, since the quantity under the radical is
essentially positive.
As to the signs, the first value of x is essentially negative, and it
is the same mtblhe first value of y. So tJtiat 4hese jralues, cdHUer-
ed independently of the sign, answer not to the proposed equationSi
but to the equations
6y-— aaj = *, a:!* — y^ = y,
in the first of which, the order of the difference between the products
ft X and b y is reversed.
The second value of x is necessarily positive ; for from 6 ]> tf ,
we deduce b i/«2-j-:^(63-i<i2) y> a 5, since the radical quami^ is
numerically greater than s.
But the secofid value of f is not always positive. In order that
it may be so, we must have the relation
aV^+4(b*^a^) >h^
whence, raising to the square
aS^«a + a^y(6a — a»)>f^,
or, transposing t?^
Transformation of huqualUvet. 145
and, dmdhig by o' (6* — o"),
By giving to fl, i, s, 5, particular values, such that 6 ]> a
and ? > -3, the problem will be slill susceptible of a direct
solution.
Finally, let ^ =:hy whence we have a* — b^ = 0,
The first set of values becomes, on this hypothesis,
and the second,
_0 _
But if we return to the equation
(a* _ J») y* — 26 *y = ** — a" q,
which, when we make a = 6, is reduced to
— 2 a 5 y = *• — a* J,
we deduce from it
— . «^g — ^^
^'^ 2as '
and the expression for x in terms of y,
X = — ii— ! — gives a? = — ^ — .
a ° 2a5
(We should arrive at the same results by imitating the process
pursued in article 102 ; that is, by making, in the equation in terms
In order that the solution
* "" ~2ii~' * "" 2as '
may be direct, we must have 3 >• -3- «
Q^ Transformations which may be performed upon Inequalities*
106. In the course of the discussion of the two preceding prob-
lems, we have had occasion to make use of several inequalities^
Bour. Mg. 19
146 Elements ofAlgebts.
and we have performed upon them transformations analogous to
those which we perform upon equalities. It is indeed what we
are often obliged to do, when in discussing a problem, we wish to
establish between die given quantities the necessary relations in
order that die problem may be susceptible of a solution direct or at
least real, and to fix by the help of tliese relations, the limits be-
tween which the particular values of certain given quantities ought
to be found, in order that the enunciation may fall in such or such
circumstances. Now, ahhough the principle^ established for equa-
tions, may in general be applicable to inequalities, there ere nev-
ertheless some exceptions of which it is necessary to speak, la
order to guard beginners against errors which they might commit
in making use of the signs of inequality. These exceptions arise
from the introduction of negative expressions, as quantities, into
the calculation.
For the sake of clearness, we will go over the different transfor-
mations which we may have occasion to perform upon inequalities,
taking care to indicate the exceptions of which these transformations
are susceptible.
Transformation by Addition^ and Subtraction. We may,
mihout any exception, add to the two members of any inequality
whatever, or subtract from ihetjfi, the -same quantity ; the inequality
still subsists in the same sense.
Thus, let 8 > 3 5 we have still
8+5>3 + 6, and 8 — 6>3 — 6.
Let there be also — 3 <^ — 2 ; we have still
— 3 + 6< — 2 + 6, and — 3-^6 < — 2 — 6.
This is evident according to what has been said (63).
This principle enables us, as in equations, to transpose certain
terms from one member of the inequality to the other ; let there be,
for example, the inequality a^ + 6^ > 3 6^ — 2 a^ ; there results
aa + 2 a^ > 363 _ Ja, or 3 a» > 26^
We may, without exception, add member to member, two or more
inequalities established in the same sense, and the resulting inequat-
"" ity subsists in the same sense as the proposed inequalities. Thus,
ftom
a>J, c>d, c>/,
there results
a + c + e > 6 -|- d -|-/.
Trahsformation of Inequaiities. 147
Bui it is not always the same, tfwe subtract, member by member,
two or more inequalities established in the same sense.
Let there be the inequalities 4 <^ 7 and 2 <] 3, we have
4—2, or2< 7 — 3, or 4.
But let there be the inequalities 9 <; 10 and 6 < 8 ; it becomes,
by subtraction, 9 — 6, or 3 > 10 — 8, or 2.
We ought then to avoid, as much as possible, this transformation,
or when we employ it, to assure ourselves in what sense the result-
ing inequality exists.
Tkansforuation by Multiplication and Divisioiv. We
may multiply the two members of an inequality by a positive or
absolute number, and the resulting inequality subsists in the same
sense.
Thus from a <[ 6, we obtain 3 a <^ 3 6 ; from — a <^ — 6, we
deduce — 3 a < — 3 6.
This principle serves to free a quantity from denominators.
If we have the inequality
a^ — b^ ^ ^ — e^
2rf ^ 3a '
we deduce from it, by multiplying the two members by 6 a d,
3a(o^ — 62)>2d(c3 — (P).
The same principle is applicable to division.
But when we multiply or divide the two members of an inequality
by a negative quantity, the inequality exists in a contrary sense.
Let there be, for example, 8 > 7 ; by multiplying the two
members by — 3, we have, on the contrary, — 24 < — 21.
In the same manner,
Q-..,- Q 9^7 7
8 > 7 gives ^^, ^f — 3 < ir3» °' — §•
So that, when we multiply or divide the two members of an in-
equlity, by a number expressed algebraically, we must ascertain
whether the multiplier or the divisor is not negative ; for, in this
last case, the inequality would exist in a contrary sense.
In the problem of article 104, from the inequality
af{a^ — l^)yi'{a^ — b%
we have been able to deduce 2 > -5, by dividing by tr (a* — b)^,
becauie we have supposed a > 6, or o" — 6* positive.
148 Elements of Algebra.
It is not permitted to change the signs of the two members of an
inequality f unless we establish the resulting inequality in a contrary
sense; for this transformation is evidently reduced to multiplying
the two members by — 1.
Tkansformation by raising to the Square. We can raise
to the square the two members of an inequality among absolute
numbers, and the inequality will subsist in the same sense.
Thus, from 6 > 3, we deduce 25 > 9 ; from a -|- i > c, we
obtain (a + b)^ > c^.
But if the two members of the inequality are of any signs what-
ever, we cannot ascertain beforehand in what sense the resulting
inequality will subsist.
For example, — 2 < 3 gives (— 2)^ or 4 < 9 ; but — 3 > — 5
gives, on the contrary, ( — "3)^, or 9 < ( — 5)^, or 25.
We ought then, before raising to the square, to assure ourselves
whether the two members may be regarded as absolute numbers.
Transformation by extracting the Square Root. fVe can
extract the square root of the two members of an inequality in abso-
lute numbers, and the inequality subsists in the same sense, between
the numerical values of these square roots.
We remark, first, that we cannot propose to extract the square
root of the two members of an inequality except when they are
essentially, positive, for otherwise, we should be led to imaginary
expressions p which we could not compare.
But if we have 9 < 25 ; we deduce from it V^j or 3 <[ V^^>
or ^.
From a®> J^ we deduce a > 6, if a and b express absolute
numbers. •
In the same manner, the inequality
fl^ > (c — 6)^ gives a'^ c — i,
if we have supposed c greater than b ; and a > i — c, if, on the
contrary, h is greater than c.
In a word, tvhen the two members of an inequality are composed
of additive and subtractive terms, we must take care to write, for
the square root of each member, a polynomial, in which the subtrac-
tions mxiy be possible. ^
106. Problem 7. Two merchants sell different quantities of the
same stuff; the second sells three ells more than the first, and both
together receive 35 crowns. H(id the second sold what the first did,
Transformation of Inequalities. 149'
he would have received 24 crowns ; had the first sold the same as
the second, he would have received ] 2^ crowns. How many ells
did each sell ?
First Ans. The first sold 15 ells, the second 18.
Second Ans. The first sold 6 ells, the second 8.
Probleni 8. A merchant owes 6240/. payable in 8 months, and
7632/. payable in 9 months. To pay these, he gives a bill of
14,256/, payable in a year. What is the rate of interest 9
Ans, 10,33 per cent.
Problem 9. A man has 13,000/. which he divides into two
parts, and places them out at interest so as to derive the same in-
come from each. Had the first sum been placed at the same rate of
interest as the second, it would have yielded 360/. per annum ; had
the second been placed at the same interest as the first, it would
have yielded 490/. per annum. What were the rates of interest ?
Ans. 7 and 6 per cent.^
N. B. The equation of this problem may be solved more simply
than by the general method.
Problem 1 0. The sum of the areas of two rectangles is q, the
sum of their bases is a, and p and p' are the areas of two rectangles^
having respectively the base of the first, with the altitude of the
second; and the base of the second unth the altitude of the first..
Required the resolution and discussion of this problem.
The base of the first is ^ [^P+9^ Vf^ Ml^
Problem 11. Required the solution and discussion of ths^fQlj-
lowing problem :■ To divide each of two numbers a and b into two'
parts, so thai the product of one part of a by one part of b may be
a given number p, and the product of the remaining part of a 6y
ike remaining part ofh may he another given number p'.
Problem 12. To find a number such that its square divided by
the product of the differences of this number and two given numbers
a and b, may be equal to - ^ required the resolution and discussion
of this problem. "
15Q EhnenU (fJlJgebra.
Questions conctming Maxima and Minima* Properties of Trino-
mials of the Second Degree,
107. There is a certain class of problems which relate to the
theory of equations of the second degree, and which we often meet
with in the Application of Algebra to Geometry. The object of
these questions is to determine the greatest or least value which the
result of certain arithmetical operations, performed upon numbersy
is susceptible of
Let it be proposed, as the first question, to divide a given num-
ber 2 a into two parts, whose product shall be the greatest possible,
or a maximum.
Let us designate by x one of the parts, the other will be
2 a — X, and their product, x{2a — x). By giving to x differ-
ent values, this product will pass through different degrees of mag-
nitude, and it is required to assign to x the value which shall ren-
der this product the greatest possible.
Let us designate by y this greatest product, whose value is at
present unknown \ we shall have, according to the enunciation, the
equation
X {2 a — x) = y.
First regarding y as a known quantity, and deducing from this
equation ^the value of x, we find
a? = a db ^a3-^^.
Now it is evident from this result, that the two values of x can
be real only while we have y <^ c?, or at most y ^z c^) whence
we may conclude that the greatest value which can be given to y,
or the product of the two parts, is a^. But if we make y = o^>
there hence results x =: a.
So that, in order to obtain the greatest product, we must divide
the given number 2 a into tioo equal parts, and the maximum which
we obtain is the square of half of the number, a result at which we
have arrived by another method (100),
A more simple solution. Let 2 a? be the difference between the
4wo parts ; since their sum is already expressed by 2 a, the greatest
of these parts will be (4) respresented by
2a-\-2% ,
^ , or a + «,
QuesiiiffU ofMaosima mi Minima. Ihl
the least by a — a:, and we shall have for the equation,
{a^ x){a — x) =y)
or, performing the calciilalions, a^ — x^ -=. y \ whence
In order thai this ^value of x may be real, it is necessary ihat y
should be al most equal to a^ ; and by making y = a^ 5 we obtain
0? = 0, which proves that the two parts ought to be equal.
This solution has the advantage of leading to an equation of the
second degree with two terms.
108. N. B. In the equations ^
X {2 a — a:) = y and (a -{- x) {a — x)=z y,
established above, the quantity x is what we call a variable^ and
I lie expression x (2 a — j:), or (a -|- x) (a — a?), is called a certain
function of the variable. This function, represented by y, is itself
another variable^ whose value depends on tliat which we give to
the first. It is for ih'rs reason ihat analysts sometimes designate
the former by the name of independent variabhj while the latter^
or y, receives values dependent upon those which we give to x.
Resolving the two equations
x{2a — x) = y and (o '\'x){a — x) = y,
with reference to a?, we have
and X :^ ± x/a^'—yf
we may in turn regard y as an independent variable^ and a? as a
certain /unction of that variable.
109. Let it be proposed, as a second question, to divide a nt/m-
ber 2 a into two such parts that the sum of the square roots of these
two parts may be a maximum.
Let us call a^ one of the parts, 2 a — a^ will be the other part,
and the sura of their square roots will have for its expression
X + V^'tt — ^ 5
it is this expression whose maximum is to be determined.
Let us put 0? -p \/2 a — ^ = y-
To resolve this equation, we must free it from the radical. We
have first, by transposing the term x to the second memberi
V5o — af« = y — ^9
152 EUmenii of Algebra*
whence, raising to the square,
or, arranging with reference to z,
2a^'-2yx=z2a—f,
an equation whence we deduce
or, simplifying.
yl ,
In order that the two values of x may be real, y^ must be at
least equal to 4 a ; then 2 t^a is the greatest value which y can
receive.
But if we make y = 2 y^a, there results x =. \/a, whence we
deduce a^ = <r, and 2 a — x^ = a.
Thus the given number 2 a must be divided into two equal parts^
in order that the sum of the square roots of these two parts may be
a maximum. This maximum is moreover equal to 2y^a.
For example, let 72 be the proposed number ; we have
72 = 36 + 36 ;
whence y/S6 + v'36 = 12 ;
this is the maximum of the values which can be obtained as the
sum of the square roots of the two parts of ^2.
For let us decompose 72 into 64 + 8 5 we have V^64 = 8,
^8 = 2 + a fraction ; whence y^64 -}- y'S = 10 -|- a fraction.
Again, let there be 72 = 49 + 23 ; we have
V'49 = 7, V23 = 4 + a fraction ;
then ^^49 + ^^23 == 11 + a fraction.
Let us consider, for the third example, the expression
ffl^g^ + n^
(m^ — n^) X*
which it is required to render a minimum (m being supposed >n).
Let us put
m^ar^ + n^ __
{m^ — n^)x^^'
whence
m^x* — {nfi — n')y . a? = — ffi;
QuestionB of Maxima and Minima. 169
wefaence deduce
Now, in order that the two values of a?, corresponding to a value
of y, may be real, it is evident that (m^ — w^)^y^ must be at
least equal to 4 m? n^, and consequently that y must be at least
. 2mn -, , 2mn . , . . - . ,
equal to --^ «. So that —k 5 is the minimum of the values
^ m'* — w* m^* — n'*
which the function y can receive.
Q «M YL
But if we make y= -s «, in the expression for x, the radical
m — n
disappears, and the value of x becomes
r
m^ — n^ 2mn n
a?= 0^2 X
2m^ m"^ — n^ m
This value, a? = — , is then that which renders the proposed ex-
pression a minimum. .tr
110. These examples are sufficient to show the steps which
must be followed in the resolution of questions of this kind.
^fter having formed the algebraic expression of the quantity
susceptible of becoming either a maximum or a minimurai we^
make it equal to any letter whatever y. Jff* the equation which
is thus obtained, is of the second degree in terms of x, (x design
nating the variable quantity which enters into the algebraic ex^
pression,) we resolve it with reference to x ; then make equal to
zero the quantity under the radical^ and obtain from this last
equation a value ofy, ivhich then represents the maximcHn or minin
mum sought. Lastly^ substituting this value ofy in the expressidn
for Xf ^e have the value of this last variable,, which wiU satisfy the
enunciation.
N. B. If it should h^ppeu that the quantity under the radical
remains essemialjy positive, whatever be the valu^ of y^ we should
conclude that the proposed expression can pass through aU pomr.
ble states of magnitude ;» in other wordsy that it would have ixfyiiity
for a maximum and zero for a rainimum.
Let there be, for a new example, the expression '•
. 4 ipS -f- 4 iP -_ 3 ^
6(2aj+l) '
Bour.Alg. 20
154! Elemmis ofAlgAra.
it may be asked whether this expression is susceptible of a maxi^
mum or of a minimum.
Let us put
4 3.8 4. 4 r _ 3 _
0(2a? + 1) "" ^'
There results from it the equation
4 a^ — 4 (3y — 1) a? = 6 y + 3,
whence we deduce
_3y-l 1
Now, whatever value we may give to y, the quantity under the
radical will always be positive. So that y, or the proposed expres-
sion, can pass through all possible states of magnitude.
In the preceding examples, the qiiantily under the radical in the
value of a?, contains only two parts, the one affected by y, or y*,
and the other a known quantity ; and it has been easy to obtain the
mcLximum or minimum of which the function was susceptible. But
it may happen that the quantity under the radical may be a trino-
mial of the second degree of the form w y^ + ^ y + jP« I^
this case the question becomes more difficult, and to be able to
resolve it completely, it is necessary to demonstrate several proper-
ties relating to these trinomials.
Properties of Trinomials of the Second Degree.
111. Every algebraical expression which can be reduced to the
form m y^ + n y + ^ is called a trinomial of the second degree ;
where m, n, and p are known quantities with any sign, and y is a
variable quaniitt/j or one which may be made to pass through vari-
ous states of magnitude.
Thus, 3y2_5y-f-7, — 9y3 + 2y + 6,
(<y._i-|-2c)y2 + 4i3y — 2ac« + 3a3i,
are trinomials of the second degree in y.
If the trinomial my^-f-ny+p be put equal to nothing, or if
my^ + ny+p=:Of
the values of y are
2m 2m
V V
Properties of TrinomiaU of the Second Degree. 155
and three different hypotheses may be made with respect to the
nature of the values ofy which have been obtained.
(1.) n^ — 4mp may be > 0, or positive ; in which case the
values of y are real and unequal, and may have any sign.
(2.) n^ — 4mp may be = O5 in which case the roots are real
and equal.
(3.) n^ — 4m p may be <[ 0, or negative; and the two roots
are imaginary.
In the first case, that is, whenever a trinomial of the second
degree is such that when it is put equal to 0, and the resulting
equation is solved, the two roots are real and unequal, every quan-
tity (positive or negative) contained between the two roots, gives a
result whh a contrary sign to that of the coefficient of y^ ; but
every quantity, which is not contained between the two roots, when
substituted for y, gives a result with the same sign as the coefficient
ofy».
Let y' &nd y'^ be the two roots (which we suppose real) of the
equation
mf + ny + p = 0, or m{f + ^y +^) =zO.
The first member of this equation may (98) he put under the
form rn{y-^y'){y — y'').
Therefore, w y* +'» y +p = «i (y — y'){y — y") 5 ^^^^l this is '
true, whatever value is given to y.
Let y' be the least of the two roots, and let a > / and! <^ y^', or
let a lie between y' and y'', then a — y' is > 0, and a — y" is <] 0,
therefore the factors a — y' and a — y" have contrary signs, and
their product is negative. Therefore m {a — y') (« ■" — y'O' °^
m a^ + n a + p, is of ihe contrary sign to that of m.
If, on the other hand, y' and y" be both greater or both less than
fl> a — y and a — y" are of the same sign, and their product is
positive ; consequently
fw (a — y') (« — y'O* ^^ «i a^ -|- n a + ji,
is of the same sign as m»
In the second case, if the two roots be real and equal, every
quantity except the root, when substituted for y in the trinomial,
gives a result with the same sign as the coefficient of y^.
For, since the two roots are equal, v? — 4mp = 0, whence
p = J— ; and the trinomial can be put under the form
156 El€fnent$ of JUgdfra*
my' + ny+^, or m(f + ly + ^y or m(y + ^J.
Now, 11 is evident that, for every value of y, except — s— i the
quantity
is positive. Therefore,
is of the same sign as m.
In the third case, where the two roots are imaginary, every real
quantity, positive or negative, when substituted for y, gives a result
of the same sign as the coefKcient of y^.
For, since the roots are imaginary,
rfl — 4 m o <* 0, or 4 mp > n^ or - "> - — 5.
nhefe i" is a qudntity which must be positive, it follows that
mt^ + ny+py or j»^y2 + ^y+y
t quantity which always has the same sign as m, whatever value be
substituted for y.
112. The second case leads us naturally to a proposition .which
is of frequent use in analj'^sis.
Whenever a trinomial of the second degree, w y^ -j- n y -|- 1?, is
a perfect square, there exists between its coefficients the relation
- fi? — 4wp = 0,
For, if this trinomial is a perfect square^ and of tlie form
the two roots of the equation my^ + n y + P = 0, must be equal*
But if they are equ^l, the quantity under the radical, or n^ — 4 mp^
must be nothing. Therefore, n^ — 4mp = 0.
Properties of Trinomials oj ike Second Degree. l^
Reciprocally, if there exist between the coefficients thet relatifm
n^ — 4 mp = O9 the trinomial is a perfect square ; for, from this
relation, we deduce
^nd
m
f + ny+p = mf + ny + ^ = Q^m + ^^^.
113. Let us see the actual use of these properties, in the reso-
lution of questions of maxima and minima.
Let it be proposed to determine whether, when wfe have made x
/p2 2a: + 21
a variable, the function — ~ — can pass through all states
of magnitude.
Let us put
afi — 2x + 21 _
6a?— 14 ""^^
whence
a:^ — 2(3y + l)a? = — 21 — 14y,
Ther^ results
0? = 3 y + 1 dz y'9y2— 8y — 20.
In order that x may be real, it is necessary that
gf — Sy — 20
should be positive. Now, if we make this quantity equal to zero,
it becomes
y^ — I y — V = ^> whence y = 2, and y = — y .
These two valaes of y being real^ it follows from the first of the
above mentioned properties, that when we give to y values compre*
bended between 2 and — y , such as 1, 0, — 1, • • . the value of
the trinomial will be negative, since the coefBcient of y^ is positive.
But by giving to y values not comprehended between 2 and — y,
as 3, 4, ... or — 2, — 3 — 4 . . . , we shall obtain a positive result.
We see then that 2 is, in absolute numbers, the minimum of the
values which y ought to receive, iir order that a? may be real. If
in the above expression of a?, we make y = 2, the radical disap*
pears, and we find a? = 7.
Indeed, the expression
a» — 2x + 21
6ar— 14
m ■'
If8 Ekmenis 0/ Algebra.
becomes, upon the hypothesis of a? = 7,
49 — 14 + 21 _ 66 _
42 — 14 "" 28 ■"
The root y =. — V ^^9 *" negcUive numbers^ the maximum of
the values which y can receive ; and the value of Xj corresponding
to this maximumj is
a: = 3X~V + l= — i- •
When having expressed x in terras of y, the coefficient of y*
under the radical sign, is negative, and when the two values of y,
deduced from the trinomial made equal to zero, are, the one posi-
tive, the other negative, the positive value is a maximumy since any
greater value would give a result of the same sign as the coefficient
of y^ ; and the negative value is a minimum among the negative
values which y can receive.
We leave to the learner the care of examining the other circum-
stances which may be presented ; for example, the case in which
the coefficient of y^ being positive, the two values of y are positive ;
and that in which this same coefficient being positive, the two
values are imaginary ; and he can moreover exercise himself with
the following questions ;
To divide a given number 2 a into two parts^ so that the sum of
the quotients which are obtained, when mutually divided the one by
the other J shall be a minimum.
[Ans. The two parts must be equal, and the minimum is 2.)
Let a and b be two given numbers of which a is the greater ; it
is required that the expression
/ (x + a)Jx-b) \
thould he the greatest possible,
{Ans. Maximum = ^ ."^ . \ and the correspondiag value of
It is required that
a + ^) (b + x)
X
should be the least possible.
{Minimum = (y'a -|- ^b)^i x = ^a b.)
Equations and PrtMeim of the Suond Degree. 159^
Oil Eqiiaiions and Problems of the Second Degree^ containing
Two or more Unknown Quantities.
114. Problem 1. To find two numbers^ the sum of whose pro-
ducts by the numbers a and b respectively^ is equal to 2 s, and whose
product is equal to p.
Let X and y be the two numbers required ; we have the equa-
tions
ax + by = 25,
xy=p.
From the first we deduce
25 — ax
y=— 6— 5
substitute this value in the second, reduce it, and it becomes
aa^ — 2sx=i — bp.
Therefore,
and
a; = - ± - \/a» — a Fpf
a a^ ^
y-T^iV^^^^'
b^ b
This problem has evidently two direct solutions, for
s > \^8'^-^ahp ;
but they cannot be real unless 5* > or = a 6 p.
Let a == 6 = 1 ; the values of a? and y are reduced to
X z=zj db yjaH^, and y = « =f y'^-.j?,
, in which the values of x are the same as those of y, with the dou-
ble sign inverted ; that is, if 5 -f- ^^^a — p be the value of a?,
s — V^s— />
\ is the corresponding value of y, and the reverse.
This circumstance may be explained, if we observe that the
equations become, in this particular, case,
x + y=:2Sj
xy:=.p}
and the question is reduced to that of finding two numbers whose
sum is 2 5, and whose product is j7, or in other words, to that of
dividing a nuniber 3 s into two parts whose product is p.
160 JBSemenU of JilgAru»
Now, by article 100, these two parts are connected with one
another by the same equation of the second degree,
a:^ — 2*a? + P = 0>
the coefficient of whose second term is the sum 2 *, with a contrary
siga, and whose last term is the product p of the parts.
115. Problem 2. To find four proportional numbers^ the sum
of whose extremes is 2 s, that of the means 2 y, and that of the
squares of the extremes and means 4 c^.
Let w, X, y, and z, be the four numbers ; then, by the funda-
mental properties of proportion, the equations of the problem are
w -{-. z = 2 5,
0? + y = 2 y,
uz = xy^
1^2 + aj2 4- y2 ^ ^2 _, 4 ^^
At first sight, it may appear difficult to find the values of the
unknown quantities ; but by means of an auxiliary unknown quan"
tity, they may be simply determined.
Let p be the unknown product of the extremes or of the
means.
Then
(1.) \^ + ^ = '^^^\ whichgive {« = ^ + V?Ef'
(2.) !^+y = ^^'l whichgive j^=i + V*IEi'
V J I xy=zp, 5 & (y = 5" — V«'2— 1>.
(See the preceding problem.)
The determination of the four unknown quantities now depends
simply on that of p.
. Substitute these values of m, x, y, and z, in the last of the equa-
tions of the problem, and we have
{s + x/s^^)\+ {s — s/W^f + (^ + s/T^^pf
+ {s' — ^W^pf = Ac^,
which, when developed and reduced, becomes •
4 5^ •+• 45^^ — 4 p = 4 c®, or «^ 4" *^' — P = ^>
and I? = «2 + *'2 -Jr- (T*.
Substitute this value of p in the expressions for u, cr, y, and z^
aad we have
Equations and ProbhrM of the Second Degree. 161
These four numbers are proportional, since
uz={s + Vcs — «'2) (5 — VcsZTTs) =Z 5» + s^'t^ c»,
xy = {8" + V^s^) (s' — V'c"^^^^) = ^ +s^ — i?.
N. B. This problem, which we have taken from the Algebra of
Lbuilier, will serve to show how much the introduction of an im-
auxiliary known quantity into a calculation facilitates the determi-
nation of principal unknown quantities. We find, in the work just
mentioned, other problems of the same kind, which lead to equa-
tions of a degree superior to the second, and which nevertheless
may be resolved with the assistance of equations of the first and
second degree, by introducing auxiliary unknoum quantities.
116. Let us now consider the case in which a problem would'
give rise to any two equations of the second degree with two
unknown quantities.
An equation with two unknown quantities is said to be of the
second degree, when it contains terms in which the sum of the ex^
ponents of the two unknown quantities is equal to 2, and does not
exceed 2. Thus,
3a5^ — 4a?-}-y* — xy — 6y + 6 = 0, 1 xy — 4ir-}-y = 0,
are equations of the second degree.
It hence follows, that every equation of the second degree with
two unknown quantities, is of the form
af + hxy + ca^ + dy+fx+g:=^0,
a, &, c • • . representing known quantities, either numerical (Mr
algebraic*
Let there be prc^osed the equations
ay' +hxy +ca^ + dy +fx + g" = 0,
a^f + b'xy + &a^ + d'y+foo+g'=zO.
We can arrange these two equations with reference to a? ; aodj
they become
c^ + {by +/) ^ + ay' +dy + g =0^
do? + {yy +/) X + a'f + d^ y + g^ = Q.
This being laid down, if the two coefficients of ar^ were the same
in the two equations, we should obtain, by subtracting these tlvo
equations the one from the other, an equation of the first degree in
terms of Xj which might be substituted for one of the proposed
Bour. Alg. 21
162 Elements of Algebra.
equations ; from this equation, we should obtain the value of x in
terms of ^, 8"d carrying back this value into one of the proposed
equations, we should arrive at an equation which would contain
only the unknown quantity y.
Now, if we multiply the first equation by (/, and the second by
c, they become
c&a^+{by +f)(/x+{af +dy + g) c^ = 0,
c& or" + (b^ y +f)cx + {a^ f + d^ y + g") c =0,
equations which may take the place of the preceding, and in which
the coefficient of o^ is the same.
By subtracting them, we find
^^bcf — cb')y+fcf—cf]x + (a& — ca')f + {dcf — cd')y
+ g& — cg' =zO,
an equation which gfves
(^ca* -^a c') y^-\'{cd' — dc')y -^ eg' — gc'
*- (^bc' — cb')y+fc'^cf
This expression for x, substituted in one of the proposed equa-
tions, would give ti final equation in terms of y.
But, without performing this substitution, which would lead to a
very complicated result, it is easy to perceive that the equation in
terms of y must be, in general, of the fourth degree ; for the nu-
merator of the expression of a? being of the form m y^ -j- n y + 1?,
its square, or the expression of a:^, is of the fourth degree ; now,
this square forms one of the parts of the result of the substitution.
Then, in general, the resolution of two equations of the second
degree mth two unknown quantities depends on that of an equation
of the fourth degree with one unknown quantity.
117. There is a class of equations of the fourth degree, the
solution of which depends on those of the second degree j they are
contained in the form a?* -j- /? a?^ + 9 == 0«
To solve this equation let a?^ = y, which reduces it to
»^ + Py + ? = 0, ory = — 1± J^ — ff.
But a?^ = y, or a; = rfc ^^y.
Therefore,
*=*j-«*j?-«-
Extradion of the S^puire Root. 168
We may see, by the resolution of this equation, that the unknown
quantity has four values, since each of the signs of the first radical
may be combined with each of those of the second ; two of the
values are equal respectively to the other two, but with contrary
signs.
Let a?* — 26 a?2 = — 144.
If we put 01^ =: y, then y^ -— 26y = — 14,4 j
whence y = 16, and y == 9.
Therefore,
(I.) a* = 16, and a? = d= 4.
(2.) a?^ = 9, and a? :;« ± 3.
Thus, the four values of x are
+ 4, — 4, +3, and — 3,
Let a?*— 7a? = 8.
Put a? z:^ y.
Then y^ — 7 y = 8, y = 8, and y = — • 1.
Therefore,
(1.) a? = 8, or a? = d= 2 v^,
(2.) ar^ = — I, or a? = d= V^^J
the two last values of a? are imaginary.
Let a?* — (2ic+;4a2)a;» = — iV.
Put ^ = y*
Then
y9_(26c + 4a^)y = — i^ca, . ,
whence y = 6 c -|- 2a^ ± 2a y'fc c-^-c^i
and
a?= d= \/6c+2a2±2aV'ftc + a»«
Extraction of tlie Square Root of an Expression which is partly
Rational and partly Irrational.
1T8. The solution of a trinomial equation of the fourth degree
gives rise to a new species of algebraical operation, viz. the extrac-
tion of the square root of a quantity of the form a d= \/b ; a and
b being either numerical or algebraical quaiitities.
Let there be the expression 3 dz y^5 to be raised to the square,
we have
( 3 d= v'o )f = 9 =t: 6 V5 + 5 = H d= 6 V5 .
364 ' Eiemmt»(fJ^firu.
Hofaoe
la like manner,
(V7=fc>v/n)2 = 7±2>v/7X 11 +11 = 18dL2v"77.
Therefore,
V18 + 2V77 ^v^Tdzy^lt.
Whence it appears that an expression, sijch as \/a ± \/i, may
sometimes be reduced to the form a' dz y^6', or \/a' =fc V'i' ; anil
this transformation should be made when it is possible, because in
that case there is only om or two simple square roots to be ex-
tracted, whereas the expression \/a =fc \/b requires the extraction
of the square root of a square root.
119. Having given a quantity of the form a ± V^, to dtstover
whether it is the square of an expression of the form a' dt -v/b',
or y^a' ± y/b'j ^^ ^^ deternmie thai root.
Call p and q, the two parts of which the square ropt of a 4" V'^
is composed ; p and q are either both irrational, or one is rational
and the other irratiooaL
We must remark, that since
<V/a + V*— P + S> (1)
it follows that
\/a — v^6 =p — J. (2)
Taking the prqduct -of these equations, we faav«
A/'^^'^^f — q^^y (3)
and, since p and q are either one or both irrationals of the second
degree, p^ and q^', and consequently p^ -^ q^ are rational quanti-
ties ; therefore, \^a^ — b is a rational quantity.
Whence we ms\y conclude, that a db \/b cannot be the squate
of an expression soch as a' ± yi/, or v<*' =t \^> unless o^ — 6 is
a perfect square.
Let this be the case^ and let y^^iTZT^ r= c* The equatioD (3)
beconoes p^ — g^ == c.
equations (1) and (2) squared, give
i^ + 9^ + ^P? = ^ + V^9
p^ ^ ^'^^pq = a •— \/b^ '
whence^ by addition, we obtain '
f^ + ^^a; (4)'
Extraction of the Square Root. 165
but we have also
f-t = c. (5)
Accordingly,
Therefore,
[a — c
or, more distinctly.
J^
(7)
Tiiese two formulas may be verified a posteriori; square both
members of the first ; then
, , a +c a — c,^ lo^^^^ ^ , /-^ — 3-
a + vi = -Y- + -2" +^ J— 4— = ^ + V^-^
But, since v^JTZTl z= c, i? z=i d^ — 6,
and
^ + V^ = ^ + Va^ — a3 4- 6 = a + V^.
In a similar way the second formula may be verified.
120. Remark. Since the formulas (6) and (7) are true, even
when c? — 6 is not a perfect square, they may still be used to give
values for the expressions
^a -|- \/6, and y'a — ^h ;
but in that case the expressions .are not simplified by such a pro-
cess, since the quantities p and q are of the same form as the given
expressions.
It is only proper to make this transformation when c? — 6 Is a
perfect square.
121. As an example of the use of these formulas, take the nu-
merical expression
94 + 42 VS, or 94 + V8820,
166 Elements of Algebra.
Here a = 94, J = 8820,
c = VoS— h = V8836 — 8820 = 4,
a rational quantity ; whence the forniula is applicable, and
= ± (v/49 + v45)
= d= (7 + 3V6),
and (7 + 3^5)^ = 49 + 46 + 42 v^ = 94 + 42 v^.
Again, let there be the expression
II I ■ I ■ I — ■■■■■y ■■■■■■■■»■ I , ^»
Here a = np + 2 m',
J = 4m^{np + m^),
f? z=, a^ — 6 = n^p^,
or, c =3 np,
whence the formula is applicable ; and the root required is
/ [ wp -[-■ 2iw^ -|- »p [ »p -f- 2i«^ — np\
. V>| 2 N 2 /
= dz (\/np + »»^ — Wl), .
and
s/np + m2 — wi)^ = np + 2m^ — 2m Vnp + v^
Let it be required to simplify the expression
Vi6 + 30v/=l + v^lG— 30v— T.
By the formulas, we find
Vl6 -f 30v'^l = 5 + 3 v/:=l,
V^ie — 30v/=l = 5 — 3 v^^l 5
therefore
Vi6 + 30 V'^=^ + ^^6— 20^311= 10.
This last example shows better than all the others the utility of
the general problem which we have solved ; for it proves that the
combination of imaginary expressions may produce real^ and even
rational quantities.
The formula may be applied by the student to the following
cases.
V28 + 10v3= 5 +v3;
a/1 + 4 yZTs = 2 + s/— 3j
Indeterminate ^naiym of the First and Second Degree. 167
\/bc + 2bx^bC'--Vi+ \/bc — 2bx/hc'-b^= ±26;
V'a6 + 4c3 — flP + 2v4a6ca — a6rf2 = y'a6 +y'4 f^ — (P.
Indeterminate Analysis of the First and Second Degree.
Introduction, — Wlien the enunciation of a problem furnishes a
less number of equations than it has unknown quantities, the prob-
lem is called indeterminate, since its equations may be satisfied by
an infinite number of values attributed to the unknown quantities.
But it frequently happens that the nature of the question requires
the values of the unknown quantities to be expressed in whole num^
bers ; in this case, one of the unknown quantities, to which we
raay first give a value altogether arbitrary, must receive only en-
tire values, and such that the corresponding value of the other, ia
each of the other unknown quantities, may be expressed also in
entire numbers. Now this condition very much restricts the num-
ber of solutions, particularly if we consider only direct solutions^
that is, solutions in entire and positive numbers for all the unknown
quantities.
The object of indeterminate analysis of the first degree is to
resolve indeterminate questions of the first degree in entire and
positive numbers. We shall see hereafter the purpose of indeter-
minate analysis of the second degree.
I. Equations and Problems of the First Degree with Two Vnknovm
Quantities.
122. Every equation of the first degree with two unknown quan-
tities raay be reduced to the form ax -^b y =. c ; a,b, c, desig-
nating entire numbers positive or negative.
fVe begin by observing, that if the coefficients a and h have a
common factor which does not divide the second member c, ^ egua-
tion cannot be satisfied by entire numbers.
For let a = A a', b =z h b* the equation becomes
A a' a? + A i' y = c,
whence we deduce
a' X -{- b'y = Tj
168 Ekments of AlgAra*
an equation which cannot be satisfied by any set of entire values of
X and y so long as c is not divisible by A.
We suppose in all that follows, that a and h are numbers prime
to each oiher. Since if they have a common factor, c must like-
wise contain this factor, in which case, it might be suppressed in
the equation.
123. For the sake of clearness we will first treat of particular
equations and afterwards generalize.
Question 1. To divide 159 into two parts^ one of which shall be
divisible by 8 and the other by 1 3.
Let us designate by x and y the quotients of the division of the
two parts sought by the numbers 8 and 13 respectively; then 8a?
and 13 y will express the two parts, and we shall have the equation
8a?+l3y=159 (1),
which, according to the enunciation, is to be resolved by entire and'
positive numbers for x and y.
We deduce, in the first place, from this equation
_ 159— 13y
X g ,
or, performing the division as far as possible,
Now, we perceive that the value of x will be entire if we give to
y such a value that ^— ? shall be a whole number ; moreover
this condition is necessary j so that it is only required that — 3-—
should be equal to some whole number. Let t be this whole num-
ber {t is called an indeterminate), we shall then have
— g— ^ z= i^ whence 6 y + 8 ^ = 7 (2),
and the value of x becomes
0? = 19 — y + ^.
Every entire value of t which, substituted in equation (2), will
give a similar value for y, will satisfy the condition that — 5 — -
should be a whole number. Thus the two corresponding values
of X and y will be entire, and moreover will satisfy equation (1),
Indeterminate Analysu of the First and Second Degree. 169
which evidently results from the elimiDation of t from the two
equations
~ ^ = ^, and a? = 19 — y + ^
The question is then reduced to resolving hy entire numbers equa-
tion (2), the coefficients of which are more simple than those of
equation (1).
From equation (2) we deduce
7 — 9t
or, performing the division partially,
1 . I 2 — 3*
y = 1— < + — g--
Every entire value of tj which makes 2 — 3 ^ a multiple of 5, will
also give for y an entire number, and consequently is suited to the
condition. Let us then put — =--3- = f^ f being a new inde-
terminate. The equation becomes
3 < + 6 <' = 2 (3),
and the value of y is reduced to
y = 1 — < -f ^'.
[Equation (2) results moreover from the elimination of if from
the two last equations.]
The question is then reduced to resolving in entire numbers
equation (3), from which we deduce
let us put
2 — 2<'
= <'',
3
and there results 2 ^ + 3 i'' = 2 (4),
and i = —i'-\^ V.
From equation (4). we deduce
» ^ 3^ -1 — < — 2-
t"
Lastly, let us make ^ = i^*', and there results i^' = 2 V" (6),
and f =i — if' — tw.
Bour. Alg. 22
170 Elements of Algebra.
As in equation (5), the coefficient of i" is unity, it follows that
^very entire value given to if'* will give likewise an entire value to
f. Moreover the two principal unknown quantities x and y, and
the intermediate unknown quantities t^ i'j t"^ and t'" have among
themselves the relations expressed by the five equations,
0? = 19 — y -{• U
y = 1 —i+t^,
t= —if + if\
<' = 1 —t"— i"\
i" = 2 i'".
So that by giving to f" any entire value whatever, and going
back from the last of these equations to the first two, we shall obtain
for X and y corresponding values in whole numbers, which neces-
sarily verify the proposed equaiion ; for, according to the reasoning
which we have pursued above, this equation results from the elimi-
nation of iy fy V'^ i"'y from the five equations which we have just
established.
But in [order, that we may assign to V" only the values to
which entire and positive values of x and y correspond, it is re-
quisite to express x and y in an immediate function* of the inde-
terminate or auxiliary unknown quantity t'^'j by means of the five
equations above.
Now the expression for f becomes, by substituting for f its
value in terms of f'\
t' = l—2V'— /''', or i' = 1 = 3 i''' ;
then going l)ack to the expression for ^,
i=^ — t' + 1f' = — \+2t'" + 2if".
Therefore ^ = — 1 + 6 1'''.
We shall have, in like manner,
y = 1 — (— 1 + 6 rO + 1 — 3 tf".
Then y == 3 — 8 1''\
And lastly,
a? = 19 — (3--. 8 r') + (— 1 + 5 1"% or a? = 15 + 13 i'".
• We call 2L function of a letter, considered as variable, every ex-
pression which contains this letter combined with known quantities.
Indeterminate Analysis of the First and Second Degree. Ill
It is easy to show by the elimination of f^'j that these two equar
tions reproduce the proposed equation. Indeed, if we multiply the
first equation by 13, and the second by 8, and add together^ the
results, it becomes .
13y + 8a? = 159.
Let us make successively f = 0, I, 2, 3, or f^^ == — 1,-2,
•—3;. the preceding formulas will give all the values of x and y
in whole numbers either positive or negative, which will satisfy the
proposed equation ; but if the enunciation requires, that we should
only consider entire and positive solutions^ V must only receive
such values as will render 3 — %i"' and 15 -f- X'^i'" positive.
Now it is evident, that there are only the values i^" = 0, and
t'" •=. — 1, which satisfy this condition j for every positive value
of V" renders y negative, and every negative value, numerically
greater than 1, renders x negative.
If we make successively
f'' = 0, r' =—1,
there results y = 3, y = 11,
a? = 15, 0? = 2.
In these two sets, a? = 15 and y = 3, a? = 2 and y = 11, are
the only values which verify the equation 8 a? -|- 13y = 159.
As to the question the conditions of which are expressed alge-
braically by this equation, sitrce 8 a? and 13 y express the two parts
sought, it follows that 8 X 15 or 120, and 13 X 3 or 39, form a
first solution; and 8 X 2 or 16, and 13 X H or 143 a second
solution; that is, the number 159 may be divided into the parts
120+39 or 16+ 143.
124. For a second example let there be the equation
17 a? — 49 y=]— 8 (1).
We deduce from it in the first place
49y — 8 ^ , 15y— 8
In order that there should be an entire value for x corresponding
to an entire value of y, it is only necessary that 15 y — 8 sliould
be a multiple of 17. Let then
15y--8_
~T7 ""^'
17t EiemenU ofAJ^Ara*
i b0iog an ioteroieAate unknown quantity ; there results frem it
15 y— 17^ = 8 (2)
and a? = 2 y + ^*
[The elimination of i from the two equations will reproduce *
equation (1)].
We deduce from equation (2)
8 + 17 ^ . , 8 + 2 ^
And the new expression — ^^^ — must be a whole number.
Putting ^-^^ = i', we obtain 2 ^ — 1 5 f' = — 8 (3),
and y :=z t ^ f.
Equation (3) gives
15f' — 8 t'
Then 5 must be a whole number.
Let us make ^ = f ', it becomes <' = 2 <'%
and i = 7 f — 4 + ^'.
Now in order to express x and y in a function of the indetermi-
nate if'f let us resume the four equations,
0? = 2y + ^}
y = i + ^,
t z= 7 ^' — 4 + ^',
If z=i2if'.
The last but one becomes
t = l X 2t" — 4 + f'j whence t= 15^'' — 4;
going back to the second, we have
y = 15 f' — 4 + 2 ^', whence y = 17 «'' — 4 ;
and finally, the first becomes
a? = 2(17^'— 4) + 15<^'— 4, or a? = 49^—12.
These two formulas reproduce the proposed equation by elimi-
nating f^ ; for if we multiply the first by 49, and the second by
17, and subtract the results, it becomes
17« — 49y = — 204+196 = — 8.
Indeterminate Andytis of the First and Second Degree. ITS
We see, moreover, that by giving to f^ any positive values what«
ever, we shall obtain positive values for x and y ; but we must not
suppose f' negative*
Let ^' = 1, 2, 3, 4 . . .
We find y= 13,30, 47, 64...
a? = 37,86, 135, 184...
The numbers of entire and positive solutions of the proposed
equation is then infinite ; and the smallest set of values is x =i 37,
y = 13.
These values verify the equation ; for we have
17 X 37 — 49 X 13 = 629 _ 637 = — 8.
In this example we have dispensed with going over all the rea-
soning which we made use of in the first example, in order to
account for all the operations ; but it is easy for beginners to repro-
duce them by following the transformations step by step.
125. We may thus recapitulate the preceding method ;
Let ax -}- by =z c (1), be the equation which it is required to
resolve. Deduce froih this equation the value of the unJcnovm
quantity which has the smallest coefficient of x for example^ and
perform the division as far as possible ; an expression is obtained
ibr X in terms of y composed of two parts, the one entire, the other
fractional, the latter of which we must endeavour to make a whole
number. Make this second part equal to a first indeterminate X;
there unU result a new equation in terms ofy and t, which we may
call equation (2), whose coefiScients are more simple than those of
equation (1) ; the value of xis then found in an entire function of
J and t, and the proposed equation results from the elimination of t
from equation (2) and the equation which gives the value of x in
terms of y and t. Deduce from equation (2) the value of y and
perform the division as far as possible. Make the fractional part
equal to a second indeterminate V ; whence results an equation (3)
in terms of t and t% more simple than the equations (1) and (2).
The value of y is thus found expressed in an entire function of t
and t^, and the proposed equation results from the elimination of t
and i' from equation (3) and the two equations which give x in an
entire Junction of y and t, and afterwards y in an entire function
oft andM.
Perform the same operations upon equation (3) as upon equa-
tions (i) and (2), and continue this series of operations untU you
174 Elements of Algebra.
arrive at a last equation between two indeterminatesj one of which
has unity for its coefficient.
Finally^ from this last equation go back to the preceding and
seek by successive substitutions to express x and y in a function of
the last indeterminate.
You thus obtain two formulas, by means of which, by giving to
the remaining indeterminate arbitrary values you will find all the
sets of entire values, as well positive as negative, which are capable
of verifying the proposed equation ax -\- b y = c.
If we seek only entire and positive values of x and y, the two
formulas indicate by their composition, between what limits the
value of the last indeterminate mu^t be comprehended^ in order that
this condition may be satisfied.
Remark 1. The process which we have indicated above must
always lead to a last equation in which the coefficient of one of
the indeterminates is equal to unity.
For in the first operation we divide the greater coefficient of the
two unknown quantities by the less, in the second the less coeffi-
cient by the remainder of the first division ; in the third the first
remainder by the second, and so on ; that is, we apply to the two
coefficients the process for finding a common divisor. . Then, since
by hypothesis the two coefficients are prime to each other (122),
we shall in the end arrive at a remainder equal to 1, which will
serve as a coefficient to the last but one of the indeterminate
quantities which have been introduced in the course of the calcu-
lation.
(2.) When we apply this process to an equation in which the
coefficients of the two unknown quantities contain a common factor,
which is not found in the second member, but which we did not at
first perceive, the course of the calculation makes known the im-
possibility of resolving the question in whole numbers.
Let there be, for example, the equation 49 a? — 35 y = 11.
(The factor 7 is common to the coefficients of x and y, but does
not enter into the second member.)
We deduce from it
49a; — 11 , ]4x— 11
y^— 35— = ^+— 35—
Putting
- — — = tf whence y = a; + ^,
Indeterminate Analysis of the First and Second Degree* 175
we have
35< + ll_ 7t + l l
X - — ^ -^t + — IT"""-
Putting
we find
— ^t — =:: ^/j whence a? = 2 < + <',
14// 11 A
7 7
This last equation is evidently impossible in whole numbers for
t and t'j since ^ is a fraction. It is likewise impossible then to
obtain a solution of the proposed equation in entire numbers for x
and jf.
126. The above process is susceptible of several simplifications
which it is important to introduce in practice.
Let us resu'rae again the equaiion ah*eady considered,
^7x — 49y ==r—S;
we deduce from this immediately
_ 49y-^ 8
We observe that 49 is equal to 17 X 2 + 15, or rather that 49
equals 17 X 3 — 2; then
17 -"^y w
thus the value of x takes the form
a? — o y Yf y
and the question is reduced to finding for y a whole number which
will make the expression •^3" — entire. Now this expression be-
comes
17 '
but the two numbers 17 and 2 zxq prime to each other* Thus, m
order that - yf - — - may be an entire number, it is only necessa-
ry that y -{• 4 should be divisible by 17.
V 4- 4
Let us then make ^ IL =: t, ^ being a whole number entirely
176 Elements o/Algeifra. ^
arbitrary ; there results
y= 17^ — 4,
and the value of x becomes
a? = 3y — 2<,
or, substituting for y its value in terms of i,
x = 49t — 12.
These formulas give all the entire solutions of the proposed ques-
tion ; for the elimination of t in the two equations, reproduces the
equation
17'a? — 49y = — 8.
By making ^ = 1,2, 3, 4, ... we find the entire and positive
values of x and y ; but we cannot make t negative or equal to 0.
We shall thus perceive the importance of the preceding modifi-
cations, since, by means of them, we have had occasion for one
indeterminate only, in the course of the calculation.
These modifications occur in almost every example ; but we
cannot explain them excepting in particular cases ; and for this
reason we will further investigate the following questions.
127. Question 2. It is required to pay 78 francs in pieces of
6 francs and 3 francs, without any other money ?
Let X be the number of pieces of 5 francs, and y those of 3
francs, we have the equation 5 a? -f- ^ y = 78, which admits, in its
solution, only entire and positive values for x and y.
This equation, resolved with reference to y, gives
78 — 52
y = — 3— ,
or, by performing tlie division,
or rather
y = 26 — a? — ^,
y = 26 — 2^7 + |.
3'
By considering the first form of the value of y, we see that the
value of y corresponding to an entire value of x, cannot be entire
2x .
unless -g- is a whole number ; and since 2 is a prime number in
relation to 3, it is only necessary that x should be divisible by 3.
Let then x=z 2t;
Indeterminate Analyiis of the First and Second Degree. 177
there results
y = 26 — X — 2 ^, or y = 26 — 6 ^
If we consider the second value, we shall see also that x must
be a multiple of 3, which gives a? = 3^,
whence there results
y = 26 — 2 0? + ^, or y = 26 — 5t.
These two formulas show that t must be positive, and cannot
have a value greater than Y, or 5|.
Let then
^= 0, 1, 2, 3, 4, 5;
there results x =z 0, 3, 6, 9, 12, 15,
y = 26, 21, 16, 11, 6, 1.
Thus, we are enabled to satisfy the question in six different
ways, namely, with 6 pieces of 3 francs, without any of 6 francs ;
wiih 21 pieces of 3 francs, and 3 pieces of 5 francs; with 16
pieces of 3 francs, and 6 pieces of 5 francs, and so on.
Question 3. It is required to find a number which, being divided
by 39, gives for a remainder 16, and divided by 56, gives for a
remainder 27.
Call X the entire quotient of the division of the number sought
by 39, 39 <r -|~ 16 is a first expression for this number. Let y be
the entire quotient of the division by 56 ; 56 y + ^7 is a second
expression for this number.
We have then the equation
29x+ 16 = 56y + 27,
or, by reducing,
39aj — 56y=ll (1).
We deduce from this
_ 56y+ll _ 17y + ll
39 "" ^ "T- 39 J
or
• . = ., _ fii-jii = „ _ ii^i).
(We have here taken the quotient in excess, because this ex-
pression presents the factor 1 1 separately in the numerator of
the fraction.)
Bour. dig. 23
178 Elements of Algebra.
As in the expression — ^ ^~ \ the factor 1 1 is prime to 39,
in order that this expression may be a whole number, it is only
necessary that %y — 1 should be divisible by 39.
there results ' 2 y — 39 ^ = 1 (2),
and consequently, a? = 2y— 11^.
Equation (2) gives
i + 1
puttmg —5— = t\
t = 2i' —I,
we obtain the equation
and consequently,
y = 19 ^ + ^.
If in this last equation we subsiitute for i its value in terms of
ff it becomes
y = 19 (2 ^' — 1) + i'j whence y = 39 ^ — 19.
Substituting this value of y and that of i in the expression for x,
we find
a?= 56^—27.
From inspection of these two formulas it is evident that i' may
have any positive value whatever.
Let <' = 1, there results
y = 39 — 19=20, a? = 66 — 27 = 29.
Subsfituting the value of x in the expression 39 07+^6, we
obtain 39 . 29 -j- 1 69 or 1147 for the smallest number which satis-
fies the enunciation of the question.
Again let i' = 2, we find
x=: 56.2 — 27=85;
then
39a7+ 16 = 39.85+ 16 = 3331,
and so on. Moreover, we may verify the enunciation by the two
numbers which we have just obtained.
Ideterminate Analym of the First and Second Degree. 17Q
N. B. The artifice, to which we have had recourse in this
question, supposes some practice, hut we cannot too much recom-
mend the use of it, since it very much abridges the determination of
the values of x and y.
]28. If we compare the formulas which will give all the sets of
values of x and y in the several questions of which we have hitherto
treated, with the equations of these problems, we shall easily per-
ceive that they have this common property, the coefficients of the
indeterminate which enter into these formulas are reciprocally the
same (except the sign of one of the two) as the coefficients tvith
which the unknown quantities x and y are affected in this equation ;
that is, in the value of Xy the coefficient of the indeterminate is
equal to the coefficient with which y is affected in the equation j and
the value of y the coefficient of the indeterminate is equal to the
coefficient of x in the equation^ taken with the contrary sign^ or
reciprocally (as it regards the signs of the two coefficients.)
In order to demonstrate this property let us resume the general
equation
ax -{-hy := c (1),
and suppose that after having applied the method, we have arrived
at the two formulas,
x = mt+A (2),
y = n/ -f B (3).
We begin hy observing that in these formulas the coefficients
m and n must be- prime to each other ; for if they had a common
factor, such tliat, for example,
m = m' A, n=i n' kj
the formulas would become
X =z m'kt '\' A^
y zzzn'kt -f B;
t'
and by making ^ = ?, we should obtain
x=im'tf -{-A, y=in'f + B;
whence it would follow that entire values of x and y would corres-
i'
pond to the fractional value t of t, which would be contrary to
the nature of the method, which proposes that all the indetermi-
nates introduced in the course of the calculation should receive
only entire values.
180 Ekmenis of Algebra.
This being premised, we have seen ulready that equation (1)
must result from the elimination of t in the two equations (2)
and (3).
Now in order to perform this elimination, it is sufficient to mul«
tjply equation (2) by n, and equation (3) by m, and then to sub-
tract one from the other ; which gives
nx — my = nA — mB,
an equation which must be identical with the equation
ax -^^ hy z=z c^
(since m and n, as well as a and &, are prime to each other), and
consequently gives
n = a, 971=^ — !)•
As we can subtract equation (3) from equation (2), we have
also
my — na? = mB — nA\
whence, by comparing it with equation (1),
n = — a, m = 6.
Which was to be demonstrated,
Othervnsc. As the values (2) and (3) ought to verify equation
(1), whatever t may be, we have necessarily
a {m t -{- A) -{- b {nt -{- B) =: Cf
or, by developing and arranging with reference to ^,
{am -}- b n) t -\- a A •\- b B =z €•
But since the supposition of ^ = in the formulas (2) and (3),
gives X z=.A and y = B^ these values inust form a particular set ;
thus we have separately
aA + 6 JS = c;
therefore the preceding equality is reduced to
{am -{- bn^t ::=: 0.
Now, in order that this equality may be satisfied for every entire
value given to ^, we must have
oi»4-Jn = 0: whence — = — r 5
' m 6 '
and since we have already seen that m .and n are prime to each
other, as well as a and i, we must have
i» = •— 6, n = a ;
or rather m = i, n = — a.
Indeierminaie Jlndytit of the Fint and Second Degree. 18t
129. We may give, moreoTer, a demonstration of this property,
which shall be altogether independent of the method which we have
pursued for obtaining the values of x and y.
Let the proposed equation always be
ax + by = c (1);
and suppose that by any means whatever we have found
X =2 a and y =^ 6,
for a first solution in entire numbers (positive or negative) ; then
we say that all the other solutions are contained in the two for-
mulas,
^""^'^IW or T'other 5y = ''T?!'
t designating a whole number altogether arbitrary.
Indeed, since a and 6 form a first set of values for x and y in
entire numbers, we have the equality
aa+h6 = e (2).
Subtracting this equality, member by member, from equation
(1), we obtain,
a{x — a) + h{y — 6) = (3),'
an equation which may be substituted for the proposed equation.
Now equation (3) becomes
a '
and in order that the value of a? corresponding to an entire value of
y may itself beome entire, it is only necessary that b (y — S)
should be divisible by a; but we have seen (122) that the coeffi-
cients a and 6 are prime to each other (otherwise the equation
could not be resolved in entire numbers) ; then it is only neces-
sary that y — 6 should be a muhiple of a.
Let us then put y — s =z at^
and there results x — a = — bt;
and from these two equations we evidently deduce
y =z6 + at,
X "=. a — b t.
As from equation (3) we may also deduce
aix — aS
1 83 Elements of Algebra.
if we put a? — « = J tj
there will result y — ^ = — aty
equations which give
X =^ a + b t,
y =- 6 — aU
It is easy to show that y = 5 + «^> af = o — hi satisfy the
proposed equation whatever be the value of ^. ' .
For, if we substitute them in tiiis equation, we find
a {a — bt •\'l{6 •\- at) = c,
or by reducing, o « -|- 6 ^ 1= c,
since a and 6 give, by hypothesis, a solution of the proposed equation.
130. Consequence. If in the forjnulas
y = 6 + at,
X ■=• a — btj
we make successively
^ = 0, 1, 2, 3, 4,..., and t=i — 1, — 2, — 3...,
they become
y = 6, 6 + a, 6 + 2 a, 6 + Sa...,
X =^ a, a — b, a — 2 6, a -— 3 6 . • • ,
and
y = 6 — a, 6 — 2a, 6 — 3a...,
a? = a + 6, a + 2 6, a4"36... •
Whence we perceive that all the entire solutions, positive or nega-
tive of the proposed question, form two progresnons by difference^
of which the ratio is for the values of x, the coeffirient with which y
is affected in the equation, and for the values of y, the coeffi4:ieni
with which X is affected in the same equation.
131. Another Method. It appears from the analysis given in
article 129, that all the difficulty in resolving completely the equa-
tion ax '{' by =z Cf consists in finding ?i first solution, since we
may afterwards obtain all the others by means of the formulas
X z=i a — 6^, y =z 6 -{- at.
This consideration leads to a second method for resolving an
indeterminate equation. It depends upon the elementary proper-
ties of continued fractions.
For example, let there be the equation already considered in
article (124),
17 jj — 49y = — 8.
Indeterminate Analysis of the First and Second Degree. 183
If we convert J J into a continued fraction, (see note at the end
on coniit^ued fractions^ we obtain the fractions
oil 8 17
TJ T> ¥> T3> TV
Now we know that the numerator of the difference between two
consecutive results is equal to k + l^if the result from which we
suhiract it, is of a place indicated by an even number ; and tok — 1,
if indicated by an odd number.
Then since {^ is of a place designated by an odd number, we
inust have
®-S = -4f>hi' "^«"^« ^^ X ^^-^^ X ^ =-^'
(an equality which may moreover be immediately verified.)
This being laid down, let us multiply the two members of the
equality thus verified, by 8 ; that is, by the second member of the
proposed equation, laken with the contrary sign, it becomes
17 X 23 X 8 — 49 X 8 X 8 = — 8,
or 17X184 —49X64 = — 8,
an equality which is exact, and which does not difier from the
proposed equation, excepting that 184 takes the place of a?, and 64
that of y ; whence we see that the proposed equation is necessarily
satisfied by
a?= 184 and y =^ 64.
This first solution being found, we have (129) for determining
the others, the formulas
a. = 184 + 49 ^, y = 64 + 17 1.
If we only want entire and positive values, we must suppose t
positive or equal to 0, — 1, — 2, — 3. The supposition t = — 3,
gives a? = 37, y = 13, that is, the lowest set of values found in
article 124.
132. In order to generalise the result, suppose the equation to
be resolved is
ax — 6y = c (1),
a and b being two absolute numbers, but c being either positive or
negative.
Let ^s convert t, which from its nature is irreducible (122),
into a continued fraction, and let us form the consecutive results.
1
164 EkiMnis of Algebra.
the last is r, and the last but one may be represented by —, which
gives the relation
a X m' ^^b X m =1 ±: I ;
that is, + ly if the result r is of an even place, and — 1, if this
result is of an odd place.
Suppose, for a moment, that it is of an even place ; we have the
veri^ed equality
a X fnf — h X ft = + I f
let us multiply these two members by c, it becomes
a X fnf c — 6 X mc = c;
a result which does not differ from the equation
ax — 6 y = c,
except that mf c and m c take the place of x and y ; therefore,
a; = m' c, and y z=i mc, form one solution of the equacion.
If the result t is of an odd place, we have
a X ^' — 6x»» = — 1|
whence by multiplying by — c,
a X — Jw'c — 6x-— w»c =*c.
Comparing this verified equality with the equation
ax — J y z= c,
we infer that
a? = — mfCf y =1 — mcj
as a solution.
If the equation is of the form
ax '{'by =: c,
that is, if the two coefficients a and b have the same sign, it may
be mjDdified and written thus,
ax — b X — y = c}
then, by forming, as above, the equality
a X nif c — b X mc =^ c,
or rather
a X — fnf c — b X — mc =• Cj
we may conclude that
0? = m^ c, y = — w» c, or a? = — mf c, and dy z=i mc^
form the solution of the equation.
Indeterminaie^Analyiit ofth6 Fini ahd Second Degree. 1B6
Thus, wbateter be the proposed equatibn, wd lilajr dways, by
means of eontinued fractions, obtain b first eoluiion pf this equation ;
and the formulas
give all the others.
135. Let us apply this method to a new example.
Let the equation to be resolved be
29 a? + 17y = 260,
The fraction f f , converted into a continued fraction, gives for
the consecutive results
ti h h V. ih
Whence we have the verified equality
29 X 7 — 17 X 12 = — l,
(here the result f-f is of an odd place.)
Let us multiply the two members of this equality by — - 250. it
becomes,
29X— 1760— 17 X —3000=260;
but the proposed equatioid may be written thus,
29X00 — 17 X— If 5=250.
Whence we see that »:n — 1750, y = SOO, form a solution.
The formulas becotne.then
x=z — 1760 — nt,
y zs 3000 -f 39 i.
If we wfeh to eonsider duly solutiobd m whole and poi^tive ntiai-
bers, we must suppose t negative ; thus by ebanging the sign of f,
we have
a? = — 1760^ + ift, y == 3000 — 29 ^
and it is evident that the values' of i aild y witt be positive 6ify
while we have
i 29 < < 3000, ^°®°^® U < '«%
or, by performing the division,
t > 102H, but < 103^|.
Then ^ =z 103 is the only value of the indeterminate that makes^^
and y positive.
For t = 103 we find a? £= 1, y =s 13, values which substituted
in the equation give
Bour. Alg. 24
188i . ^ r-^ -,'EleihtenUcJ Algebra.
! g9 X 1 + 17: X, 13^^= 2^ + 221 = 360.
. ! ^ ' ; I
' We see with "what precision the preceding methbd gives all ihe
solutions of the equation.
134. In some circutn^nces ^ first solution m2Lj be obtained
without the necessity of reducing r to a continued fraction.
(1.) If one of the two coefficients a and b is £iX\ exact svhffiultiple
of the known quantity c, the equation gives immediately a first solu-
tion. . .
r or' example, let the equation be 6j? + 3y = 78 ; the coeffi-
cient 3 divides 78 and gives for a quotier^t 26.
Then if we put a? ::;= arid y = 26, the equation is satisfied,
since it becomes > ^
5 X +G'X 26 :^ 78;'
the other solutions are found an the formulas
1. .'' • -y '•■•.•• X = 2t, •' • .
y = 26 — 6 ^
Let there be the equation" ^^ -"
. I6r+d5y = 156.
156 is divisible by 1-2 aiui gives f6r a- quotient i 3, so that a? = 13,
y =5..0i*gi^e ^^ifirst 9et.Jof.mluesiw^d we haye foff the others
a? = 13 — 36 ^, y: =?: 12 i.
(2.) Whenever we perceive fay inspection of the equation, that
the sum or the difference of tbe two coeffideiits a and &, multiplied
reap^K^tiv^ly by two i|umber3) gives a divisor of the second menb^r,
tb^ first aolutioii is obttiined imroe^iiately.. \ .
For example, let the equation be
.^6,^-r.-.16.y:;= 12.
26 X 2— 16 X 3 = 2,
it follows, that .by; multiplying the two membess of* this verified
equality by 6, the quotient 0f 12 divided by 2,
26 X 12— 16X 18= 12,
Whence we conclude . that a? = 1% arid y = 18, satisfy the pro-
posed equalioD. - .'• ...
Again, let there be the equation
it is evidently satisfied by
a? = 0, y = <!•
> . I
• )'
'. ..'..!•;• n I u
I
Indeterminate Anafysis oftheFim and Second Degree. 167
So that the general formulas are ^ ' '^
a? = 47^, y =: \St.
These means of finding a first Solution are peculiar to cer-
tain equations, while the conversion into a continued fraction ^isiA
method by which we are fiiways sure of arriving. at it. . i •
We recomme^nd to beginners to familiarize tberaaelves. equally
with both these methods for resolving the equation a. « -j* iy sx i^*
135. By a mere inspection of the signs of the equation
ocr + iy = c,
we can determine whether the number of isolutibns in whoh and
positive numbers^ is limited or infinite, /
(1.) Whenever b is positive, (a being always supposed, positive,)
the number of solutions is ZtmtVec/.
For, from this equation we deduct ,
c — by
a '
This being laid down, if c is negative, wfiateVer positive value is
given to y, the corresponding value of a? willbe negative ; so thtit
in this case the equation admits of no solution. ' '' '
If cjs positive, we cannot give to y positive values greater than
T, Otherwise x would be negative.
(2.) While bis negative, whatever be the. sign; of. c,rthj^:S^Mapt^Q^
of sQlutions is unlimited^ • ^ ..<. .:. . ;
l^or, the formulas x =: a — bty yzs^-j-a^, becon)^^;}ifrJ^D
we give to b its sign, ^ .
X =: a -^ bM^ y^=^ <S^ + a t.
Now by takinjg the most unfavorable casa, tfaatiiQ^whieh it:and S
are two negative numbers, it is sufficient in order that o^iand y^inay
be positive,.to suppose, for t nu(m6cicai vfli^as: greater dian those of
^ and -. Thus, we may give to t any entire values whatever ij^nifeh
are greater than these two quotients.
In the case where the number of solutions is limited, wb inay
always fix the Irm its between which the values of the indeteriiiiriate ^
r ought to be comprehended, By''dbhsiflerin4'tffe^tipi^tf*f6rmulas,
which in this case are ' '
x:=ia — bt: • " ^ - •:••- '■''
yz=z6 + ai: '' '^ v^w^^u■u.,•^-
188
Ekmeaii of Algebra
For this it is sufficient to express the inequalities
, a — bt>0,
6 + atyo,
and to deduce from them, according to the transformations of two
other inequalities, explained in article 105, in which the two first
members contain onlj i. We obtain thus the whole number of
aobitions of which the question is susceptible.
II. Cf Equations and Problems with Three or more Unknoicfi
Quantities,
136. Let us consider, in the first place, the case of two equa-
tions with fhree unknoum quantities.
For example, let there be the two equations
5x + 4y+ z = 272 (1),
8a? + 9y +3^= 656 (2),
in one of which the unknown quantity z is affected by a coefllcient
equal to unity. We commence by eliminating it.
For this purpose let us multiply tha first equation by 3, and sub-
tract the second from the first ; we have
7a? + 3y = 160 (3),
an equadon which may take the place of equation (2).
Applying to equation (3) the first method, we find the two
formulas
x= 1 — 3 ^,
y = 51 + 7 ^.
Subsdtutiog these two expressions for x and y in the first equa-
tioo, we obtain ,
5(1—30 + 4(61 +70 +« = 273,
or by re^Mcing
« = 63 — 13^.
The three unknown quantities are thus found expressed in an
entire Junction of the indeterminate t. So that by giving to t aiyr
entire yalyj^,w^ever, we shall obtain similar ones for a;, y, and z.;
and these values will satisfy the two proposed equations^ since
according to what we have just ^d tjhe set of three formulas is
equivaknt to two equations*
Indeterminate Analym tf iH Fint and Second D^ree. M9
If we seek entire and positive values for x, y, and Zj it is evident
that t cannot bo positive, for x. would be negative ; but we may
suppose
^=0,-1,-2, ... to ^ = — V, or~7f.
By making
^ = 0, — 1, — 2, — 3, — 4, — 5, — 6, — 7,
we find
a?= 1, 4, 7, 10, 13, 16, 19, 22;
y = 61, 44, 37, 30, 23, 16, 9, 2;
z = 63, 76, 89, 102, 115, 128, 141, 164;
whence it will be seen that the problem is susceptible of e^A^ de-
ferent solutions.
Let us verify only the extreme solutions
/ix^_i .._«i . — flQ „;voi^- 1+4.51 + 1. 63ri= 272?
(l.)x-.l, y^ 51,^ = 63, give I g 14:9.51 + 3. 63:=fl56l
/o\ 00 o i^>i '• .(5.22 + 4. 2+1.164=272.
(2.) x = 22,y = 2,z = 154,give{3 22 + 9. 2 + 3.154 = 656..
137. Let there be, for example, the equations
6« + 7y + 4r = 122 (1),
lia?+,8y — 6-r = 145 (2).
In order to eliminate z from these two equations, multiply the
first by 3, and the second by 2, then adding the results, member to-
member, we have
40 a? + 37 y = 666 (3),
^ equation for which we find, according to the first ipethod,
a? = 37 ^ + 9,
y = 8 — 40 ^
Substitute these expressions of x and y In equation (1), and it
becoiqes
6(37^ + 9) + 7{8 — 40t) + 4z = 122,
or, by performing the calculation and reducing
2 ^ — 29 < = 6 (4).
Here the unknown quantity z is not, like x and y, expressed in
ap entire function of the indeterminate t Sp that it is necessary
tp. apply to equation (4) one of the two known methods.
Wq b^ve fpr the formulas relative to this equs^iipn
;» = 29 ^ + 3.
190 Elements ofAlgeira.
Since, moreover, every entire value of tj substituted in the expres-
sions of X and y, will give similar values for the unkoowa quand*-
ties, it follows, that if we put 2 f in the place of t in these expres-
sions, which give
0? = 74 ^' + 9,
' y = 8 — 80^,
these formulas, combined with
z = 29tf + S,
will comprehend all the sets of entire valuesr of a?, y, and ar, which
can verify the proposed equation.
If we wish for only direct solutions, it is evident that f cannot
be positive, since y would then become negative, and f cannot be
negative, because z and a? would then be negative. But the hy-
pothesis t^ = 0, give% a? =3 9, y .=: 8, ^r = 3 ; this set then is the
only one which can. satisfy the two equations. "
By reviewing the steps which .we have taken, we derive the fol-
lowing general rule ; Eliminate one of the unknovm quantities from
the proposed equations^ and seek for the equation resulting from
elimination^ the two formulas which give the unknovm quantities
which enter into them j in an entire function of an indeterminate
U Substitute these expressions in one of the proposed equations^
^which gives a new equation containing only t and the unknovm
quantity which we have just eliminated. Determine^ for this new
equation, the two formulas which give the expression for the two
unknown quantities which enter into it, in an entire function of a
second' indeterminate V. Lastly, substitute (he expression fot'iin
those of the two first unkndwri quantities. The values of the three
unknown quantities are thus expressed in an entire function oi f }
after which it is only required to determine for t' the limits between
which these values must be found, in order that the principal luir
known quantities may b^ entire and positive, ■
N. B. Wlienever one of the unknown quantities has unity for
its coefficient in one of the equations, it is more simple^to elimi-
nate this unknown quantity, because that after having expressed
the two other in an entire function of the same indeterminate, if
we' substitute these values in the equation in Which the third un-
known quantity is affected with a coefficient equal to unity, we
obtain immediately this third unknown quantity in an entire func-
tion of the same indeterminate, so ^ that in this case only one opera-
Indeterminate Jlndyiis of the lint aiad Second Degree. 191
tion is necessary. The two equatioos of article 136 aflbrd an
example of this.
138. The following are the steps which we must take for three
equations with fonr unknown quantities. After having elimiriaied
one of the unknown quantities, we express by means of the two equa*
tions tokich result^ and according to what we have just laid down^
the three other unknown quantities in an entire function of the
same indeterminate^ and substitute these values in one of the prO'
posed equations. If in the new equation^ the coefficients of the two
unknovm quantities which enter into it, differ from unity, we estab-
lish two formulas which give these unknown quantities in an entire
function of a second indeterminate, we then substitute in the express
sions for the three first unknown quantities the first indeterminate
in a function of the second, and we thus obtain the four primitive'
unknown quantities in an entire function of the second indeter^
minate.
The reasoning is the same for four equations with five unknowrr
qyantitie3^ and so on. We propose the following questions as an
exercise for the learner.
Question 3. A coiner has three kinds of silver. In 8 ounces or
1 mark, the first contains 7 ounces of pure silver, the second 6^
ounces, and the third 4^ ounces. He wishes to make a mixture of
30 marks in weight, which contains 6 ounces of pure sitver for
every 8 ounce weight. How many marks (in whole numbers)
ought he to take of each sort ?
(x= 10, 12, 14, 16, 1%
Answer. ^ y = 20, 15, 10, 5, 0,.
(^= 0, 3, 6, 9, 12^
that is, 5 solutions by admitting for the value of y and z.
Question 4. It is required to find three whole nunAers euch
that the sum of their products by the respective numbers 3, 6, and 7,
may be equal to 560, and the sum of their products by the squares
rf these nt^ers9, 25, and 49, may equal 2920.
Cx= 15, 50,^
Answer. ^ y = 82, 40, > that is, two solutions.
Cx= 15, 50,)
•.^y = 82, 40, V
(z = 15, 30,)
Question 5. li is required to find a number N, which being
divided by II ^iM remainder is. 3, being divided by 19, gives the
remainder 5, and difJidedby^Of gives the remainder 10 ?
Answer. JV*=: 412Q + 6061 1, so that 4]$i8 is the smallest
number that satisfies the enunciation.
litt^ EUmentio:
QvesfioQ 6« Tojmdsuck a value ofx that the eccprunons
3x — 10 llx + 8 16x — 1
7 ' 17 * 6 '
laii^ he whole wamiert^
Armoer. ar = 211 -f ^^^^5 ^ being an indeternnaale.
139. If in the sixth question we designate the quotients
3a; -^10 ll2; + 8 16 g--^!
7 ' 17 ' 6 *
by 3^, Zf and v, we have for the equations of the problem
3a?— 10= 7y; lla? + 8 = 17;r; 16a?— I = 6v;
or 3a? — 7y = IO5 lla?— Har = — 8; 16a?— 5t;r= 1.
We must then apply to these equations the process laid down in the
preceding article for three equations with four unknown quantities.
But we will now proceed to develope another more simple method
for determining the value of a?, which is here the principal unknown
quantity. This method is moreover applicable to all questions of
the same kind.
In the first place, if we consider the third expresHOO r — f
X — 1
, it may be reduced to 3 a? -| ? — ; so that in order that it should
be entire it is ooly necessary that a? — - 1 shoidd be a multiple of &•
Let ua put
» — 1
— g — = t^ there results a? = 1 + 6 ^
Evei^ entire value of t, substituted in this formula, will give for a? a
number which will satisfy the third condition of the enunciation.
Let Qs now substitute this value of x in the first expression
3x — 10 . ,
= — ; It becomes
1^— , or *^— 1 +^;
whence we see that this new ezpression:- will be entife if we sUp^
pose t =z7 f, moreover this condition is necessary. So that in
order that the first and (bird expresisioos should be entire, it is
necessary that we should have a? = 1: -f- ^ ^9 ^ being of the form
t x: 7i^ ; which gives a? =: 1 -{. 36 f'.
Let ua carry this new value into the second expression
llai + 8^ . , • 885f + 19
— ly — J; It becomes — 7-jy» >
hideterminate Analysis of the First and Suond Degree. Id5
or
Now we have
2 — 6^=2 (1 —Zf)y
moreover 2 is prime to 17 ; then in order that the second member
may be a whole number, it is only necessary that 1 — 3 f should
be divisible by 17.
Putting
we deduce from it
1-17 <".
^ — a '
or, bj performing the dirision,
\
Let
<" 4- 1
<" 4- 1
* T^ * 4111
3 ^^ '
we obtain ^' = 3 if" — 1^
whence we deduce
^ = — et'' + ^'', or fz^^n If" + 6.
Carrying this value into the expression
a? = 1 + 36 1',
we obtain, all reductions being made,
a? = 211 —b9bif".
Such is the formula which will give all the values of x capable
of satisfying the enunciation.
Let i"' = 0, we find a? == 211 ; this is the smallest of all the
numbers sought. By supposing for i'" any negative, valtie8:'We
shall obtain other solutions.
N. B. We remark that 595, the coefficient of f" in the for-
mula, is the product 7 X 17 X 5 of the denominalors of the thfee
proposed expressions. It will be easy to take account of this pr<>-
perty, which is modified when the denominators are not prime 4p
each other ; for, in this case, the coefficient is equal to the most
simple multiple of the denominators.
Bimr. Alg. 25 '
194 MSenmis ^.AlgAra.
140. It still remains to speak of the problems which are said to
be more than indeterminate^ that is, of problems for which the num-
ber of equations is les^ by two or mx>re units than the number of
unknown quantities.
In the first place, let there be the equation with three unknown
quantities
««+ fry + Cj» = d.
If we transpose the term c 2: to the second member, it becomes
, ax -^ by z= d — c^:, oraa? + iy = C,
(designating by & the quantity d — ez, which for the present we
consider as known.)
This being premised, we establish for the equation
'ax -^ by =z </,
the two formulas x =z a — bt, y ts^ 6 ^ at. After this we
restore in « and 6, for ;€^ its value d — cz; then x and y are
found expressed in an entire function oj the iiideterminate and of
the third unknown quantity z.
'Let it be proposed, for example, to pay 187 francs unth pieces
of 5 francs, 6 francs, and 20 francs, without any other coin.
Let us designate by £, y, and z, the number of pieces required
of each sort ; we have the equation
5«-f 6y + 202f =5: 187,
which is reduced to
5x + 6yzsiet — 20z=2&.
Deducing from this equation the value of x^ we have
• 1 • *
C — ey
*= 6 '
or
1
Fattiaf'
• • ' * •
6 -'•
W9 deduce from it
y=«'-^sr,
whence
x — — &4-6t.
< » rf
Restoring for & in these formulas its value 187 — 20 z, we find
'y= 187 — 20 ;r — 5^,
a? == — 187 + 20^+ 6^.
Inieterminaie Awdym cftke Firrt md Second Degree. 195
So long as we admit for x and y whole mimbers, posttiire or
negative, we may give to z and t values altogether arbitrary, but if
we wish to satisfy the enunciation directly^ the same form of the
proposed equation
6a? + 6y + 20«= 187,
shows that z cannot receive values greater than V/, or 9^\, sinc^
otherwise x or y would be negative.
Let us put successively
Z m U, 1, «, «5, • • • • • o, t7*
If we make ;; = 0, the values of x and y become
a? = —187 + 6^,
y= 187 — 6^5
formulas which show that t must be > *|^, but<^ ^f'', or> Sl||
but <^ 37| ; then i can have six values, nameljr,
32, 33, 34, 35, 36^ and 37.
Thus, tor z zsi Of we have
t = ^2, 33, 34, 36, 96, 87,
0?= 6,11,17,23,29,35,
y = 27, 22, 17, 12, 7, 2.
Let 2; =: 1, we find
a? = —167 + 6^,
y = 167 — 5 ^ ;
whence < > *|% or 27|, but < *|7, or 331, which gives these
six values,
28, 29, 30, 31, 32, and 33.
Thus, for « = 1 , we have
i = 28, 29, 30, 31, 32, 33,
« = 1, 7, 13, 19, 26, 31,
y = 27, 22, 17, 12, 7, 2-
For z = 2,
For « = 3,
t = 25, 26, 27, 28, 29,
« = 3, 9, 15, 21, 27,
y = 22, 17, 12, 7, 2.
i = 22, 23, 24, 25,
a: = 5,11,17,23,
y = 17,12, 7, ».
196 EkmeiUi pfJUg^u^
For 2r = 8y the formulas would be
ir = — 27 + 6/,
y= 27 — 6/;
whence / > V or 4^, but < y or 5|. So that i can only receive
the value t = 5, which gives cr = 3, y = 2.
Finally, from the hypothesis 2r = 9, there is no solution, for the
formulas become
0? = — 7 +6/, y = 7 — bt,
whence t^iox 1^, but / < | or If.
141. It will thus be sufiBciently plain what is necessary to be
done in the case of two equations with four unknown quantities,
and three equations with five unknown quantities. But, we will
give a complete resolution of one question of this kind in order to
dk>w, how, by the aid of some particular considerations, we may
frequently simplify the calculations.
Question 7. JL farmer bought a hundred head of cattle for
100 doUarSj namely , oxen at 10 dollars each^ cows at 5 dollars^
calves at 2 dollars j and sheep at 1^ dollar mch How many of
each did he purchase ?
Let X, y, Zf Uf be the numbers sought, we have the equations
*+ y+ ^+ ** = 100,
10 a? + 5y + 2z+ lu =: 100,
0^9 reduced,
a?+ y + z + u = 100,
20a? + 10y-|-42f4-t* = 200.
By subtracting the first equation from the second, we obtain
I9x + 9y + 2z = 100,
an equation which must be treated in the same way as that of the
preceding article. But in the first place, we observe that it is
preferable to express y and z in an entire function of a?. (1.) Be-
cause it is evident that x ought not to have values greater tiian
y/ or 5j%. (2.) Because the coefficients of y and ^r have a'
common factor, which necessarily introduces 1 condition for deter-
mining the proper values of <r. -.
This l^eing premised, let us transpose the term' 19 a?, it becomes
9y + Sz = 100— 19 a?,
or rather
« J 100 — 19x
IndeterminaU Awdgns of the Second Degree. lOT
Now, since we require for c, y, x, u, positive whole numbers, it Is
necessary that ;t should be whole and positive ; but evi-
deotly 0? = i and a? = 4, are the only values which can satisfy
this twofold coudition. So that already we can only have the
values 07 = 1 and x =z 4,
Let 0? = 1, there results
3 y + ar = 27, or ar = 27 — 3 y.
Substituting these values of x and z in the first equation propos-
ed, we find u = 12 + 2y.
The first of these two formulas shows that y cannot be > 9 ; sO'
that for 0? = 1, we have
yz= 0, 1, 2, 3, 4i 6, 6, 7, 8, 9,
;r = 27, 24, 21, 18, 16, 12, 9, 6, 3, 0,
u = 72, 74, 76, 78, 80, 82, 84, 86, 88, 90.
Let 0? = 4, it becomes
3 y + ^ = 8, whence z =z 8 — 3y,
and u = 88 4- 2 y.
This expression for z shows that y cannot be ^ 2 ; ^o that tor
0? = 4, we find
y= 0, 1, 2,
z =i 8, 5, 2,
tt = 88, 90, 92.
Whence we see that the proposed question is susceptible only of
thirteen and of ten solutions, if we except the solutions of 0.
III. Cf Indeterminate Analysis of the Second Degree.
142. We propose, in this part, as in Indetermine Analysis of
the First Degree, to resolve, by entire numbers, problems, wliich
lead to a number of equations less than the number of unknown
quantities. But as, in general, an equation of the second degree,
with two unknown quantities, gives one of them in an irrational
Junction of the other, it follows that the question consists, (1-) '°
determining, for one of the unknown quantities, rational Values^
which have the property of giving similar ones for the second.
(2.) To choose among the values of the first unknown quantity the.
entire values which. >will give similar ones for the second. We per-
ceive, therefore, that indeterminate analysis of the second dqgred
196 Ekmenis o/J3]^^a,
mfast present greater difficulties than that of the first d^ree. It is
indeed one of the most difficult theories of algebraic analysis, and
would lead us aliogeiher beyond the elements' of the science. We
refer the student, for further information, to the Tkeorie des JVom-
bres of Legendre.
We will explain, however, the resolution, in entire nun^bers, of a
series of questions with two unknown quantities, whose equations
contain only the rectangle or product of the unknown quantities^
without including any of the two squares.
These questions, which are in themselves very curious, are takea
from the Algebra of M. Lhuillier, a work from which we have
already taken the enunciations of several problems.
143. Question 1. It is required to find in entire numbers the
sides of a rectangle, whose surface contains four times as many
square feet as its perimeter contains feet.
Let X and y be the sides of the rectangle expressed in feet, a?y
expresses its surface, and 2x -{- 2y \is perimeter.
From the nature of the enunciation, we have the equation
a?y = 8a? + 8y.
From this equation we deduce
0? = ^— Q, or a: = 8+^^—^.
From the form of the value of x and y, it is evident that for ai>
entire value of y, we cannot obtain a similar one for a?, except when
y — 8 is a divisor of 64.
Let us suppose then, that we have determined all the divisors of
64, and that we take them with the signs -f- and — , we shall have
y— 8=1, 2, 4, 8, 16, 32, 64, | —64,— 32,— 16,— 8,— 4,— 2,-1 ;
whence
y=9,10,12,l6,24,40,72,|— 56,— 24,— 8, 0, 4, 6, 7 ;
Substituting in the expression for a?, instead of y — 8, its differ-
ent values^ and reducing, we shall find
» = 72,40,24, 16, 12,10,9,1 7, 6, 4, 0,'— 8, — 24, — &6.
If we look at the last two lines of the calculation, we shall see that
the values of x are the same of those of y, but in the inverse order,
which ought to be the case, since the equation of the problem does
not change when we put x in the place of y, and ibe reverse.
Whence we conclude, leavtog out of considerauoii negative soksH
tionsy that tbe sets of values realty diHexent are
IndeterminaU ^Sndb/sii cf <Ae Second Degree. 199
y = 9, 10, 13, 16,
X = 72, 40, 24, 16,
Thus the qucslioD is susceptible of 4 solutioos.
Let us %'erify ibe set y = 10, a? = 40.
The base of the rectangle contains 40 feet, and its altitude 10,
and its surface is 400 square feet. On the other hand its perime-
ter is equal to 2 (40 + 10) or 100 feet ; now 400 is quadruple of
100.
Let us generalise this questions and propose to * ourselves to
determine a rectangle whose surface contains m times as many
square yards as its perimeter contains yards.
We have the equation
xy z=:m (2a?/+ 2y) =: 2 ma? + 2»y*
Whence, deducing the value of x by performing the division,
a? = 2 w H X—
* y — "^
We remark, in the first place, that it is useless to take any notice
of the negative divisors of 4 m® ; since if y — 2 m is negative and
numerically smaller than 2 m, y is then positive ; but then when
" ■- ^ is negative and numerically greater than 2 m, the corres-
ponding value of X is negative. The contrary would take place if
y — 2 m was negative, and numerically greater than 4 m^ ; that is,
y would be negative and x would be positive. Now it is only pro-
posed to admit direct solutions of the question.
This being premised, let rf, d'^df^\ d''\ be the divisors of 4 m*,
we have
y— 2m = d, d^d^d''';
whence
y = 2 m -I- d, 2 m + rfS 2m -I- d'^ 2 m + d*",
and consequently, by designating by j, J^ J?''? j'^S the entire quo-
tient of 4 m* by d, d\ d*\ d''',
X =2m + q, 2 m 4- g', 2 m -f j^', 2 m -|- y''',
The number of solutions at first appears equal to the number of
the divisors of 4 m^ ; but as the equation is symmetrical* in a? and y,
* We call a symmetrical function of two or more quantities any
expression which contains these quantities combined in the same
aiuin^; th%t is, sudi that when we change these quantities the one
into the\other,.tlw ;exprGis8ioB does not change «xfl^t kk the order of
the terms.
SOO EkmentiofJ^thra.
the values of t, deduced from the equation first resolred with refe-
rence to y, will be the same as those of y taken in the inverse order.^
Thus, the whole number of distinct solutions is really equal to only
half the number of divisors, if this number is even, and to half the
number plv^s unity i( h is odd.
As a second application, let
m = 3, whence x = 6 H — '■ — -^ ;
seeking the divisors of 36, we have ,
y_6= 1, 2, 3, 4, 6, 9,12,18,36;
whence y = 7, 8, 9, 10, 12, 15, 18,24, 42.
-^ = 36,18,12, 9, 6, 4, 3, 2, 1.
Then x = 42, 24, 18, 16, 12, 10, 9, 8, 7 ;
"which gives five distinct solutions.
144. Question 2. Having given the side Rofa square, to find
in whole numbers the sides of a rectangle whose perimeter shaU be
to that of the square in the same ratio as their surfaces.
Let X and y be the sides of the rectangle ; 2 (j? -j~ y) ^od x y
represent it8^ perimeter and its surface. Moreover, * 4 a apd a^
express the perimeter and the surface of the square ^ so that we
have the equation
X y _ 2 (xjfy)
or simplifying 2 a? y = a (a? + y)-
This may present two cases, when a is an even and when it is
an odd number.
(1.) If a is an even number and equal to 2 a', it becomes, by
suppressing the factor 2,
xy = a' (a: + y), whence a? = a' -j 7,
y —— a
and the question returns to the preceding.
(2.) If it is an odd number, we have
ay a c^
^ =27=-^' or a: = g + 2^2y_„).
, In ord^r that this expression of x may be entire, it is evidently
only necesaary that:2y — a should be a divisor of a*i,v
Bydesignatiiig by d, d*, d"s these divisors, and by yy ff,'^f^'^iam
quotients*of the division of €? by d, d\ df^, (the quantitiei^ i, d'^ d'^j
Indetepninate AnahfsU of the Stamd Degree. 301
9' 9^9 9^^ are necessarily odd numbersj since by hypothesis a is an
odd number), we shall have
2 y — a = rf, d\ i"^ odd numbers ;
whence
y = — 2 — J — 5 — > — 2 — > entire expressions.
^y_a ~ S'j J'j S^'j o^d numbers,
and consequently, ,
a 4- 9 « + 9' « + 9"
a? = — ^-^, — jr-i , — 2" > entire expressions.
In the first place, let a = 20 ; the equation is
2a?y =20(a? + y),
or, dividing by 2, and resolving with reference to a?,
seeking the divisors of IOC, we find
y— 10 = 1, 2, 4, 6, 10, 20, 25, 60, 100;
whence
y = 11, 12, 14, 15, 20, 30, 35, 60, 110;
100
yZTTo == ^^^' ^^' ^^' ^^' ^^' ^' 4, 2, 1 ;
then
a? = 110, 60, 35, 30, 20, 15, 14, 12, 11,
which gives fine, different solutions.
^ In the second place, let a = 15; we have the equation
2a?y = 15 (a: + y) ;
whence we deduce
15y 15 , 225
^ = 27^T5' or ^ = -2" + 2(2y-15)-
Seeking the divisors of 225, we obtain
2 y— 15 =2 1, 3, 5, 9, 15, 25, 45, 75, 225;
whence
y, = 8, 9, 10, 12, 15, 20, 30, 45, 120;
225
5^-—, = 225, 76, 46, 26, 15, 9, 6, 3, 1 ;
Bour. Alg. 26
202 JBSemenU ofAlgAra.
then
X = 120, 45, 30, 20, 16, 12, 10, 9, 8;
ID all, five d^erent solutions.
145. Question 3. Havmg given the side a of a cube j in a whole
number, it is required tofin^f^ also in whole numbers^ the side of the
base and the altitude of a rectangular par allelopiped, with a square
base, such that their solidities may be to each other as their surfaces.
Let X be the side of the base, and^y the altitude of (his parallel-
epiped, a^y and 2 a^ -^ 4xy will represent the solidity and the
surface of this solid ; moreover, a^ and 6 a^ are the expressions for
the solidity and the surface of the given cube ; we have then the
equation •
2z^ + 4xy "" 6aa'
or, by freeing from denominators and reducing,
^^yzziax-^-Zay.
From this equation we deduce
®""3y — a"" 3 +3(3y — a)'
or, 3 a? = 2 tt -|-
3y — a*
This being premised, let us designate by d^ df^ $\ eU the divi-
sors of 2 a', and let us put t
3y— a = <?, i, d''\
there hence results
a'\- d a-^d* a4-e{^'
^ = "T^' -3—' —3 —
Moreover, let y, j', j'^, be the quotients of 2 a* by d, d', d"^ we
have
2^3
S^IZTa = ?'?'' J"'
whence
3aj = 2a-|-g, 20 + 3^, 20 + 9^'.
Then
2a + g 2a + g' 2a + 9"
« = — 3— , —3 , g--2-.
By taking account of entire expressions only in ibfi two series of
values of x and y, we shall obtain the sets of values in whole num-
bers which are capable of verifying the equation.
Formation ofP^w^rs and the Extraction of Roots. 308
For example, let a =: 8, the equation becomes '
128
seeking the divisors of 128, and making them equal to 3 y •*<-^ 8^
we have
3y — 8 =5 1, 2, 4, 8, 16, 32, 64, 128,
whence
371-8 = ^^^' ^^' Q' ^5
then 3 a? = 16 + 128, 16 +|[32, 16 + 8, 16 + 2,
and consequently,
a? = 48, 16, 8, 6.
Thus, this particular equation admits of the sets
X = 48, 16, 8, 6,
y = 3, A, 8,24.
We propose for a new exercise the equations
(2.) 8 a?y = 6 0? + 5]y + 12 ; the only set, y = 6, cr = J .
(3.)4cry = 3a. + 2y-12 {y^e.'J
(4.) The general equation jwa?y = aa? + ^y + <^'
Question 4. It is required tojind, in whole numbers, the rectan-
gular parallelopipeds with square bases, such that their solidities
may contain Jive times as many cubic feet as their surfaces contain
square feet.
Side of the base, x =z 220, 120, 70, 60, 45, 40, 30, 28, 25, 24, 22, il,
altitude y = 11, 12, 14, 15, 18, 20, 30, 35, 50, 60, 110, 210.
Twelve solutions.
Of the Formation of Powers and the Extraction of Root9
of any Degree whatever.
Introduction. — As the resolution of equations of the second
degree supposes the process for the extraction of the square root
ahready known, so, the resolution of equations of the third and
S04 Elements of Algebra.
fourth degrees requires that we should know how to extract the
roots of the third and fourth degrees of a quantity, either numerical
or algebraic. (See article 2, for the definitions of the words,
power and root,)
The raising of powers, and the extraction of the roots of every
degree, and the calculation of radicals, form the principal subject
of this section, which, with tlie first and a part of the third, com-
prehend together the operations which we may have to perform
upon numbers expressed algebraically.
Although any power of a number may be obtained by means of
the rules of multiplication, either arithmetically or algebraically,
nevertheless this power is subject to a law of composition which
must be known if we would return from the power to its root.
Now as the law of the composition of the square of any quantity,
ehher numerical or algebraical, is founded upon the expression of
the square of a binomial, so the law relative to a power of any de-
gree whatever is deduced from the expression of the power of a
binomial of the same degree. It is then by the determination of
the developement of any power of a binomial^ tliat we ought to
commence this new theory.
I. Binomial Theorem of JSTewton and the Consequences which are
derived from it.
146. If we multiply the binomial x + a several times by itself,
we arrive at the following results.
(x + ay zzix -|- a,
{x + af=:a?^ + 2ax + a^
(r + a)« = x3+3aa:»+ 3a^x + a',
lx + ayz=:x^ + 4ax^+ 6a^a»+ 4a^x+a\
(x + ay=zx^ + bax^+l0a^z^ + l0a^x^ + 5a^x + a^.
By looking at these different developements, we easily recognise
the law according to which they proceed, as it regards the expo-
nents of X and a ; it is not the same with respect to the coefficients.
But Newton arrived at a law, by means of which, the degree of the
power being given, we are enabled to form this power of a binomial
without being obliged to go through the inferior powers. He has
left no traces of the reasoning by which he was led to this conclu-
sion, but the truth of the law has since been established in the most
Formation of Powers and the Extraction ofRooti^ 205»
rigorous manner. Of all known demonstrations, the most elemen-
tary is that which is founded upon the Theory of CombinatiQns^
Still, as it is somewhat complicated, in order to simplify the expla-
nation, we shall begin by resolving some problems relative to com-
binations, from which it will be easy to deduce the binomial for^
mula, or the developement of any power whatever of a binomial.
147. Preliminary conniderations. We know already (Lacroix's
Arithmetic, article 27), that the product of a number n of factors
a, 6, c, d, is not changed in whatever order we perform their
multiplication. Now it is proposed to determine the whole number
of w'c ys in which these letters are capable of being placed by the
side of each other. The results answering to the changes ihua
made among these letters, are called permutations.
For instance, two letters a and 6, which give only one. product
ah, furnish two permutations, ab and b a.
In like manner, the three letters a, i, c, give only one product,
ahcy but furnish six permutations,
a be, acb, c ab, b ac, boa, cba,
Let there be now a number m of letters a,b,c, rf, . ,. ; if we* place
these one after the oiher, 2 and 2, 3 and 3, 4 and 4, in every pos-
sible order, in such a manner, that in each result the number of
letters may be less than the number of letters given, and the whole
number of the results thus obtained is sought. These results are
what are called arrangements.
Thus, ab, a c, a d, b a, b c, b d, c a, c b, c d, . . , J ^ve the arrange-
ments of m letters, taken 2 and 2.
Also, ab c, ab d, b ac, bad, a cb, acd, . ,, , are the arrange*
ments, taken 3 and 3.
Finally, when we have thus disposed the letters by the side of
each other, 2 and 2, 3 and 3, 4 and 4, we may require that n6 two
results thus formed, shall be composed of the same letters, that is,
that they shall differ from each other, at least by one letter, and we
may seek the whole number of results which can be obtained in this
way. In this case the results take the name of comiinations.
Thus, ab, ac, be, ad, bd, , . , , are combinations of 2 and 2,
in which any two of the results differ at least in one of the letters.
. Also ab c, ab d, acd, be d^ , , . , are combinations of 3 and 3.
There is then an essential distinction iri the signification of the
words permutation, arrangement, and combination.
206 EUfnenis of Algebra*
We give the name permutation to the remits which we obtain
hy dispaeing one after another^ and in all possible orders^ a deter-
minate number of Utters^ in such a manner that all the Utters enter
into each result^ and that each letter is found in this result but once*
The name arrangements is applied to the results which we
obtain by placing one after another, and in all possible orders^ a
msmber m of letters, taken 2 and 2, 3 and 3, 4 and 4, . . . n and n,
m being ^ n, that is, the number of letters in each result being
less than the whole number of letters made use of. If, however,
we suf)pose mzizn, the arrangements n and n become simple per-
mutations.
Finally, we. call combinations, the arrangements in which any
two differ from each other at least in one of the letters which com-
pose ihem»
It is important that the learner should be well initiated in these
definitions in order to understand clearly the resolution of the fol-
lowing pr')blems.
148. Problem 1. To determine the whole number of permuta-
tions of which n letters are susceptible.
In the first place, two letters, a and 6, evidently give two permu-
tations, a b and b a. Thus, the number of permutations of two
letters is 2, or i X 2. .
Let there be three letters, a, 6, c. Take any one of these letters,
c, for example, and write it on the right of the two arrangements
a 6, and 6 a, which the two letters give, and there results two per-
mutations of three letters, ab c, bac. But as we can take sep-
arately either of these letters, it follows that the whole number of
permutations of three letters is equal ^o 2 X 3, or 1 X 2 X 3.
In general, let there be a number n of letters, a, 6, c, d, and let
us suppose the whole number of perjnutations of n — 1 letters
already known, and let this number be designated by Q.
Let us consider separately one of the n letters, and write this
letter on the right of each of the Q permutations which the n — 1
other letters give ; there will result Q permutations of n letters,
terminated by the letter which we at first insulated. But as we
can thus take separately each one of the n letters, it follows that
the whole number of permutations of n letters is equal to Q X w.
Let n = 2, Q will then designate the number of permutations
that a single letter can give; then Q= I, and we have in this
particular case, Q x » = 1 X 2.
Formation of Powers and the Extraction of Roots. 207
Let n = 3, Q will then express the number pf permatations of
3 — 1, or of 2 letters, and is equal to 1 X 2. So that Q X n is
reduced to 1 X 2 X 3.
Farther, let w = 4, Q will designate in this case the number of
permutations of 3 letters, and is equal to 1 X 2 X 3. Then
<2 X » becomes 1x2x3x4.
N. B. We see then, that the formula Q X w includes all the
particular cases of the proposed problem. By observing the above
reasoning we can immediately determine the general case, and
thence deduce at the same time all the particular cases.
149. Problem 2. A number m of letters^ a, b, c, d, being given
to determine the whole number of arrangements which can be formed
with these ni letters^ taken n at a timCy m being svpposed greater
than n.
To resolve at once this general question, let us suppose already
known the whole number of the arrangements which can be formed
with the m letters, n — I at a time, and let us designate this num-
ber by P.
Let us consider any one of these arrangements and set down at
the right hand each of the letters which do not enter into this
arrangement, and of which the number is necessarily
m — (n — 1) or m — n + 1 5
h is evident, that we shall thus form a number m — n -f- 1 of
arrangements of n letters, each differing from the others by the last
letter.
Let us consider a new arrangement of n — 1 letters, and write
at the right hand the m — n + 1 letters which do not enter into this
arrangement, we shall still obtain a number m — n + 1 of arrange-
ments of n letters, differing from each other, and from the preceding
arrangements at least by the disposition of one of the n — 1 first
letters. Since we may consider separately each of the P arrange-
ments of the letters, taken n — 1 at a time, and write successively
at the right hand the m — n '\- I other letters, it follows that the
whole number of arrangements of m letters, taken n at a time, is
expressed by
p^rn — n+ 1).
To find now, as a particular case, the whole number of the
arraDgecoeots of m letters, taken 2 and 2j 3 and 3, 4 and 4.
306 MemenU of Algebra.
Take « = 2, whence we have m — n+l=wi — 1; P ex-
presses, in this case, the whole number of arrangements taken
2 — 1 at a time, or taken one at a lime, and is consequently equal
to m; then the formula becomes m[m — 1).
Let 71 ^= 3, whence m — n+ 1 =»i — 2;P then expresses the
number of arrangements, taken 2 and 2, and is equal torn (m — 1);
then the formula becomes m{m — 1) (w — 2).
Further, kt n = 4, whence m — ,n4- 1 =»w — 3;P expresses
the number of arrangements, taken 3 and 3, or is equal to
m{m — I ) (»i — 2) ;
then the formula becomes m{m — I ) (»* — 2) (»i — 3) ; and so on.
N. B. According to the manner in which particular cases have
'been deduced from the general formula P {m — » + 0> ^® ^^^1
conclude that this formula developed, becomes
m(m— 1) (m — 2)(m — 3) (m — n+ 1);
that is, it is composed of the product of the n consecutive and
decreasing numbers, which are comprehended from m to
m — (n — 1) or m — n+ 1, inclusive.
This being laid down, it is easy to deduce from this formula
developed, the formula of the preceding article, that is, the value of
Q X w also developed.
We have seen (147) that the arrangements become permuta-
tions when we suppose the number of letters which enters into each
arrangement equal to the whole number of letters considered.
Thus, to pass from the whole number of arrangements of m let-
ters, taken n at a time, to the number of permutations of n letters,
we have only to make, in the above developement, m = n ; which
gives
n(n— l)(n — 2) (n — 3) 1,
Reversing the order of the factors and observing that the last factor
being 1, the last but one is 2, the next 3, we have
1.2.3.4 (n — 2){n — l)n
for the developement of Q X n.
This is only the natural order of the numbers, comprehended
between 1 and n inclusive.
150. Problem 3. To determine, the whole number of different
combinations which can be formed with m letters taken n at. a time*
JFamuition ofFo^wen and tAe Emirudion ofRooU. SM
Let us designate by X the whole number of arrangements whiclji
can be formed with m letters, taken n at a time by Y, the number
of permutations of which n letters are susceptible, and by Z thi^
whole number of different tombinations^ taken n at a time, the
number which it is required to determine.
It is evident, that in order to obtain all the possible arrttngdments
of m letters, taken » at a time, it is sufficient to give to the n letters
of each of the Z combinations, all the permutations of which these
letters are susceptible.
Now a single combination of n letters gives by hypothesis Y
permutations ; then Z combinations of n letters must give Y,xZ
arrangements, taken n at a time; and as we have already designated
by X the whole number of the arrangements, it follows that th^
three quantities X, F, Z, are connected by the relation
whence we deduce Z = ^
But we have found (149)
X = P(m — n+1),
and (148) F = Q X n.
Then, finally,
QXn ~Q^ n
Since P expresses ihe whole number of arrangements, taken
It — I at a time, and Q expresses the whole number of permuta-
p
tions of n — 1 letters, it follows that ^ expresses the number of
different combinations of m letters, taken n -— 1 at a time.
Accordingly, if we wish to find as particular cases, the number of
combinations 2 and 2, 3 and 8, 4 and 4 • • . }
let us make » = 2, in which case ^ expresses the oitmber of
combinations, taken 2 — 1 at a time, or 1 at a time, and is equal to
m ; the formula above becomes
^ m — 1 m(m — 1)
. . P
Let us make n = 3, in which case^^ expresses the number of
Bout. Mg. 27
910
Ekmnkj^AlgAra.
m{m — 1),
the formula be-
combinations, 2 and 3, or is equal to
■
comes
m {m — I)«(to-— 2)
1.2.3 '
We may find, in the same manneri
m(m— 1) (m~2)(iii — 3)
1.2.3.4
for the number of combinations, 4 and 4, &c., and in general, for
the numUer of the combinations, n and n, we have
ffl(wt — l)(m — 2) (m — 3)...(m— ■!»+ 1) ^
^1.2.3.4.5...n — 1?» '
that iS| the expression
— ^-^ — *" - developed.
151. Demonstration of the binomial formula. In order to dis-
cover more easily the law of the developement of ihe m** power of
the binomial a? 4" ^9 ^^ shall begin by observing the law of the
product of several binomials
iC + a, a? + &, a? + c, a? + d, . . .
having a first term common, and whose second terms are different*
cc + b
X -{- ab
1st prpduct
a^ -{- a
+ h
a; + c
2d product
^3d product
0? -|- a
--6
+ c
X '\' d
— ac
+ bc
X + abc
0?* -|- rt
x^ + a 6 c
4- bed
X + a6cd
" ac
--ad
-be
"bd
+ cd
These multiplications being performed according to the common
irules of algebraic multiplication, we discover by the three preced-
ihg products the following law ;
Formation of Powers and the Bwtraetion of Roots. 2lit
(] .) With respect to the exponents, the exponent of « is eq^al
to the number of \he binomial factors. This exponent goes on
decreasing by unity in each term till the last, where it is equal to
zero.
(2.) With respect to the coefficients of the different powers of Xf
the coefficient of the first term is unity ; the coefficient of the
second term is equal to the sum of the second terms of the bino-*
mials ; the coefficient of the third term is equal to the sum of the
different products of these same second terms mukiplied two and
two ; the coefficient of the fourth term is equal to the sum of the
different products, three and three. Following this analogy,\9e
say that the coefficient of a term, having n terms before it, is equal to
the sum of the different products of the second terms of the bino-
mials, taken n at a time. Finally, the last term is equal to the pro-
duct of the second terms of the binomials.
To be convinced that this law is general, let us suppose it aUea4y
verified for the product of a number m of binomials, and s&^ii it
applies when we introduce a new factor into the product.
Let then
the product of m binomial factors (JVa?'*~* represents a term which
has n terms before it, and JMa5"^"+^ that which immediately pre-
cedes it.)
Moreover, let a? + JE'be the new factor introduced; we!4iave^
for the product, arranged with reference to a?.
+K
+AK
4-BK
+MK
X
W— W4-1 I
+ UK.
Already the law of the exponents is evidently the same.
As to the coefficients, (1.) that of the first term is unity, ^2.)
A-\- K^ or the coefficient of a?"*, is also the sum of the second terms
ofthem^Y' 1 hinomials.
(3.) B is, by hypothesis, equal to the sum of the different pro-
ducts of the second terms of the m first binomials, taken 2 and 2 ;
AK expresses the sum of the products of each of the second terms
of the m first binomials, multiplied by the new second term JT;
then B + AK is also the sum of the different products of the second
terms ofihem-\'l binomials^ taken two and twom
In general, since JV expresses the sum pf the products of the
second terms of the m first binomials, taken n at a time, and MK
aia Ehuimis of Algebra.
itepresMts Ae sum of the products, of these second tetms, ttftcQ
ni'^^^ I at a utne, jniiliiplied by (be new second term K^ k followsi
that JSf^MKf or tbe coefficient which io the pdyooinial of tbo
degree m -}- I, having n terms before it, is equal to the sum of Me
jprodutia ^f the second terms of the m -^ I binomiah^ taken nai a
time. Tbe last terno, U X K, is moreoTer equal to ihe product of
the m + 1 second terms.
. Thus tbe law of composition, supposed true for tbe product of a
iMMnber m of biiXMentals, is so likewise for a number m-^ li then
il is geneffftl.
Let 09 suppose that in the product of m binomial factors
a? -j- fl, X ^ b, a? -[- c, a? -j- rf . . . ,
we make a = i = c = £?;
the expression indicating this product,
(» + a) (a? + 1) (op + (^ + ^) • • • 's changed iato (x -f- a)*.
As to its developements, the coefficient being
a -j- J + c 4- <f +•••> aJ + flc -j- ad +...3 abc -f- <ibd -j- aed -}-...|
(1.) the coefficient of »*"S ora + i' + «+i+»«*f becpnaee
that is, a taken as many times sls there are letters «» fr» e • • . , and ia
reduced consequently to ma. (2.) Tbe coefficient of a*"^, oc
a & + a c -|* • • • is reduced to ir^ + ^ + <^* • • • » or rather to as
many times a^ as we can form different combioationa of the m let*
ters, mnkiplied 2 and 2, or (150) in short to
m — 1 g
•fi • — ^ — a •
(3.) The coefficient of a?*"^ is reduced to the product of a',
multiplied by tbe number of different combinations of m letters
taken 3 and 3, or rather to
m — 1 m— 2 g^
m • — 5 — • — ^ — • a ;
i^nd so on.
In general, if we designate by ^a^"""* the term which has a
number n before it, the coefficient JV*, which in the hypothesis where
the second terms of the binomials are different, is equal to the sum
of their products, taken n at a time, is reduced, when we suppose
ibem all equal, to a" multiplied by the number of different combioa
Formation ofPow^i and ih IktracHon of Roots. 313
tiofi».giren hy m letters, taken n at a time. Thus (ISO),.
Finally, then, we have the formula
{x + a)" = a;"* + m aa;**"^ + m ~ o^ x*^
152. If we look at tbe different terms of this developement, we
shall discover a simple law according to which a coefficient of what-
ever place is formed by means of the preceding coefficient.
2 lie coefficient of a term of any place is formed by multiplying
the coefficient of the preceding term by the exponent of x in this
iermy and dividing the product by the number of terms which pre-
cede that which is considered.
For, let us take ^the general term
— ^— ! — ' a*a?"^*,
Q . n
(we call it the general term^ because in taking successively
n =r 2, 3, 4..., we can deduce from it all the others) ) tbe term
which precedes it by one place, is evidently
-P
since (150) -^ expresses the number of combinations, taken n — 1
at a tinie.
Now we see that the coefficient
P(f„ — n+ 1)
il.n
P
is equal to the coefficient ^ which precedes it, multiplied by
m — n -^ 1, the exponent of a? in this term, and divided by n, the
number of the terms which precede that which we are considering.
It is in tlHs law, for which we are indebted to Newton, that the
binomial formula principally consists. It serves to develope a
paiticular power even without our being obliged to have recourse
to the general formula.
For example, let it be proposed to develope {x + a)®. We
shall find, according to this law,
(« + a)« = cc^ + (}aa? + 16aV + 20aV + 15aV + &a^x + a\
314 Elements of Algebra.
After having formed the two first terms, in which there is no
difficuhy, acbording to the terms of the general formula
we multiply 6, the coefficient of the second terra, by 5, the expo-
nent of X in that term, then we divide the product by 2, which
gives 15 for the coefficient of the third term. In order to obtain that
of the fourth, we multiply 15 by 4, the exponent of x in the third
term, and divide the product by 3, the number of terms which
precede the fourth, which gives 20 ; and so on for all the other
terms.
We should find, in a similar manner,
{x + qY^ = x^^ + lOaa^ + 45aV + 120aV + 2l0aV
+ 252a V + 2l0a^x^ + 120aV + 45a8r^ + iOa^x + a}^
We shall resume this subject in the sequel with reference to the
powers of algebraic expressions-
Cpnaeguences of the Binomial Formula and of the Theory of
Combinations.
153. Consequence First. — ^The expression {x + a)"* being com-
posed in the same manner of a and a;, it ought to be so likewise in
its developement ; then if this developement includes a term of the
form Ka^x^"^, it ought to have another equal to iTx* «**"*, or
Ka^^^x^i These two terms are evidently at an equal distance
from the two extremes in the developement ; for the number of
the terms which precedes any term whatever, being marked by the
exponent of a in this terra, it follows that the term Ka^x^'^^ has
n terras before it, that the terra ira"*~"a?* has a nuraber m n
before and consequently n after it, since the whole nuraber of terms
is (m -f- I.)
Thus, in the developement of every power of a binomial the coef-
ficients at equal distances from the two extremes are equal.
N, B. In the terras ^a*^'^«, Jro"»-»a?«, the two coefficients
express the numbers of different combinations which can be forraed
whh m quantities taken n at a time, and m — n at a time ; thus we
may still conclude that the nuraber of different corabinations of m
Formation of Powers and the Exiradion of Roots. 8}$
quantities taken n at a timei is equal to the number of combinations
of these same quantities taken m — n at a time. >
For example, twelve quantities combined, 5 and 5, give the same
number of combinations as these twelve quantities combined, 12 — 5
at a time, or 7 and 7. Five quantities combined, 2 and 2, give
the same number of combinations as five quantities combined,
5 — 2 at a time, or 3 and 3.
154. Consequence Second. If in the general formula
(a? + a)"* = 0?" + m a a?""^ + m —^ — a® a?"*^ + ^'C.,
we suppose a? = 1, a = 1, it becomes
(1 + 1)"* or 2*= 1 +m + *'*— "o h ^'^ — •""§ H^^c-J
that is, the sum of the coefficients of the different terms of the bino-
mial forimila is equal to that power of 2 whose exponent is m.
Thus, in the particular formula
{x + ay = cc^ = 5ax^+ 10 a^ a:^ + 10 a^ a^J ^ 5 ^4^ _j. ^5^
the sum 1 + 5 + 10 + 10 + 5 + 1 of the coefficients equals 2*,
or 32. In the tenth power, developed article 162, the sum of the
coefficients in equal to 2^®, or 1024.
165. Consequence Third. fVhen we have a series of numbers
decreasing by unity from one term to the other, of which the first is
m, and the last m — p (m and p are whole numbers), if we make a
single product of all these numbers, this product is divisible by the
product of all the natural numbers from 1 ^0 p + !• That is, we
have
m (nt — I) {^ — ^) (m — 3)...(m — p)
rr^ : § ; 4 p + i
equal to a whole number. For, it follows from what has been
said (150), that this expression represents the number of differ-
ent combinations which may be formed of m letters, taken p -j- 1 at
a time. Now this number of combinations must be, by its nature,
a whole number ; then the above expression is necessarily a whole
number.
We recommend to the learner to seek a demonstration of this
property independently of the Theory of Combinations and of the
binominal theorem, apprising him that ihe question which is suffi-
<dMtly easy with respect to the fif^st expressions
m{m — 1 ) m (wi — 1) (m — 2)
1.2 • TTST^ '
becomes more difficult in the general case.
II. Of the Extraction of the Roots of Particular Numbers.
Although we have explained in our Arithmetic the principles of
the extraction of the cube root, it seems necessary to introduce here
this process ; firsts because next to the extraction of the square root
it is an operation for which we have the most freqtient occasion ;
secondly, because the satne reasonings may be applied to the ex-
traction of the 4^**, 6^, and in general of the n^^ root.
156. We call the cube or third power of a number the product
of that number multiplied twice by itself, and the third or cube root
the number which, raised to the cube, will produce the proposed
number.
The ten first numbers being
1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
we have for their cubes
1, 8, 27, 64, 125, 216, 343, 612, 729, iOOO.
Reciprocally, the numbers in the second line have for their cube
foots the numbers of the first line.
We discover, by inspecting these lines, that among numbers
of one, two, and three figures, there are only nine which are perfect
cubes ; each of the others has for its cube root a whole number
plus a fraction which cannot be expressed exactly by means of unity.
For, let us suppose for a moment that ti an irreducible fractional
number, is the root of a wbo!e number .AT; it would follow that the
a cfi
cube of J or rg, would be equal to JV. Now that is impossible, for
a aiid b being prime to each other, il is the same with a' md b^ $
then p cannot be equal to a whole number.
157. The difference between two consecutive perfect cubes is so
much the greater as the two roots are greater, aad this difference
may be easily estimated.
FarmaiiM ofP(nmi% ^mi Urn, EaMoMum ofRofOi. 9ir.
For^ \tA « aa&^a «|u 1 be twd ete«eraiive ivImIv nirtbcliv i* wt^
have (162),
iviienoe (n + 1)» — a» = 3«*-f- 3a + 1;
that is, /Ae difference of two consecutive cubes is equat to three times
the square of the smallest rooty plus three times this same root,
plus t.
Thus the difference between the cube of 90 and the cube of 89
is equal to
3 (89)2 ^ 3 X 89 + 1 = 24031.
1.58. Let us DOW seek a process for extracting the cube root of
a. whole number.
First, if the number has only three figures at most^ its r06t.
it itnmediaielyi obtainti by r^erenee to the cubes of the nine first
numbers. Thus the cube root of 125 is 5 ; die cabe foot of 7d h
4>'ghs a fraction^ or is 4 within unity, the cube rool of 841 ifli> 9r
within uniQT) since 841 falls between 729 or the cubd of 9,. and':
1000 or the cube of 10. Let us then consider a nuoiber of wve.
than three figiufes.
For example, let 103823^ be the nuisbor proposed.
10S.823
47
64
'48
48
398. 23
47
48
47
384
329
192
188
2304
2209
48
47
18432
16463
S^16
8836
110592 103823
This number being comprehended between 1000^ irtiicliris^lhe
cube of 10, and 1000000, which is the cube of 100, its root is-
necessarily composed of two figures, or of tens and units. Let ugi
designate by a the tens and by b the units, we have (46)
Whence it follows that the cube of a number, composed of tens
and units, contains the cube of the tensj the triple product of th$
Bour. ^Ig. 28
\«
Sm Mc- \\ ^ . dHNMM»)df'vi%«iwNi.«
»<
tquar^ cf^dhaitens^iftthe itmti^Ae tf^le-proAitct of lAa iigfiMfi^ of
units by the tens^ plus the cube of the units.
This being laid:dewa, the cube of tile tei^s grring-at least thou-
sands, the three .IfstJsignifictDt figures bo the right do not enter into^
it, and it is in the portion 103 (which we separs^te from the thrpe
last figures by ^ point) that we find tlie cube of the lens. Now the
r6ot of the greatest cube contained in 103 being 4, the root of 64,
4 is the figure of the tens of the root soughl ; for 103823 is evi-
dently comprehended between (40)^ or 64000, and (50)^ or
125000 ; then the root required is composed of 4 tens, plus a cer-
tain number of ubits less than ten.
••The'figuifd of thie tens Ifeing obtained, tet us subtract the cube
64 from 103 ; there remains 39, which, with the reiiiaining portion
839, gives '99823; and this resuk contains still tkree times the
syukre (af'theteris inuiHplied by the units, plus th^ two other parts
a)x)ve Aiamed. Now the square of a number of tens containing no
filgure inferior to hundr^s, it follows, that this triple product can be
f&tmd-i^nly in the part 3d8, on the left of the two last figures 23,
(whfeb we' separate for that reason by a point). ' On the oth^r
hand, we can form the triple square of the 4 teias, whicH gives 48 ^
then if we divide 998 by 48, the quotient 8 is the figure of the
units of the root, or a figure too great, becao&e the 398 hundreds
are composed of the triple product of the square of the tens multi-
plied by the units, and the tw9 remaining parts.
In order to aspertatn if this figure 8 is not too great, we can, as
for the square root,' fbrm, by the aid of this figure 8 and of the
figure 4 of the tens, the three parts which are contained in 39823 ;
but it is much more simple to raise 48 to the cube (as we did in
the foregoing table).* • Now, we find for this cube 110592, a num-
ber greater than 103823 ; so that the figure 8 is too large. In
forming the cube of 47, we obtain 103823; thus the proposed
number is a perfect cubei and has for its cube root 47.
'N.'B. 'We cannot immediaitelv find the figure of the units ; for
the'tiibe of the units (156) giving tens knd even hundreds, these'
tehs'^d hundreds are involved with those which proceed from the
other parts 'of the cube.
r, *
If
1'
\ •'.'.' I « I V ■>..».'« ■• '• • \ • • > ' • . • •
I M» ^" ■'>'' ' .*\
Formation of Fo/wert mi ih^'Bkiraction of Roots, « ^Sld
Again, let it be required to extract'tfae cube- root- of -49954.
^ \'
47 . 954
27
36
•
27
'
209
36
1
t
36
47954
216
46656
i 108
1298
1296
« •
36
7776
f
* 3888
:'^ '.!•
. .:ir<.^'
■Li I'.. I-
1 1 *
. * J
.:!•'» t
46666
The number 47954 being less than 1000000, its rp^^.jbft^nly
two figures, that is, only tens and units. The citil^e of ,t|}e ,^Qff,^s
found in the 47 thousands, and we can prpye asbefore^i ^hat,3|{fJbj9
root of the greatest pube contained in 47, e^^presses thp t^nf^fif the
root. Let us subtract ihe cube of 3 or 27 from 47, tbe,^e. rejx^^qs
20 ; bring down to this remainder the single figure. 9 of ti^p P^f^.^
954 ; the number 209 hundreds is composed of the triple, proc^iiiii^t
of the square of the tens multiplied by the unjts, ,,ajgi^, ^figqr^
arising from tjie two other portions. Then if \^e forp th^.jtfijdie
squa're of the tens 3, which gjve^ 27, and diyidA 20p by.:J^J^,
the quotient 7 is the figure of the units of the ro9t, or a figure too
great. By raising 3f to the* cube, we find "^ 50653, a number
greater than 47954 ; if we form then the cube of SB, We obtain
46656', a number which subtracted from 47954738 id seen* in ttte
foregoing table), gives for a remainder 1298. Tbu^, the proffyoiieil
number is not a perfect Guhe^ but its tddt, tdihm unityj is 36,
For the dt^erenoe between the .prQpQSB4 piM^ber und tb0 eute
of 36, is, as we have just seen, 1298, a am;nber ifroaller than
3 X (36)8+3 X 36+ 1, ^ , „ I,
since we have obtained, in the course of th^ verification, 3888 for
three times the square of 36. . .
159. Let it be now proposed to extract thie cube root of a num-
per of more than 6 figures, of 43725658, for exampiQ. ...
■Ai.-- '^ ^
. •»'! • •>'.'•.' *
.I-' M, .
Ti
r.ii
,i.i f '•'-.. ^-^
]9t0
fibm«Mf ^4%Bfaa.
rem.
«3. 725.658
27
fm
•
27 3675
167
35
352
35
352
43 725
175
704
■ 42 ^75
105
1760
8506
1235
35
1056
123904
43725658
43614208
6125
3675
352
247808
111450
42875
^19520
371712
43614208
Whatever may be the root sought, it has necessarily more than
one figure, and we may consider it as composed of units and teqs
only (as the tens may be expressed by one or more figures).
Now the cube of the tens gives at least thousands ; thus, it is
necessarily found in the portion at the left of the thre^ last figures
'658. If now we extract the root of the greatest cube contained in
the portion on the left, 43725, considered in its absolute value, we
shaB have the whole number of the tens of the root ; for, let a
ht this root of 43725, within unity, that is, let o^ and (a + If
'include 43725, then also 43725000 is comprehended between
oP X 1000 and (a + l)^ >^ 1000 j
;|Dd as thesp two last quoibers differ from ei^ch other by more tbf^i
)000, it folkws tb9t the proposed i^gmber jts^lf} or 4^7956^8^ js
.fomjpirebeDded between
tf» X lOGO and (a + if X lOQO j
ifaus (he root sought is comprehended betweeen those ^
a' X 1000 and (a + 1)^ X 1000 ;
that is, between
a X 10 and (« + 1) X 10.
Then in short, it is composed of q tens and a certain niinibeir of
iinits less than ten.
The question is then reduced to extracting the cube root of
43735 ; but this new number having more than three figures, its
Formation of P,0mn ^09d Ifo. £§f^Mtion of RooU. iSSl
i^OQjt has oMMre 4ban oxk^ Uutt is, it cootaim ieot and ipntta* la
order to obtaio th9 teui, we must separate the ibree fosl figiMre^
7^1 and extract tbe rool of the greatest cube contaiited hi 43.
(It is sufficiently evideot what would beneceaaary^ if tbb nfttr «iiitti^
b0r bad Qiore tban three jSgnres). ,
Tbe greatest ciibe ccmtaiaed iu 43 is 27, oi wfaicb the roclt ia Zr
and this figure eicpresses then the tens of lh« root of 4372^ (or ifae
figure of tbe buodfeds of the whole root), SubtraetiDg the' cube of
3, or 27, from 43, we obtain 16 for the remaioder, to which we
iDusC bring down tbe first figure 7 of tbe portion 725, whicb ^v«s
167.
Forming the triple square of tbe tens 3, we find 27 ; and if we
.di?ide 167 by 27, the quotient 6 is the figure of the units of tbe
root of 43725, or a figure too large ; it is easy to foresee that tbis
£gure is indeed too great ; hence we must make trial of 5^ and
for that purpose let us raise 35 to the cube ; the result is 42875, a
number which, subtracted from 43725, gives for a remainder 850,
a number evidently smaller than
3 X (35)«+3 X 35 + 1.
So that 35 is tbe root of the greatest cube contained in 43725 1 it
is then the number of the tens of the root sought.
In order to obtain the units, we bring down to tbe remainder
850, tbe first figure 6 of the last portion 658, which gives 8506 ;
we form moreover the triple square of the tens 85, (tbis is easily
done, since in the preceding verification, we have already formed
the square of 35) ; then we divide 8506 by this triple square,
3675 ; the quotient is 2 ; let us make trial of it by raising 352 to
tbe cube, which gives 43614208, a result smaller than the pro-
posed number ; by subtracting it from the latter, we obtain for a
remainder 1 1 1450. Thus, 352 is the cube root of 43725658,
within unity.
General Rule. In order to extract the cube root of a whole
number^ separate the number into portions of three figures each,
proceeding frofn the right, till you arrive at a portion oj one, two,
or three figures at most (the number of portions is equal to the
i>umber of figures of tbe root) ; extract the root of the greatest cube
contained in the first portion on the left, and subtract this cube from
the first portion; bring down to the remainder the first figure of
the second portion and divide the number thus formed by the triple
^itquare cf the ;pgute mf the root alreoAf found f set d^wnrihe.'fm"
iient on the right of this figure^ and raise the two figures to the
cube; if this cube is greater than the two first portions, diminish
4he quotient by one or more units, till you obtain a anbe^'which may
be subtracted from the two first portions; the subtraction being
tfujtdejto the remainder bring down the first fgvre of the third por^
#M»n, then divide the nimber thus formed, by the triple square of the
two figures ^dready found ; the quotient, if it is not too great,
oiught io be such, that in writing it on the right of the two first
figures of the rooty and raising the number which results from it, to
the cube, we may subtract this product from the three first portions'^
this new subtraction being made, bring down to the remainder the
first figure of the fourth portion, and eontinue the same series of
operations iill you have brought down aU ike portions.
Remark. Frequently, in tiie course of operations, we may imag-
ine that one of the quotients of which we have just spoken, is bdch
too large, and chat we may diminish it by two or more units ; bi^ by
raising to the cube the root already foond, together with this figure,
if we subtract this cube fronx the amount of the portions already
•considered in the given number and obtain a very great remainder,
we are led to think that the last figure obtained in the root is too
small. We are guided (157) by this sign; thai the remain^der
equals or exceeds the triple square of the root obtained plus three
times the same root, plus one. In tliis case, we increase the root
by one or several units of the order of the last figure obtained.
The following examples are for practice.
3
^483249 = 78 with a remainder 8697 ;
3
\/^i(i32506()41 = 4608 with a remainder 20644129;
a
3^
\/329773402l8432 = 32068 exactly.
160. Process for extracting the root of the vfi" degree of a whole
number. In order to generalise the process of the extractiqp of
roots, let us designate by JV the proposed number, aqd by n th^
degree of the root to be extracted ; if JV has not more than n
figures, its root has only a single figure, and we may obtain, it
immediately, by forming the n^^ power of each of the whole, pujut
hers from 1 to 10 inclusive, of which the n^ power is 10", or th^
gmallest number of n -|- 1 figures. . , \.
Formation of Powers and ih$ Etctraetian of' Udofr. 2tS
If .^ iseomposed of more than n figares, ius root hafs'more than
obe figure and may then be regarded as containmg ten^ and units^
Designating its tens by a and its units by 6, we have (16-1)
' I
JV = (a + 6)* = a* + na*-^ + n ^^^ a*"^.6^ +• ifce. ;
i *
that is, the proposed number contains the n^ power of (he iens^plusr
n times the product of the n — 1 power of the tens multiplied by the
unitSy and a series of other parts which it is unnecessary to take'
ajccount pf.
Now the n^*" power of the tens not giving units of an order infe-
rior to unity, followed by n zeros, the last n figures on the right do
not enter into this power. We must then separate them and extract
the root of the greatest n*^ power, contained in the portion on the
left ; this root will express the tens of the root sought.
If this portion on the left contained still more than n figures, we
should separate the last n figures on the right, and extract the. root
of the greatest n^ power contained in the new portion on the left^
aod so on.
After having separated in this manner^ the number N into por^
iions of n figures (the last portion on the left having only n figures
at the most), we extract the root of the greatest n^ power contained
in that portion on the leftj which gives the figure of the greatest
units of the whole root, or the figure of the tens of the root of the
number formed by the two first portions on the left. Subtracting
ike vf^ power of this figure^ from the first portion on the left^ we
obtain a remainder, which, followed by the second portion, contains-
still n times the product of the n — 1 power of the figure, by the
following figure, and a series of other parts. But this first part
evidently cannot give' units of an order inferior to 10*~^; thus, the
last n — 1 figures of the second portion cannot enter into this por-^
tion. It is sufficient then to bring down to the remainder corres-
ponding to the first portion^ the first figure of the second portion ;
and if after having formed n times the n — 1 power of the first
figure of the root, we divide by this result, the remainder followed
by the first figure of the second portion, the quotient will express
the second figure of the root, or will be a figure too great. In order
to make trial of it, we write it on the right of the first figure, and
raise the two figures ^to the n^^ power; then we subtract the product
obtained from the t$^dfint por/tions^ which will^ive a new remain-
<kr, to wkickwbrif^ d^^m ihefa'9iJigurecftheMrdpai4ia/fif toe
then divide the number thus formed by n times' the a—* 1 power of
4he two figures of the root already found.
We may cootinue this series of operations till we have brought
-down alV the portions.
The learner who has well understood the preceding steps majr
apply them to the extraction of the 4tb, 5th, and 6th roots.
161. Remark. Whenever the number expressing the degree
of the root to be extracted, is a multiple of two or several others, as
4, 6 ... , the root may he obtained by a series of extractions of
roots of a more simple degree. To illustrate these simplifications,
we remark, that '
(a«)* = a» X a» X a^ X a^ = a^+^^s = ^3x4-, ^la^
and that, in general,
(o**)" = a« X «"* X a» X «* . . . = a«x« (16).
'Tbem, Ac a*^ power of the m^ power of a number is equal to the-
m n^ poufer ^ that number.
Reciprocally, the m n*^ root of a number is equal to the n^ root of
ike' m** root of tkxi nunAery or algebtfaic&Dy, we have
inn
V^ = S^a.
For, let there be
J V Va = a' ;
"we nm the two membera to the n^ power ; it becbmes
j^a = a'* ;
{foe, 9PCordiog to the definition of a root, we have
Let vts raise again the two members to the m^ power ; we obtain
a = (a'")« z= a'*».
Sxtrdctfbg the mnf^ root of the two membeiRS,
but w« have already
f A m% f m
\t \^a =3 «^ ; tbeif j^'Os a V V^'
Fortnation of Powers and the Extraction ofRoots^ 325
We shall find, according to this principle,
4
6 3 3
V^986984 = VVJS85984 = V'A'^^^ = 12;
6 3
Vi771561 = VVi77l5Gi = 11 J
8 4
^1679616 = ^1296 = VVi296 = 6.
. N. B. Although the successive roots may be extracted in any
order whatever, it is preferable to extract first the root of the
most simple degree, because then the extraction of the root of the
highest degree, which is a more complicated operation, relates to a
number having less figures than the proposed number. This is
done in the second and third of the foregoing examples.
Of the Extraction of Roots by Approadmation.
162. When the whole number of which the n*** root is required,
is not a perfect power, the process of art. 1 60, gives only the whole
portion of the root, or the root within unity. Ad to the fraction
which is^to complete the root, it cannot be obtained exactly ; for
the n^ power of an irreducible fractional number ^ being ^, this
new number cannot be reduced to a whole number. But we can
determine the root to any degree of approximation we wish.
Let it be, in general, proposed to extract th^ n^^ root of a whole
number a, to within a fraction - ; that is, in such a manner that the
error may be less than -.
Let us observe that a may be expressed under the form ' .
If we designate by r the root of ap", obtained, to within unity, the
number — -^ or a is then comprehended between
^ and (l+il" ;
pn pn »
n
then also j^a is comprehended between the roots of these two last
Bour. Mg. 29
326 ElemenU of Algebra.
numbersi that is, between
^ and "i+l.
P P
r . . 1
Therefore, - is the required root, to within the fraction, -.
General Rule. In order to extract (he n*^ root of a whole num-
ber, to within a fraction -, multiply the number by p*; extract tlie
n^ root of the product, to mthin an unit, then divide the result by p.
163. In the second place, let t be a fraction or a fractional num-
ber of which we wish to extract the n^ root.
Let us multiply the two terms of the fraction by i*~^; we have
a a 6*-^
6"" 1b^'
Let r be the n^ root of a i*~^ , to within unity ;
a6»— 1 a
n
r* (v 'I l^* tn
is comprehended between v^^ and ^ T^^ ^ 5 then, Wr is itself
r r —t— 1
comprehended between t and T^ . Thus, after having render-^
ed the denominator of the fraction a perfect n power , we extract the
n^ root of the numerator ^ to within unity, then divide the result by
the root of the new denominator.
If we wish to have a greater degree of approximation than that
1
indicated by the fraction t, we may extract the root of a 6"~^ , to
1 m
within any fraction whatever, - ; let r' -| be that root, then
r' '\- " will designate the required root to within a fraction indicated
if
b
Such are the principles of the extraction of roots by approxima-
tion, which we are about to apply to the particular case of the cube
root, since that is the operation most frequently used.
164. Let it be proposed to extract the third root of 15, to
within tV-
\
\
Fortnation ofPou>er$ and the Extraction of Roots. 227
We have
16 X 12' = 15 X 1728 = 26920.
Now the cuhe root of 26920, to within unity, is 29 ; then the re-
quired root is f f or 2y'f . (See article 1 62.)
Let it be proposed to extract the third root of 47 to within ^^.
We have
47 X 203 = 47 X 8000 = 376000.
Now the cube root of 376000 to within an unit, is 72 ; therefore,
V47 = 11 = 3jf wilhin j\.
3
Let it be proposed to find the value of \/^ within 0,001.
We must multiply 26 by the cube of 1000, or 1000000000,
which gives 25000000000. Now the cube root of this number is
2920; then a^25 = 2,920 within 0,001. (See article 162.)
In general, to extract the third root of a whole number within a
decimal unit^ we must add three times as many zeros to the right of
the number J as we wish to have decimal figures in the root ; extract
the root of the new number within an unit, then separate towards
the right of that root the number of decimal figures required,
165. Let it now be proposed to extract the cube root of a deci-
mal fraction. Let us seek, for example, the cube root of 3,1415.
. As the denominator 10000 of this fraction is not a perfect cube
we must render it such by multiplying it by 100, which amounts to
adding 2 zeros to the right of the proposed decimal fraction, and
we have 3,141600. We next extract (163) the cube root, within
unity, of 3141500, that is, of the number, leaving out the deci-
mal point, which gives 146 ; then we divide the result by 100
3 3
or -v/lOOOOOO; and we find -v/3,M15 = 1,46 wilhin 0,01.
If we wish a greater degree of approximation, we add three times
as many zeros more to the number, as we wish the number of
decimal figures in the root to be increased.
In order to extract the cube root of a vulgar fraction to within a
decimal unit, the most simple method consists in reducing the prO"
posed number to a decimal fraction and continuing the operation
until we obtain three times tlie number of decimal figures which we
wish the root to contain. The question is then reduced to extract-
ing the cube root of a decimal fraction.
3S8 Elements ofJllgAra.
166. Lastly, let it be proposed to extract the sixth root of 23
to within 0,01. '
By applying to this example the rule given in article 162, we
roust multiply 23 by 100^, or add to the right of 23 twelve zeros,
and then extract the sixth root of the result, to within unity, and
divide this root by 100, or separate two decimal figures towards the
right.
\ But we have (161)
6 3
V23 -X (100)6 = y^y23 X (100)6 ;
so that after having subtracted the square root of 23 X (100)® to
within unity, we must extract the cube root of the resuh, and then
divide the new result by 100 or separate two decimal figures
towards the right.
6 _
We shall find by this method -v/23 = 1,68 to within 0,01.
The following examples will serve as an exercise ;
3 3
V473 to within ^V = Vt J V79 to within 0,0001 = 4,2908;
6 _
Vl3 to within 0,01 = 1,53 ;
3
V3,0()II5 to within 0,0001 = 1,4429 ;
3 3
VO,00101 to within 0,01 = 0,10 j ^H to within 0,001 = 0,824.
III. Formation of the Powers and the Extraction of the Roots of
Algebraic Quantities. Calculus of Radicals.
Let us first consider simple quantities.
167. Let it be required to form the fifth power of 2 a^b^y we
have (2)
(2a»i«)« = 2a3J3 >< 2 a^b^ X 2an^ X 2a^b^ X2a^b^;
whence we see that the coefficient 2 ought lo be multiplied four
times by itself, or that it should be raised to the fifth power ; (2.)
that each of the exponents of the letters should be added to itself
four times or muhiplied by 5.
Therefore,
(2 o3 b^y = 25 . a3X5 jsx 5 -- 32 ^w jw
In like manner,
(8 a« bh)^ = 8' • a^xs jsxa c3 = 6 12 a® 6 V.
Formation of Powers and the Extraction of Roots, 2S9
Thus, in order to raise a simple quantity to any given power,
we must raise the coefficient to this power, and then multiply the
exponents of the letters by the exponent of the power.
Reciprocally, in order to extract the root of a simple quantity to
any degree whatever, we must (1.) extract the root of the coeffir
dentf (2.) divide the exponent of each letter by the index of the root.
Thus,
3 v 4
We see, according to this rule, that in order that the quantity
may be a perfect power of the degree of the root to be extract-
ed, its coefficient must be a perfect power of this degree, and the
exponents of its letters must be divisible by the exponent or
index of the root to be extracted. We shall presently see how we
can simplify the expression of the root of a quantity which is not a
perfect power.
168. Hitherto we have paid no regard to the sign with which the
simple quantity may be affected ; but if we observe that whatever
be the sign of the simple quantity, its square is always positive^ and
that every power of the degree designated by 2 n may be consid-
ered equal to the n*^ power of the square, that is, that a^ = (a^)*,
we conclude, that every power of a quantity, either positive or nega^
tive, of a degree indicated by an even number is essentially positive*
Thus (± 2 a^ b^ cy = + 16 aH^ c^.
Moreover, as every power of a degree indicated by an odd num-
ber 2n + A> is the product of the power of the 2 n degree by the
first power, it follows that every power of a degree indicated by an
odd number is affected with the same sign as the quantity itself
Then (+ 4 a^bf = + 64 aH^; {—4 a^ if = — 64 a« bK
It is evident from this, (1.) that every root of a simple quantity,
of a degree indicated by an odd number, must be affected with the
same sign as the quantity.
Thus,
3 _ 3 5
V^-f 8a3 = -f 2 a ; V— Sa^ = — 2a; s/^ 32aio js = — 2 c? b.
(2). That every root of a positive simple quantity, of a degree
indicated by an even number, may have indifferently the sign -f-
or '^^
230 Elements of Algebra.
Thus,
4 6
(3.) TTiat every root of a negative simple quantity, of a degree
indicated by an even number, is an impossible root ; since there is
DO quantity which when raised to a power whose exponent is an
even nunaber, can give a negative result. So that
i 6 8
\/ — a, \/ — 6, \/ — cj
are the symbols of operations which cannot be performed ; these
are imaginary expressions like \/^^, \/— h, (86)'
169. We have already seen how we may raise a binomial X'\'a
to a power of any degree whatever; but it may happen that the
terms of the binomial are affected with coefficients and exponents.
Let it be proposed, for example, to develope (2 a^ + 3 a b^) ;
let us put, for the moment,
2a^ = a?, 3ai = y;
we have
(2a2 + 3ai)3 = {x + yf = s? + ^a^y + ^xf + f.
Restoring 2 a^ and 3 a 5 in the place of x and y,
(2a2+3ai)3=(2o3)3+3(2a2)2. (3a6)+3(2a2j. (3aJ2+(3a6)3.
or, by performing the operations according to the rule given in
article 167, and that for the multiplication of simple quantities,
(2 a» + 3 a 6)3 = 8 a« + 36 aH + 54 a* b^ + 27 a^ b^ ;
we shall find, in like manner,
(4fl24 _ 2abcY = (x + yY =:x^ + Ax^y + 6jcY + \xy^ + y*
= {Aa%Y + A{Aa%f ( — 3a6c) + ^Aa%f{ — Zobcf
+ A{Aa%) ( — 3a6c)3 + ( _ ^abc)^
= 256a%* — 768a''6^c + SMa%\^ — 432a56V .j. Sla^iV.
(The signs are alternately positive and negative.)
Let it now be proposed to develope (-2? + y + *)^ J first, let us
put 0? 4" y = ^ 5 It becomes
{u + zf = ti3 4- ^zu^ + 3;^9tt + x?-,
or, by replacing x -^ y instead of «,
Formation of Powers and the Extraction of Roots. 281
or, by developing anew the operations indicated,
{x + y + zy = x^ + 2a^y+Sxf + y^ + 2a^z+6xyz
+ 2fz + 3xz^ + 3yz^ + z^j
This expression is composed of the cubes of the three terms, plus
three times the square of each term multiplied by the first power of
the other two, plus six timss the product of the three terms.
This law is easily verified in the case of any polynomial (86).
-In order to apply the preceding formula to the developement of
the cube of a trinomial whose terms have coefficients and expo-
nents, we must, as in the case of binomials, designate each term by
a single letter, develope, and then restore the terms in the place of
the letters introduced, and perform the operations indicated.
We shall thus find, all the operations being performed,
{2a^ — 4ab + Si^f zi: 8a«— 48a«6+ 1 32 aH* — 208 a' 6'
+ 198 a^ 6* — 108 ai^ + 27 6«.
We may develope by an analogous process, the fourth, fifth, and
in general, any power of any polynomial.
170. With respect to the extraction of the roots of polynomials,
we shall confine ourselves to explaining the process for the cube
root, it will be easy afterwards to generalize.
Let JV be the given polynomial, R its cube root. Let us sup-
pose, as in the case of the square root (87), that these two polyno-
mials are arranged with reference to a certain letter, a for example.
According to the law for the composition of the cube of a polyno-
mial, the cube of iJ contains two parts which cannot be reduced
with the others ; these are the cube of the first term and three
times the square of the first by the second ; for it is evident, that
these two parts contain the letter a with an exponent greater than
in the triple product of the square of the second term by the first,
in the cube of the second term, in the triple product of the square
of the first by the third, &c. Then these two parts form necessa-
rily the first and second terra of JV*. Thus, by extracting the cube
root of the first term of N, we shall have the first term ofR, then
by dividing the second term of N by three times the square of the
first term of R, (the triple squarie is easily formed) we shall have
the second term of R. By knowing tlie two first terms of R, we
form the cvbe of this binomial, and subtract it from N. The re-
mainder N' contains still the product of the triple square of the
332 Elements of Algebra.
first term of -R by the third, plus a series of other parts containing
a with an exponent less than that which enters into this product,
and which is consequently the first terra of the remainder JV.
Then, if we divide the first term ofW by three times the square of
the first term of R, we shall necessarily have the third term ofR.
By forming the cube of the trinomial already found for the root,
and subtracting this cube from N, we shall have a new remainder
N^^, upon which we may perform the same operation as upon the
remainder N', and so on.
If now we bring together the parts of the preceding reasoning,
pat in italics, we may deduce a general method for the extraction
of the cube root of any polynomial whatever, and we can apply it
to the polynomials developed in article 169.
We shall content ourselves with indicating to those who wish to
discover the process for the extraction of the n^ root, that if we
designate the difi^erent terms of the root by Jp-}~y4~^4~^***
the expression for the n^ power contains among other parts,
a?" -[- ^ a?"~^y . . , + n x^^^z + . . . + n a:"~^w ;
and it is these parts which, presented successively in the proposed
polynomial, serve to find tlie different terms of the root.
Calculus of Radicals.
171. When the quantity either simple or polynomial, whose root,
of a certain degree, is required, is not a peifect power, we can only
indicate the operation (2) by placing the sign \^ before the quan-
tity, and above this sign the number which indicates the degree of
the root to be extracted. This number is called the index of the
radical.
We can frequently subject radical expressions to certain simpli-
fications, founded upon a principle analogous to that of article 84,
namely, that tAe root of the n^ degree of a product is equal to the
product of the n roots of its different factors, ^
In algebraic terms, ,
» n Ik % n
\/a bed* • • = \^a X \/b X \/c X \^d . . ^
For, if we raise each of these two expressions to the n^ power,
we shall find for the first
Formation of Powers and the Extraction of Roots. 233
^ abed • • •)* =:a5cd«..,
and (or the second^
W^ X V* X Vc-)" = (V«)** (V*)"- (VO* . . . =s a 6 c d . . .
Tbeo, since the n powers of the expressions are equal, they must
themselves be equal (176).
3
This being premised, let there be the expression \/54a5"P^,
which cannot be replaced by a simple rational quantity, since 64
is not a perfect cube, and moreover, because the exponents of a
and c are not divisible by 3, we have
3 3 3 _^ 3
V54 a^b^c^ = V27 a^fiS . ^2 ac« = 3 a i V2ac«.
And, ia tbe same manner,
- 3 3 4 4 . •• ' «. '
V^;S?= 2v^J \/4^tia^b^(fi = 2ab^ej^J^;
^ 6^ ^ 6 6
V192a76ci» = v^64tf6cM X V^oJ = 2 a c^ \/Sal.
In the expressions
3 3 4
3a6v'25^, 2Vo^j 2a6»cV3ac9,
the quantities which precede the radical, tvith the sign ofmultipli'
cation^ are called coefficients of the radical.
172. The principle demonstrated in article 161, affords anoAer
kind of simplification.
6
If we hav€i, fpr example, the radical expressbn ^id^ ; since by
tfajs principle
fi 3
y^4 a* = V V^ «^*>
3
and as the quantity under the radical V is a perfect squat'e, we
-may perform this extraction of the square root, which gives
6 3
V4a5 = V2a*
' And, in the same manner,
4
V36 a^ (^ = VVseS"^ = 4i/6ab.
In general.
- I--
a* = \\/a* =
y^a* = \\/a* = \/a,
Bour. jiJg. 30
£34 ElemenU of Algebra. \
that is, when the index of the radical is a multiple of a certain num-
ber n, and the quantity under the radical sign is an exact power of
the degree indicated by n, we may^ without changing the value of
the radicfilf divide its index by p, and extract fAe.n^^ root of the
quantity under the sign,
' The above proposition is the reverse of one not less important,
which is this, that we may multiply the index of a radical by a cer-
tain, number^ provided that we raise the quantity under ihe sign to
the power ^ indicated by this number.
mn
Thus, ^v^ = \/a» . For a is the same as \/a*; then
m tn mn
This last principle serves to reduce two or more radicals to the
same index, which is frequently very useful.
For example, let tri^fe" be the two radical expressions
- 3 • ^ 4-
" \^2a and V(a + 6),
which we wish to reduce to the same index.
If we multiply the index of the first by 4, the index of the
secpnd, and raise the quantity 2 a to the fourth power ; if, also, we
multiply the index of the second by 3, the index of the first, and
raise the quantity (a -{- b) to the third power, we do not change
the values of the two radicals, and we obtain by these operations
«
3 IS la 4 la
General Rule. In order to reduce two or more radicals to the
same index, multiply the index of each radical by the product of all
the other indices, and raise the quantity under the radical sign to
the power indicated by this products
This rule, which is very similar to that of the reduction of frac-
tions to a common denominator, is susceptible of the same modi-
fications.
Let there be, for example, the radicals
4_ 6 8
to be reduced to the same index.
Since the numbers 4, 6, 8, have common factors, and since 24
is the most simple multiple of these numbers, it is evidently suffi-
cient to multiply the first by 6, the second by 4, and the third by
n
Formation ofPotoert and the Extraction of Roots. 33S
3, provided that we raise the quantities under the radical signs to
the sixth, fourth, and third powers respectively, which gives
4 94__ [6 24 8 94
X/a = \/cfi ; V56 = V5464; yaS + 69 = ^(03+ 69)3.
These principles being established, we propose to perform upon
radical quantities the operations of Arithmetic, which are six in
number, including the formation of powers and the extraction of
their roots.
173. Addition and Subtraction of Radicals, Two radicals are
said to be similar when they have the same index, and when tbe
quantity under the radical sign is likewise the same.
This being laid down, in order to add or subtract two similar rad-
ical quantities, we must add or subtract - their coefficients^ and put
the sum or difference as a coefficient before the common radical.
Thus,
3 3 3 3 3 3'
3 v6 + 2 v* = 5 %/*! 3 -v/6 — 2 vi = V*>
1-4 4 4
Sa\/b ±2c\/b =z {Sa±2c)x/b.
Frequently two radical quantities are not at first similar, but imay
be made so by applying to them the simplifications of articles 171
and 172. For example,
3 3 2 3 3
%/48a62 + 6 v^5a = 4 J V3a + 5 b x^3a = 9b ^/Sa }
3 3 . 3 3 '___
^^8 036 + 1604 — v2>^ -f 2 a 63 = 2a\/h + 2a — bx/h + 2a
3
= (2a — J) V6 + 2a,
6 3 3 3 3
3v4a9 + 2 V2a = S\/2a + 2^2a = 5 v2a.
If the radicals ar^ not similar, we can only indicate their addi-
tion or subtraction by means of the signs + and — .
174. Multiplication and Division. Let us first consider the
case where the radicals have the same index.
Let it be required to multiply or divide \/a by ^6, we say that
we shall have
n
n n n n n 1^
V« X v6 = Vo b and x/a-^x/b = Ir-
388 E^menis ofAlgdn^a.
Thus,
3
4/8 a3 being equal to Si
\/8a3 being equal to Nv8a^> i^ ^s reduced to \/5o.
In like manner,
3 6
It is moreover evident, that we have
» at
since each of these expressions is equal to x/a^
176. The rules which we have just established for the calcula-
tion of radicals are founded chiefly upon the principle, that the
n^ root of the product of several factors is equal to the product of
the n^ roots of these factors, and the demonstration of this princi-
ple depends upon this, if the powers of the same degree of two
expressions are equals the expressions themselves are equal. Now
this last proposition, which is true wiien we consider only absolute
numbers, is not always so for the different* expressions to which
Algebra may lead us.
In order to verify the truth of this assertion we will prove that
the same number may have algebraically several square rootSj
several cube roots, several fourth roots, &c.
For, let us designate by x the general expression of the square
root of a number a, and by p the numerical or arithmetical value
of this square root, we have the equation
a»=:a, ot x^ = p^
from which we deduce x =: dt p. Whence we see that with
whatever sign we affect the arithmetical value p of the square root
of a, its square is equally a, a result conformable to what has been
said article 85.
In the second place, let x be the general expression for the cube
root of a, and let us designate by p the numerical value of this
cube root ; we have the equation
a?3 =: fl, or a?^ = p\
Now we observe that we may put x^ ^ p^ under the form
a? _ p8 -. 0.
Formation of Powers and the Eogtiraciion of Roots. 339
We have seen (31) that the expression 0?^ — jp* is diviaible by
X — p and gives for an exact quotient
the above equation can then be transformed thus,
{x —p) (o;^ -f pa? + p^) = 0,
an equation which nriay be satisfied either by putting
a?— j? = 0, whence a? z=:jp,
or, by putting
J? '\- p X ^ p^ =. 0^
whence a? == — g =*= f V^^>
or
. = . (-4^0-
We see then that tlie cube root of a admiti of three different
algebraic values^ namely^
Let it be further required to resolve the equation a?* =: p^
4
(p designating the arithmetical value of the \/a). This equation
may be put under the form x^ — p^ = 0. Now the expression
a?* — p^ is reduced (19) to
% (x^ — p^) (a? + P^)»
Then the equation itself is reduced to
(aj2_/) (a« + p9) = 0,
and we can satisfy it eiiher by putting ■
a?^ — jp^ = ; whence a? = db ji,
or, by putting
ap^ 4- p^ = J whence a? = =fc V--p = ± p V— !•
We thus obtain four expressions algebraically different for the
fourth root of a number a.
Let us propose to resolve the new equation a? = p\ which may
be put under the form cfi — p® = 0.
Now, a?' — p^i& reduced (19) to
(a|3_p3)(^3^p3) -.0.
240 Elements qfMg^bru.
1t4ius i^e equation feeeonies
Already the equation a^ — 1>' = 0, resolved in the preceding
article, has given
a? = i> and X =z p t ^ 1.
Let us consider the equation a:^ + 1}^ = 0, and observe that if
we substitute for the present — p' instead of p, it becomes
whence we deduce
0? = ;/, and x = p' (^ ^ ~ ) 5
or, by restoring in th« place of jp', its value — p,
<c = — p, and X ;= — j? I ^ )•
Thus, the equation x^ — jp^ =: 0, and consequently the sixth root
of a adn^its of six values 3 '
p, «p, a'p, — p, — «i>, — t/p,
by putting, for the purpose of simplifying.
We may conclude from analogy (which, moreover, will be demon-
strated hereafter in a more complete manner), that every equatioa
of the form of
a?" — a = 0, or 35** — p" = 0,
is susceptible of m different values, that is, that the m^ root of a
number admits of m values, algebraically different,^
177. Consequence, If in the preceding equations and the
results which correspond to them, we suppose, as a particular case,
a = I, which gives ^ = 1, we shall obtain the square roots, the
ctiba iTOots, the fourth roots, &c., of unity. Thus, + 1 and *^ 1
-^ are the two square roots of unity, since the eqiiaiion o?^ *— 1 =3: (^
{ives aj :3 d= 1.
In like manner,
a.1 -1 + V/-3 — i^^^a
Formation of Fawtt$ Mil the Eofif action of Roots. til
are the three cube rool» of anity, or the roots of «* — 1 =0.
+ 1,^ — 1, + V— 1» — V — 1,
are the four fourth roots of unity, or the roots of 2?^ — 1 =: 0.
ITS. It follows from the preceding analysis^ that the rules for
the calculus of radical expressions, which are exact, so long as we
con6ne the operations to absolute numbers, may be susceptible of
certain modifications^ when the operations are performed upon
expressions or symbols purely algebraical^ and especially when we
apply these rules to imaginary expressions, these modifications are
necessary, and they are a consequence of what has been said m
article 176.
For example, if we demand the product of V-^ by V— aj the
rule of article 174 gives
^ — a X ^ — a^= V+ ^^•
Now, the v5 IS equal to db a (176). There is here then appa-
rently a doubt as it regards the sign with which a should be alflect-'
ed in order to answer the question. Siill, the true answer is — a,
for the reason, that in general in order to raise ^m to the Square,
it is sufficient to suppress the radical sign y now, v — a X \^-^a
is reduced to (v/ — a)^ and, consequently, is equal to — a.
In the second place, let it be required to form the product
^_ a X V-^ 6 ;
we shall have, according to the rule of article 174,
^ — a X j^ — 6=^i^-^-ad.
Now, \,^ ^n zh p (176), p designating the arithmetical value* of
the square root of a & ; but we say, that the true result ought to be*
— 2?, or — %/a6, so long as^ we consitJer each of the two radi-
cals v— a and v--^> »s preceded by the sign -j—
For we have
y"iri zi: \/a . \/— 1, and \/^^h == \/h . \/ — i ;
therefore,
A^ZTa X V^ = Vo • V^^ X \/h . \/^i = VoX. (V— I)*
= \/aI X — I =2 — V«^*
We shall find, according to these principles, for the different
powers of s/^l]
Bour. Alg. 31
^43'. Ekmentt <^AigAra.
V— 1 = 'V/-X, (V— 1)' = — 1,
(^— 1)3 = (v/— 1)8. V— 1 = — v^— 1,
and
(V~ 1)*= (V- i)». (V- l)** = - 1 X - 1 = + 1.
. Asjthe four following powers are obtained by multiplying the
fourth + 1, by the first, second, third, and fourth, we shall still
find for these four new powers,
+ V— 1, — 1,— v/— 1 + 1;
therefore, in general, all the powers of \/ — 1, taken /otir and four ^
form a period equal to
s/—h — h — \/—\, + 1.
4 .
Let it be further proposed to determine the product of \/— a
4 ^ 4
by V — ^j which, according to the rule, will be -v/-|-a6, and, con-
sequently, will give the four values,
4 4 4. 4
+ \/a bf — v^a 6, + \/ab . y — 1, — x/ab . v— !•
. In or)]tBr to determine the true product, we observe that
4 4 4 4 4 4
V — a = \/a . v' — 1> V — ^ = V^ • V — 1 5
but
4 4 4
then Vr- a X %/— ^ = Va6 . \/— 1.
Let u^ apply the preceding calculations to the verification of the
expression
— 1 +vir3
as a root of the equation a^ — 1 = ; that is, as a cube root of 1
(177J. According to the formula
(a + 6)3 = a3 + 3 aH + 3 ai^ + J^
we have
(_ 1)3 4- 3 (— 1)« . ^-ZZs + 3 (- 1) . (V- 3«) +{V~3)^
— 8
Formation ofPoumrs and the Extraction of Roots. 343
— ] 4.3v/-r3— 3 X — 3 — Sy/irg ^ 8 _
— 8 — 8 ■" ^*
We may verify, in the same manner, the second value,
— 1 — ^ZTs
IV. Theory of Eocponenis of whatever JVature*
179. We must here make known two new kinds of notation of
great use in algebraic calculations, namely, fractional exponenti
and negative exponents ; they derive their origin from the niles
established for the extraction of roots and the division of simple,
quantities.
Let it be required to extract the n*^ root of such a quantity as
a". We have seen (167) that if w is a multiple of n, we must
divide the exponent m by the index n of the root. But if m is not
divisible by n, in which case the extraction of the root is not possi-
ble algebraically, we may agree to indicate this operation by indi-
cating the division of the two exponents.
n J? .
Then \/a« = a*, according to the convention founded upon the
rule for the exponents in the extraction of the roots of simple
quantities.
Thus,
3 ?. 4 1
V'a2 = a^ ; y/a' = o*.
In like manner, if it is required to divide a" by a*, we know
(23) that we must subtract the exponent of the divisor from that of
the dividend whenever we have m ^ n, which gives
a**
a*
But if 7» <[ n, in which case the division is impossible algebraically,
we may agree to indicate tins division by always subtracting the
exponent of the divisor from that of the dividend, help be the
absolute difference between n and m, we have then
n = TO + p, whence -^j^ = a"* ;
«
moreo.vfeEi.-^Tr'js^^uced to — , by &uppressiflg4lie lactor a^ com-
mon to the two terms. Tberefote, ar^ = -—.
aP
The expression ar^ is theo the symbol of a division which can-
not be performed, and its true value is the quotient arising from
the division of unity by the same number a, vnth the exponent p
tcJcen positively. Thus,
V
The notation of negative exponents has the advantage of pre-
serving an entire form to fractional expressions.
From combining the extraction of the root with the impossible
division of simple quantities, there results another notation, that of
negative fractional exponents.
< 1
Let it be required to extract the n^ root oi
a*
In ihe first place, we have
n
- = a-«, then ^- = ^a^^ = a — ^
Igr. substituting for the ordinary radical sign a fractional ^xpq^Q^
Tbe e:^p;:6Ssions
m m
a*, a~P, a »,
are therefore, accordiog to a convention founded upon the precede
ing established rv/e^, notations equivalent to
n
Thus, we may substitute the first of tliese expressions for the sec*
end, and the reverse,^ as circumstances render it convenient.
As in common language o^ is said to be the p power of o, p
being a positive whole number, so also, by analogy,
m
a*, arP^ a »,
are called the — power, the — p power, and the power of
a ; whicb has led algebraists to generalize the word power. But
Formation ofPomra and the E$itraciion of Roots: 345
it would perhaps be more proper to employ only the denominations
Oj with the exponent -, with the exponent — p, with the expo-
nent , restricting the use of the word power to designating the
product of a number mulliplied several titxies by itself (2)*
180. These ideas concerning the origin and signiGcatipn of
quantities affected by any exponents whatever, being established,
let us Bee how we are to perform upon them the operatious of
Arithmetic, beginning with multiplication.
MukipUcation. In the first place, let it be required to multiply
a^ by a^. We say that it is sufScient to add the two exponents^ and
we shall have
a^ X a^ = a^ ■*" ^ = o^*,
For, we have seen (179) that
hICII
3 5 2 3
3 3 5 3
or, by peribrming the multiplication, according to the rule of
article 174, ^
3 11 « 1 S.
3
Again, let it be required to muhiply * by a'^, we say that we
shall have
for
1 fl ^ • «
therefore,
4 ,. 12 12 ,^
■ a 5 1 1 • 1 1 1 a • tfllO is 1
a *Xa^=^^XVa*=V^XVa''=V'5J = V«=«^^-
In general, let it be required to multiply a " by a^ ; we have
m p m , p np-^mq
346 Elements ofJllgebra*
for
therefore
_5 "|l - «
n ng
\
np" 'M^
Thence the general rule for multiplying two simple quantities
having any exponents whatever, is as follows ; we must add together
the two exponents of the same letter ; this is the same with that
established in article 16, for quantities having entire exponenls.
. We shall flnd, according to this rule,
ah~^ c-i X a^bK^ = a^ b^ c~^ ;
30-^6^ X 2a~h^(? = 6a~'**6^ A
Division. In order to divide the one by the other two simple
quantities having any exponents whatever, we must follow the rule
for quantities having entire and positive exponents, that is, for each
letter we must subtract the exponent of the divisor from that of the
dividend.
For, the exponent of each letter in the quotient must be such
that when aided to that of the same letter in the divisor, the sum
shall be equal to the exponent of the dividend ; then the exponent
of the quotient is equal to the difference between the exponent of
the dividend and that of the divisor.
We shall find, according to this rule,
a 3 3 ( 3\ 1 7
a^ :a * = a^ ^ ^^ z= a^^ ;
3 4 3 4 _ ' 1. * 1
a* : a^ =za* ^ =a ^^ ; o^ : a^ = a^ ;
3 3 1 7 1
an*: a ^J^ = a^vj t.
Formation of Powers. In order to raise a simple quantity hav-
ing any exponent whatever to tlie m^ power, CQnformably to the
rule given in article 167, we must tmultiply the eorponent of each
letter by the exponent m of the power, since raising the quantity to
the w^ power, is to multiply it m — 1 times by itself ; then, ac-
cording to the rule for multiplication, we must add the exponent of
each letter m — 1 times to itself, or multiply it by m.
Formation of Powers and the Extraction of Roots. 247
(3\ 5 Iff / S\ 3 6
(2a ^6*) =64 0-^6^5 U V =cr^\
Extraction of Roots. In order to extract the n^ root of a sim-
ple quantity, conformably to the rule in article 167, we must divide
the exponent of each letter by the index n of the root.
For, the exponent of each letter in the result must be such, that,
when multiplied by the index n of the root to be extracted, the
product shall be the exponent which the letter has in the proposed
simple quantity ; therefore, the exponents in the result must be
respectively equal to the quotients arising from the division of the
exponents in the proposed simple quantity, by the index n of the
root.
Thus,
3 4 a
Va^ = o^ ; Va^'^ = ^^^" 5 ^a~^ = a~* ;
3
I ~^ 1 a
• • •
The last three rules have been easily deduced from the rule
relative to multiplication ; but they might be demonstrated by going
back to the original quaotilies having any exponents whatever.
We conclude, by an operation equivalent to a demoDstration,
which embraces implicitly the two preceding.
. "~ ■ r
Let it be required to raise a" to the power, we must prove
that we have
^a" / = a* * = a "*.
For, if we go back to the origin of these notations, we find that
r
""") = l7V'= 1^^ = 1^
s n
^^a"*''
n» ntr
= .Lbs =vor^ = i
ns.
M8 ISmsnit of Mgebm*
The advantage derived from the employment of exponeotfi, of
any nature whatever, consists principally in this, that the calcula-
tion of expressions of this kind requires no other rules than those
which have been established for the calculations of quantities having
entire exponents. Moreover these calculations are reduced to
simple operations upon fractions, operations with which we are
very familiar.
- Of Progression by Differences and by Quotients^
Progression by Differences.
181. We give the name of equidifference^ arithmetical progrep*
sion^ or progression by differences to a series of terms, each of wliich
exceeds or is exceeded by that which precedes it, by ^ constant
quantity which is termed the ratio or diffefimce of the progression*
Thus, let there be the two series
1, 4, 7, 10, 13, 16, 19, 22, 25 . . .
60, 66, 62, 48, 44, 40, 36, 32, 28 . . .
The first is called an increasing progression, of which the ratio'
is 3, and the second a decreasing progression^ of which the ratio
18 4.
Let us designate, in general, by a, 6, c, rf, c,/. . . the terms of
a progression by differences ; it is usually expressed thus ;
-r- a. 6 >c>d.e.f,g,h,i»k^,,
and we read it in this manner,
a is to 5, as b is to c, as c is to e/, as (2 is to e ... ,
or, more concisely,
a is to & is to e is to <^ is to e •• .
This is a series of continued equidifferences, in \i(hich each term is
at the same time both consequent and antecedent, with the excep-
tion of the first term, \Vhich is only an antecedent, and the last,
which is only a consequent.
182. Let us call r the ratio of the progression, which we will
suppose increasing, in all which is to foUour. (If it were decreas-
ing, it would be sufficient to change r to — r in the results.)
Progression iy D^remu ani^iy Quotients. MO
This being laid down, we have evideiidy, according to the defi-
nitioa of the progressbn,
and, in general, a term of any place whatever is equal to the Jirst^
plus as many times the ratio as there are terms before that under
consideration. Thus, let / be this term, and n the whole number
of terms to this, inclusive ; we have for the expression of this
general term,
Z = a + (n — l)r.
For, if we suppose n = 1, 2, 3, 4, • • • successively, we shall
find the first, second, third, . • . term of the progression.
If the progression were decreasing, we should have, on the other
hand, •
I =z a — (n — 1) r.
The formula, Z = a -f- (n — l)r, serves to give the expre&r
sion of a term of any place whatever, without being obliged to
determine ^11 ihpse wbichprciceda it. • ■ '
Thus, if we seek the 60*^ term of the progression
-r- 1.4.7. 10.13.16.19...
we have, by making n = 50,
Z= 1 +49.3z= 148.
iS3. A progression by differences being giv^, it may be pro<-
posed to determine the sum of a cehrtain nikmher of terms.
Let there be the progression
-r- a ,0 . c . d . e ,f. , . i . k .1,
continued to the term I inclusive, and let n be the number of ternsrsi
and r the ratio.
We begin by observing, that if x designates a term which has p
terms before it, and y a term which has p terms after it, we have,
according to what has just been said, the equalities
X =z a + p X r^
y = l — p X r;
whence we deduce, by adding
X + y =:a + 1;
Bout. Jllg. 32
which filbows» ibat in ey&ty progression, the sum ^'a^ two ierms^
taken at equal distances from the extrames, is equal ta the snm of
tlie extreftjss j^ or the two. ^xtremesi and two terms, takm at espial
distances from these extremes, form an equidifference (in the order
in which they are wrillen.)
This being admitted, let us suppose that we have written the
progression under itself, but in an inverse order, thus
'. Let US; call S tbe sum of the terms of the 6rst progression, 2S
will be the sum of the terms of the two progressions ; and we sbaU
faaVie, by uniVmg the terms in tbe same vertical cokimn,
or indeed, since all the parts a + Z, 6 + A:, c -{-«*«•• 9 are equal,
Bind in number n.
2 S = (a + Z) n; then fioaUy S 5;: ^^-^^;
that is, the sum of the terms of a progression by difference is equal
to the product of the sum of the extremes multiplied by half the
number of terms.
If, in this formula, we aubstitute for I its value a -|- (n — 1) r,
we obtain further
o ^ ^ ^— •
but the first expression is most used.
Applications. We require the sum of the first fifty terms of the
pmgression,
2 . 9 . 16 . 23 . 30 . . .
We b^ve, ^rst, for the fiftieth term,,
Z == 2 + 49 . 7 =3 945,
then
S = i?+^UH=34TX25 = 8675.
We should find also, for the hundrelh term,
/ = 2 + 99 . 7 = 696 ;
Progression iff D^itmom mCi by ^mitents. Sftl
Had for the sum of the first hundred terms^
S = (1±J^ = 34850.
184. The fbrmulas
contain five quantities a, ^, n, Z, and 5, and, consequently, give rise
to this general problem ; Any three of these Jii)e quantities being
given^ to determine the other two. This problem is subdivided into
as many separate problems as we can, with the five letters, forni
different combinations, taken three and three or two and two. WW
we have found (150) for the numbers of combinations, taken 3 and
2 and 3 and 3,
2 ®"^ 273 •
Making in these formulas m = -5, we find
5X4 _ ,n^^^ ^ X 4 X 3 _.,^,
— ^ — , or iO and ^ - ^ — or lU ;
whence we see that 5 letters combined 3 at a time, give (he same
number of combinations as 5 letters combined 2 at a time. (This
result agrees with the consequence of article 153, by which the
tkumber of the combinations of m letters, taken p at a timej is ^qtial
to the number of combinations taken m — j» at a time;)
We see then that the above problem is divided into ten i^eparat^
problems, of which the following are the enunciations :
Having given 1st. a, r, n, to find I and S ;
2d. ff, r, Z, . • .
. n and iS ;
3d. o, rj Sf . .
. • n and Z ;
4th. a, n, Z, • .
. . r and S ;
5th. a, n, S, • • .
, . r and Z ;
6th. a, Z, S, • .
. . r and n ;
7tb. rjnj l^ . . .
; . a and S ;
8tb. f , ti, jS>, • • •
. . a and Z ;
9th. r, Z, S, . . .
• a and.n;
10th. w, Z, S, . . .
. . a and r.
The first problem is already resolved, for the two formulas give
ioimediately Z and £• in a functipn of a, r, n. As to the other f>ro]^
lems, their resolution presents no difficulty ; but we advise begift-
26S EUmenUof^^d^a.
ners to perform them sucoegsirely,: this exercise being very' well
fitted to make them acquainted with the resolution of equations of
the first and second degree ; for it is worthy of remark, that
although the quantities a, r, n, Z, and 5, enter only to the fir$t de-
gree in the two formulas, we are however led to the resolution of
an equation of the second degree, when" a and n, or rather I and n,
are unknown, because a and n, or I and n, enter at the ,same time
into the tvtfo equations, aud are multiplied together in the second.
, 185. We will confine ourselves to the resolution of tlie fourth
problem ; it is the case in which, by knowing a, n, and Z, it is
required to determine r and S.
; The formula
7 — — a
Z = a + (n — 1) r gives r = J^ZZJ^
and the formula
S
(a + Z) »
shows immediately the value of 5.
From the first expression,
I— a
»— 1'
we deduce the solution of this question; ta insert beiufe^n two
given numbers a and b, a number m of arithmetical means (we thus
designate the numbers comprised between a find 6, and forming
with the latter a progression by difference.)
In order to resolve this last question, it is sufficient to determine
the ratio ; now, by substituting, in the above formulas, b for Z, and
m •{- 2{or n, which actually expresses the whole number of terms,
we find
b — a b — a
r =: — rir^ — Tj or r = — --2 ;
w + 2 — ^ ^-^K
that is, the ratio of the required progression is obtained by dividing
the difference of the two given numbers a and b by the number of
terms to be inserted, phis one.
The ratio once obtained, we form the second term of the pro-
b — n
gressioD, or the first arithmetical mean, by adding r, or , to
Tim ~T** l
tiie first term a ; the second mean is obtained by increasing this by
T, and so on.
Progremon by iHfftrtnua and by Quotients. 25S
Let it be required, for example, to insert 13 means between
12 and 77. We have
r — jjj — ^^ — 5,
which gives the progression
-f- 12 . 17 . 22 . 27 . 32 . 37 . . . 72, 77.
Consequence. ]^ between all the terms of a progression, cou'
sidered two at a time, we insert an equal number of arithmetical
means, these terms and the arithmetical means united form but one
and the same progression*
Finally, let -r- a . 6 . c . rf . c ./ be llie proposed progres-
sion, and let m be the number of means which we wish to insert
between a and b, b and c, c and d .. .
The ratio of each partial progression will be, according to what
has just been said, expressed by
6-— o [c — d d — c
m+T m+l' m + i *'
all equal quantities, since a,b,c,,.. are in progression ; thus the
ratio is the same in each of the partial progressions, and as more-
over the last term of the first forms the Jirst term of the second,
and so on, we may conclude tliat all these partial progressions con-
stitute one uniform progression.
186. The following are the enunciations of several problems:
Question First. To determine the first and the number of the
terms of a progression by difference, of which the ratio is 6, the last
term 185, and the sum 2945.
{Ans. First term = 5, whole number of terms = 31.)
Second Question. To insert between all the terms of the pro^
gression -r-2.5.8.11.14..., nine arithmetical means.
{Ans. Ratio or r = 0,3.)
Third Question. To find the number of m^n contained in a
triangular battalion of which the first rank is I, the second 2, the
third 3, and the ninth n. In other words, to find the expression
of the sum of natural numbers 1, 2, 3 . • .from 1 to n.
(.ins. 5 = "^%+^) .)
Fourth Question. To find the sum of the n first terms of the
progression of the uneven numbers, 1, 3, 5, 7, 9, . . .
{Ans. S =: n^, or the square of the number of the terms.)
S5t SklimU ^^Igiilra.
Fifth Q^uestion* An menue of trees is diiiafU from a heap of
sand 40 yards , and 100 loads are necessary to coder ii^ being placei
at intervals of 6 yards from each vther% Required the distance to,
be passed over by the teatu^ the first hoad brni^ deposited 40 yards
from the heap^ and the team returning at last to the place from
which it started, Ans, 67400 yards.
Sixth Question. A foot soldier travels 1 leagues a day ; a
horseman sets out at the same time^ and travels only 3 leagues the
first day, but every succeeding day he travels 2 leagues more than
on the preceding. It is required to find in how many days the
horseman will overtake the foot soldier^ and how much of the jour-^
ney each will have performed.
(Number of days, 8 j distance, 80 leagues.)
Of Progression by Quotients.
a
187. We cM geometrical progression or progression by quotients
a series of terms of which each is equal to the product of that which
precedes it, by a constant number which we call the ratio of the
progression ; thus the two series
3, 6, 12, 24, 48, 96...,
64, 16, 4, 1, T) T7 * * * )
of which the first is such, that each term contains that which pre-
cedes it twice, or is equal to twice that which precedes it, and of
which the second is such, that each term is contained in that which
precedes it four times, or is equal to the fourth of that which pre-
cedes it, are called progressions by quotients ; the ratio is 2 for the
first, and i for the second.
Let a, 5, c, d, e,/. . . be numbers in a progre^sioti by quotiedts;
we write it thus,
lir a : b : c : d : e :fig. . . ,
and we enunciate it as a progression by difference, although there
is this distinction to be made, that one is a series of equal differ-
ences, and the other a series of quotients or equal ratios in which
each term is at once antecedent and consequent, except the first
which is only an antecedent, and the last which is only a consequent.
188. Let us designate by q the ratio of the progression
•^a:6:c:d...,
Progression bf JDifioreifce^ tmi iy QuoUenis. Wk
q being ]> 1, when the progresBiOD is increasing^ and <[ 1, if the
progression is decreasing ; we deduce from tlie same definition the
series of equatioi^
b ss aq, c ss bq ^:^ aifj d:scq :s. aq\ e z=z dq :=z a^ .. .f
and in general, a terni of any place whatever n, that is to say,
which has n — 1 terms before it, has for its expression a j*"^ .
Let I be this term, we have the formula I =z a f^f by means!
of which we can obtaia the value of any tefm whatever, wkbout
passing through all the terms which precede it. For example^
the eighth term of the progression
-77- Sr : 6 : 18 : 54 : . . . ,
IS equal to
2 X 3^ = 2 X 2187 = 4374.
In a similar manneri thd twelfth term of the progression
4r- 64 : 16 : 1 : J . . • ,
i»c<]ual tp
fid /iV ^ ^ - L - ^
\4/ ■" 4" "" 48 ■" 6553()*
189. Let it now be proposed to determine the suca of the n first
terms of the progression
I designating the n^ term^
We have (187) the equations
h =1 aq^ c =z b q, d =z cq, e=idq.,.k=iiq^ ZrsAXj;
whence we deduce, by adding them, member by member,
i+,c -|-d[ + e +...-[- A -f-Z = (a + 6 + c4- d -]-..,+ * + A) J>
or rather, representing by 5 the sum required,
S — a=z{S — l)q = Sq — lq,
1 n _- n
or iSj: — ,S =1 Iq — a; tbea S = - ;
that is to S9y, in order tp obtain the sum of a determinatie number
of terms of a progression by quotients,, we must, multiply the last
term by the ratio^ subtract from the product the first term^ and
divide the difference by the ratio diminished by one,,
When the progression i» decreasing, we have q<Cli iK,fi9
and it is proper to put the above formula under the form
256 Elemmts if^Are^
1 — q
so that the two terms of the fraction may be positive.
I The two expre'ssionsf of S become moreover, by the substitution
of a 5*"^ in the place of Z,
q—l' l—q
We shall 6nd, according to the preceding formulas^
L For the sum of the first 8 terras of the progression
-^ 2 : 6 : 18 : 54 . . . : 2 X 37 or 4374,
5 = ^^-^=^^^^;-^ = 6560.
q — X »
2. For the sum of the first 12 terms of the progression
4 "" VV "' S5536'
1 /1\" 1
64 : 16 : 4 : 1 : T : . . : 64 I -7 J or
^ -- l — q — nin — ^3 ' — °^ "t- 19660&*
We see that the principal difficulty consists in determining the
Dumerical value of the last term, a very laborious operation, when
the number of terms is considerable.
190. jRemarA;. If in the formula
^- q^l »
«
^e spppose J = 1, it becomes 5 = ^.
This result, which is sometimes the symbol of indeterminateness,
proceeds also frequently (72) from the existence of a common
factor, which becomes nothing, by a particular hypothesis made
upon the given terms of the question. This is, indeed, what takes
place in this instance^ for we know (31) that the expression
5* — 1 is divisible by y — 1, and gives for a quotient
2"-"' + 3""^ + 3"^ + ..- + ? + 1 5
if we perform the division, the value of S takes the form
S = a g""^ + a j*"^ + a j"^ -f- ... + «? + ct;
whence, making 3=1,
Progression by Diffirences and by ^oiienis. 357
We can arrive at the same result by going back to the proposed
progression -^ a : & : c . . . /, which, in this particular case of
9=1, is reduced to -H- a : a : a : a • • • o, a series of which the
sum is equal to n a.
The result |, which the formula gives, may further be regarded
as showing an insufficience of this formula, to give the expression of
the sum, in this particular case. Indeed, the progression being
composed of terms all equal among themselves, is not more a pro-
gression by quotient than a progression by difference. Thus, in
seeking the sum of a certain number of terms, there is no more
reason to make use of the formula
than of the formula
q ^ (<» + ^
relating' to progressions by difference.
191. Of Infinite Progressions by Quotients. Let ther6 be a
decreasing progression
••' (z • • c • a • e IT • . .
of an indefinite number of terms ; if we consider the formula
which gives the sum of a number n of terms, it may be put under
the form
s = ^
aq*
1-g 1-g
Now, since the progression is decreasing,p ? ^s a fraction ; j* is
also a fraction, which will be smaller in proportion as n will be
larger ; so that the more terms we shall take in the progression,
the more
X S'
1-g
will diminish ; consequently the more the partial sum of these terms
will approach to being equal to the first part of <S, that is, to
|--— . Indeed, if we take for n a number greater than any
Bour. Alg. 33
358 Elements of Algebra*
assignable quantityi or if we suppose n = oo,
a
will be less than any assignable quantity, or will become equal to
; and the expression ^ will represent the value of the whole
series.
Whence we may conclude, that the sum of the terms of a prO".
gression decreasing to infinity^ has for its expression
This is, properly speaking, the limit towards which all the
partial sums which we obtain are coptinually tending, by taking a
greater and greater number of terms in the progVession.
The difference between these sums and . , may become as
small as we wish, and can never become entirely nothing, except
when we take an infinite number of terms.
Applications. Let there be the progression decreasing to infinity
*.• T • t • TT • TT • • • •
We have, for the expression of the sum of the terms,
«_ a _ 1 _ 3
The error which is committed by taking. this expression for the
value of the sum of the n first terms, is marked by
« n 3 /IV
Let there be at first n =1 5 ; it becomes
2\3j —8.34 - 162'
For n =s 6, we find
11 1
~ 162 ' 3 "" 486*
Whence we see that the error committed^ when we take | for
the sum of a certain number of terms, is as much smaller as this
number is greater.
Progression by Differences and by Quotients, 259
Furtfaer, let there be the progression
i.i.i.i. 1 • 1.
I • T • i* • t* • TTT • 3'a • • • •
We have
l—q 1_^
... • ft
The expression S = ^ may be obtained directly according
to the progression
-H- a:b :c: die if: gi •••
Let us take again the equations
b =: aq^ c m bq^ d =i cq^ e =z dq . . .
of which the number here is indefinite, and add them member by
member ; it becomes
b + e + d+e + ...=i^a + b + c + d+...)q.
Now the first member is evidently the proposed series, diminished
by the first term a, and consequently it has for its expression
5 — a ; the second member is equal to q multiplied by the whole
series, since there is no last term, or that this last term is nothing ;
the expression of this second member is then q S^ and the above
equation becomes
S — a =: qS. whence we deduce S = ^ .
Indeed, if we develope . mto a series, by the process of
division, we find for a result,
which is no other than the proposed series when we have substi-
tuted bj Cf d . , . for their values in a function of cu
Again, let there be the expression
-^ aiaq :ag^ la^ . .,
and let us make
S = a + aq + a^ + a^ + aq* + a^+...j
whence, multiplying the two members by q^
\
300 JElemenU of Algebra.
Let us subtract the second of these two equations from the first ;
we have
S — q S =1 a; and lastly, S = ^ .
192> When the series is increasing, the expression
l—g
can no longer be regarded as a limit of partial sums ; for the sum
of a determinate number of terms being (199)
1—3 1—2
the second part r— -^ — increases more and more nuraerically in pro-
portion as n increases ; that is, the more terms we take, the more
the expression of the sum of these terms differs numerically from
a
The formula S = rr-^—
iSf in this case, simply the algebraic expression, which, by its de-
yelopement, gives the series
a + ^J + flS^ + fl?*--*
Here is presented a very singular circumstance. Since . _^
13 the fraction which generates the series of which we have just
spoken, we must have
a
9
Now, by making in this equality a =;= I, 9 = 2, we find
--!-_ or — 1 = 1+2+4 + 8 + 16 + 32 + ...,
an equation of which the first number is negative, while the second
is positive and so much the greater, according as q is greater.
To interpret this result, let us observe that when in the equation
we stop the series at a certain term, it is necessary in order that the
equality should subsist to complete the quotient. Thus, in stopping,
Progression by Differences and by Quotients. 261
for example, at the fourth term, a ^,
a
1st remainder + aq
2d + af
3d + « 9^
4th + a 2*
l-q
a + aq + af + af + ^^
we must add to the quotient obtained, the fractional expression
l-q
, which gives rigorously
a , o ^ I ** 9
4
If now we make, in this exact equation, a = 1, y = 2, it
becomes
— 1 = 1 +2 + 4 + 8 + -^^ = 14-2 + 4 + 8—16;
an equality ^hich is verified of itself.
In general, when an expression in terms of a?, which we shall
designate by /(a?), and which is called a function of x, is developed
into a series of the form
a + 6af + ca^ + (ia?^+...,
we have rigorously
/(a?) = a + 6a? + ca^ + da^ + . . . ,
only when on stopping at a certain term in the second member, we
suppose the series completed by a certain expression in terms of a?.
If in any particular application the series is decreasing, this
expression, which serves to complete it, may be supposed as small
as we please, when we prolong the series as far as it is convenient ;
but if the Series is increasing, the contrary takes place, and this
expression must not be neglected. For this reason, increasing
series will not serve for obtaining the approximate value of num-
bers. It is for the same reason, that algebraists give the name
converging to series whose terms go on diminishing, and diverging
to series whose terms go on increasing. In the first, the more
terms we take, the more the sum approaches numerically to the
expression of which the series is the developement ; while, on the
contrary, in the others, the more terms we take, the more their
sum differs from the numerical value of the expression reduced to
a series.
262 Elements of Algebra.
193. Remark. We will condude the priociples relating to infi-
nite progressions, by the following observation. It results from the
definition of progression, by quotients (187), that we may regard
them as recurrent series of the first order, of which the scale of
relation is the ratio of the progression (art. 189). This approxi-
mation is fitted to make known the origin of progression prolonged
to infinity. They owe, as well as recurrent series in general, their
origin to the developement of an algebraic fraction into a series.
We have given (190 and 191) the method for finding this generate
ing fraction for particular progression^. We shall see hereafter
the method for resolving the same question for all recurrent series.
194. The consideration of the five quantities a, ^, n, Z, and S,
which enter into the two formulas
obtained (arts. 188, 189) gives rise to ten particular problems of
which the enunciations do not differ from the enunciations which
relate to progression by difference, except that the letter q takes
the place of r. But we propose, as in progressions by difference,
to determine q and S, a, I, and n being known.
Now the first formula gives
9*"^ = -, whence j = |i ;
by carrying l)ack this value into the second formula, we should
obtain the value of S.
The expression
""^«
gives the method of resolving this question ; Insert between two
given numbers a and b, a number m of mean proportionals^ that is,
a number m of quantities which form with a and b, considered as
extremes f a progression by quotients.
It is sufficient for this to know the ratio ; now, the number of
terms to be inserted being tn, the whole number of terms n is
equal to m -|- 2 ; we have moreover I =: b ', thus the value of q
becomes
Progression by Differences and by Quotients. 263
- Ja^
that is, it is necessary to divide the two given numbers b and a, tJie
one by the other, then of the quotient to extract a root of a degree
indicated by the number of the terms to be inserted plu>s unity.
The progression is then
m+l OT+l TO+1
-^ a I a VLia \^\a _:...o.
\a Sa^ \a^
Hence let it be required tonnsert six mean proportionals between
the numbers 3 and 384.
We have m = 6, whence
7 7
q = ^384 = ^128 =2;
whence we deduce the progression
4f 3 : 6 : 12 : 24 : 48 : 96 : 192 : 384.
We shall soon show a more expeditious method in numerical ap*
plications, for calculating the number expressed by
We shall not stop to demonstrate, that if between all the terms
of a progression by quotients, considered two by two, we insert the
same number of mean proportionals, all the progressions thus form"
ed, constitute a unique progression. The demonstration is analo-
gous to that of 185.
195. Of the ten principal problems which may be proposed con-
cerning progressions, ybt/r are capable of being easily resolved.
The following are the enunciations, with the formulas which
relate to them :
1. a, q, n, being given, to find I and 5.
* q — 1 q — 1
2. a, n, I, being given, to find q and S.
m-l %-i »-i
1/ o V'* — V^
264 Elements of Algebra.
3. a, », Z, being given, to find a and S,
a =
I g^ ^(y— 1)
4. 9, n, 5, being given,. to find a and Z.
5" — 1 ' q* — 1
Two other problems depend on the resolution of equations of a
degree higher than the second ; they are those in which we sup-
pose the quantities a and q, or rather Z^and q unknown.
Indeed from the second formula we deduce
a =. Iq — Sj-f" ^f
whence, substituting this value of a in the first formula I = oj"^^,
or rather
(^S — l)q^—Sq^'^ + 1 = 0,
an equation of the degree n which we have not yet learned to
resolve.
The same will take place, if we wish to determine I and q ; we
shall arrive at the equation
aq* — Sq + S — a = 0.
196* Finally, the four other problems lead to the resolution of
equations of a very peculiar nature ; they are those in which n is
the unknown quantity, as well as one of the four other quantities.
At first, the second formula easily gives the value of one of the
quantities a, 3, Z, and S, in a function of the three others; so that
the whole is reduced to determining n by means of the formula
Z = a 9"-! .
Now this equality is reduced to g" = ~, an equation of the form
a* = t, a and 6 being known quantities. We call these equations
exponential equations, to distinguish them . from those which we
have hitherto considered, in which the unknown quantity is raised
to powers indicated by known numbers.
Let us now attend to the .resolution of these equations, to which
is attached one of the niost important theories in the mathematics,
the theory of logarithms.
Progression bjf Differentes and hy Quotients. 966
U. Cfthe Theory of Exponential Quantities and Logarithms.
<
197. Resolution of the Equation a^ = b. The question con-
sists in finding the exponent of the power to which we must raise a
given number a, to produce another given number b.
Let us at first consider some particular cases.
Let it be required to resolve the equation i* = 64. By raising
2 to its different powers^ we soon discover that 2^ = 64 ; tbea
X :=6 satisfies the equation.
Further, let there be tlie equation 3^^ := 243. We have for a
solution 0? = 5. In short, so long as the second member b shall be
a perfect power of the given number a, x will be a whole number
which we shall obtain by raising a to its successive powers, begin-
ning from the degree 0.
Let it now be required to resolve the equation 2* = 6. By
making x = 2 and a? = 3, we find 2^ =: 4 and 2^ = 8 ; whence
we see that x has a value comprehended between 2 and 3.
Let us put then
0? = 2 + -, . . • (a?' is then > 1 ).
We have, by substituting this value in the proposed eq|i^tian»
9.1 1 i
2 ^^ = 6, or (art. 180) 2» X 2^ = 6 ; then 2»' = |.
or, raising the two members to the power ^j(n) =^ ^«
In order to determine a/, let us make successively
a?' = 1, «/ = 2; we find (^ = g,
a number smaller than 2, and f ^ j == 79 ^ number greater than 2 ;
hence a/ is comprehended between 1 and 2.
Let us put then ^
I
a/ = 1 -| — 77 . . . (a/' is also >• I).
We obtain, by substituting in the esQMnential equation o[x^,
Bout. Alg. 34
366 El&nenU if Algebra.
or reducing
Q«" _ 3
The two hypotheses
a number smaller than I, and
iri
/4Y _ 16 _ ^ r
\Z) - 9 - T^ 9'
a number greater than f ; hence a/' is comprehended between
1 and 2.
Let then a/' = 1 + -77-, ; there results
/4Y+ ?" _ 3 4 /4\;^' _ 3
whence reducing,
/9Y" _ 4
If we make successively x'" == 1, 2, 3, we find, for the two last
hypotheses,
'9\2 81
V8y 64 ^ ^64'
a number <; I + J, and
'9\3 729 . . 217
W "" 512 "" ^ + Si2'
a number > 1 + i ; so that tx/" is comprehended between 2 and 3,
Let oj'^' = 2 ^ — -i the equation in a/'' becomes
a?'
a- ^
W ~ 3' "64^,8/ ~3'
and, consequently.
ii'256\«"_ 9
V,343/ — 8*
Progression by Differences and by Quotients. S67
By proceeding in this exponential equation as in the preceditig,
we should find two whole numbers, k and k -{-1^ between which o?*^
would be comprehended. Putting
*" = * + F'
we should determine x^ as we have determined a^^', and so on.
Let us bring together the equations
we obtain the value of x under the form of a continued fraction
1
» = 2 +
1 +
1
i+i
2 + -T;-
Now, we know (Arith. art. 175), that in a continued fraction the
greater number of integral fractions we take, the nearer we ap-
proach to the value of the nuniber reduced to a continued fraction ;
so that we can, by these means, find the value of x fitted to verify
the equation 2' = 6, if not exactly, at least to any degree of ap-
proximation that we wish.
For example, by forming the first four results according ta the
established law (Arith. art. 169), we find
9 3 5 13
1 J T> T» 19
and the result y differs (Arith. 174) from the value of x only by
■ ■>
a quantity less than j^ or ^. But the approximation is still
greater ; for if we calculate the value of x*^ according to the
equation
V243/ - 8'
we shall discover that o?*^ is comprehended between 2 and 3 ^ so
that
then the 5^ result is
13 X 2 + 5 31
6x2 + 2' ^' 12*
S6e .EhmmUw ^Algebm,
So U)«t V iitkn from the value of a;, by a quantity less than
1 1
or
12 X 5' ^' m'
The result |^ differs from it less than
1 1
General Method. Let a' = i be the equation to be resolved*
By forming the successive powers of a, we find that h i^ compre-
hended between a* and a^+^j then we make
' x'
Substituting this value in the equation, we obtain
a ' = 6,
an equation which returns to
L /hY
c» X o*^ = h whence yZn) = «5
or putting, for the sake of greater simplicity^
— - = c t • . c^ = a.
a*
Proceeding with the operation in this equation as m the proposed
equation, we shall discover that a/ is comprehended between nf
and n' -^^ I, which will give
^ = «' + i77;
substituting this value in the equation of a/^ we shall further be
ted to resolve an equation of the form d^"* =• c{d having for its
value -j;j), and so on. Then, finally^ we shall obtain for the value
of X an expression of the form
. 1
^ n' + _£
By carrying on the series of operations to a convenient length, we
shall have the value of x with any degree of approximation that we
Progre$sion by Differences and by Quotients. 269
wish ; and this degree can always be estimated, since it is indi-
cated (Arith. art. 174) hy the quotient of one divided by the square
of t/ie denominator of the last result at which we have arrived.
198. Remark 1. If we suppose, in the equation a* = &, & <^ a,
« as we have a^ =^ I (24) and a^ = a, it follows that x is compre-
1
hended between and 1, and we must then put a? = -7.
CO
2. If & is a fraction and a is greater than unity, we must put in
the equation
o* = 6, a? = — y,
which gives ary = 6, whence (179)
ay =1 t; and as r is > 1,
we shall determine y, according to the foregoing method, and the
corresponding value of x will be equal to that of y taken negatively.
By means of these remarks, the application of the method offers
no difficulty ; only the calculations, in order to obtain a great de-
gree of approximation, are very laborious.
We can, for the remainder, perform the foflowing examples as an
exercise.
3» = 16 . ^ . . « == 2,46 to 0,01 nearly,
10* = 3 .... a? = 0,477 to 0,001 nearly,.
5» = f . . . . a? = — 0,26 to 0,01 nearly,
(7 \* 3
j^\ = 2 .... a? = 0,63 to 0,01 nearly.
We suppose here that we have converted to decimal fractions,
the results furnished by the method.
199. It may be asked, whether by following the preceding
method we shall be led to a continued fraction of a limited number
of integral fractions, which will give for x a commensurable number
and equal to the last result of the continued fraction ; or indeed,
whether the number of integral fractions will be unlimited, in which
case X will be incommensurable.
In order to answer this question, let us suppose hi the equation
' c^ =1 b^ X equal to a commensurable number — , and let us see
what relation will exist between the numbers a and &, in order that
this value may be admitted, that is, in order that x may be com*
mensurable.
370 Elements of Algebra.
In the fint place^ let a and b be two whole numbers ; we have
s
the equation a* = 6, which we may put under the form a* = 6*.
It is at once evident that this equality can only su]^sist when a
and b are composed of the same prime factors ; for if we suppose
in i a prime factor which is not found in a, and divide the two
members by this factor, the second member will be a whole num-
ber, and the first a fractional number, which is absurd ; then, if we
have, for example,
we must also have
b = a^ 6^/ ^.
Substituting these values in the equation cS^ = 6", we change it to
this,
«"V ^ ;.*»• 5"* = 1A^' 6^ Y^' S^\
This new equality can evidently subsist only as long as the pow-
ers of the same prime factor are equal in the two members ; for, if
they were unequal, by dividing the two members by the highest
power, we should still be led to this absurd result ; viz. a whole
nutrd>er equal to a fractional number.
Thus, we ought to have separately
mp = np^, mq =^ nq'f mr znnr^y W5 = n^,
whence we deduce
m p' ^ W 5'
n p q r s*
Then in order that the value of x may be commensurable, it is
necessary and sufficient that a and b be composed of the same prime
factors^ and that the exponents of these factors form between them-
selves a series of equal ratios. If these two conditions are satisfied,
the value of x is equal to the constant ratio which exists between
the exponents.
Let us suppose, in the second places that a and b are fractional
h k
and equal ^ rji-nl ^^ equation a^ = (* becomes
f-rjj = ^^7) , whence A*i'* = A'*i».
Now, h and A^, k and A/ being always considered as prime
among themselves, it is the same with h^ and A'**, k* and A;'" ; so
Progression by Differences and by Quotients, 271
tbat| in order that the preceding equality may exist, we must
have separately
A* = A», A'» = k\
which leads to conditions like those above, between the numerators
and denominators respectively compared.
Examples. 1. Tf a and 5, being whole numbers, contain but one y
single prime factor, x is necessarily commensurable.
Let there be the equation 4* = 32, which becomes 2** = 2* ;
there results 2 jc = 5, whence a? = |.
Further, let there be 27* =2187, or 3^ = 3^; we have a? = f
2. ^ a 19 composed of prime factors onZy, raised to the first
power, it is necessary that b should be a perfect power of a, in
order that x may be commensurable ; thus x is, in this case, a
whole number, or rather, it is incommensurable.
Finally, let
a =: aSydi whence 6 = «?' 6*' /* 5*' j
the equation o" = 6* becomes
a" ^/" d" = a^* 6»'* /'» 5'** ;
whence we deduce
m z=i p^ n ^=1 ^ nj= r'n :=z sf n*,
or rather pf = q^ = r* = sf -
then
and, consequently a? = p'.
Thus, let a = 10 = 2 X 5 ; it is necessary that b should be
a perfect power of 10, in order that x may be commensurable.
Theory of Logarithms.
.i
200. Introduction. If we suppose that in the equation a* = y,
a preserving always the same value, we replace y by all possible
absolute numbers, we can, for each value of y, determine by the
method of (108) the value of a?, if not exactly, at least to as great
a degree of approximation as we wish.
Let us suppose at first a > 1.
3T3 Ekmmts ofAlgdfra.
If we make successively
a? = 0, 1, 2, 3, 4, 6 ... ,
there results
y = a% or 1, a, a^ a', a*, a* ... ;
then, aS <Ac vaZwe* ofj greater than unity are produced 6y powers
^ofz^ of which the exponents are positive, whole or fractional ; and
the value of tl is so much the greater as that ofjis greater.
Let us make tbeuy
a? = 0, — 1, — 2, — 3, — 4, — 6 . . . ,
there resuks
_ ,11111
y _ a,or 1, -, ^, ^, ^, ^•..;
theu, aS ^Ae t;aZu6« of j less than unity are produced by potoers of
a, of which the exponents are negative ; and the value of x is so
much the greater ^ negatively y as the value ofj approaches nearer to
zero.
On the contrary, let there be a <^ 1 and equal to a fraction -7 ;
by making
X := 0, 1, 2y 3, 4, 5 . • • ,
we find y = (^ ,
,1.1 1 1 1
^'^' ?' Ifl' ^' 5^» i?5---»
and if we make
a? = 0, — 1, — 2, — 3, — 4, — 5 ... ,
we obtain y = (^^ ,
or 1, a^ fl'^, a'', a'*, o'*...;
that is, on the hypothesis of a <^ 1, all the numbers are produced
with the different powers of a, in an inverse order from that in
which they are, when we suppose a ]> 1.
But there results no less this general consequence, that all pop-
sible absolute numbers may be produced vnth any absolutej but invar
riable number whatever j by raising ii to suitable powers.
• N. B. We must always suppose a different from WMfy, for we
know that all the powers of 1 are equal to 1.
Theory of Logarithms. 27^
201. This being laid down, let us suppose that we biive formed
a table containing, on one hand, all whole numbers^ and on a line
with these numbers, the exponents of the powers to which we must
raise an invariable number in order to form all these numbers ; we
shall then have an idea of a table of logarithms.
We generally call the logarithm of a number the exponent of the
power to which it is necessary to raise a certain invariable number^
in order to produce the first number.
The invariable number may at first be taken altogether ar{)itra-
rily, (provided that it be > or <^ 1) ; but when once chosen it
must remain the same for the formation of all numbers, and it id
called the base of the system of logarithms.
Whatever may be the base which we may have chosen, tie;
logarithm of the base is unity, and the logarithm ofiis zero^
For we have,
1. a^ = a, whence log a =i 1 ;
2. a^ = 1, whence log 1 =ii 0.
(We generally designate, for the sake of brevity, the word log-
arithm by the three first letters log^. or pimply by the first letter l^
after which we place a period, and then the nuiQber whicb we, are
to consider.)
Let us see what properties actually belong to a table of logar
rithms, with reference to numerical calculations.
202. Arithmetical Multiplication and Division^ Let there, bd,
in the first place, a series of numbers, y, yf, y", yf'* ... to be mul-
tiplied \;i^th each other. Let us designate by a the base of a sys-
tem of logarithms (which we suppose already calculated) \ and by
X, a/, a/', x!" ... the logarithms of y, yf, yf*, y"' . . .
We have, according to the definition (208), the series of eqiiar
tions
y =:a«,Y = a*^, t=(^\ i^" = a^\..
Multiplying these equations, member by member, and applying th^
rule of the exponents (180), we find
Therefore,
log yy'y'' ... = x + if + x'[ +... =2= log y 4- log yf + log yf' +... }
that is, the logariJthm of a product is equal to the sum of the hg(h
rithms of the factors of that product. • ' ■
Bour. Alg. 35
ST4 Memenis of Algebra.
Let there be, in the second place, two numbers, y and y', to be
divided the one by the other, x and a/ being their logarithms $ we
have the equations
y z= a', / r= a*' ;
whence we deduce (180)
Therefore,
y^
•og ^/ = a? — a/ = log y — log y' ;
that is, the logarithm of the quotient of a division u equal to the
d^erence between the logarithm of the dividend and the hgarithm
of the divisor •
Consequences of these two properties. If we have a muldplica-
tion to perform, by taking in the table the logarithms of the factors,
and adding together these logarithms, we shall have the logarithm
of the product } then, by seeking this new logarithm in the table,
and taking the number which corresponds to it, we shall obtain the
product required. So that by a simple addition we find the resuU
of a multiplication.
In like manner, if we wish to divide one number by another, we
Subtract the logarithm of the divisor from that of the dividend, then
we find what number corresponds to the difference; this is the
required quotient. Thus, by a simple subtraction^ we obtain the
quotient of a division.
90S« Formation of Powers and extraction of Roots. Let there
be, in general, a number y to be raised to the power — ; designat-
ing by a die base, and by x the logarithm of y, we have the equation
niieiuie, by raising the two members to the power -,
M M
— — • *
y* wm a* .
Therefore,
M
log y» = ^.a! = -.logy;
n
that 18, the logarithm of any power whatever of a numi^ is equal
Theory of Logarithmi. 9Tfr
to the product of the logarithm of the number by the eafponent ef
the power.
Let there be, as an example, n = 1 ; there results
log y* = m . log y,
an equation susceptible of an enunciation analogous to the proc-
eeding.
Let there now be nt = 1 , n being any number whatever ; there
results
I n 1
log y» or log vy = ^ • log y 5
that is, the logarithm of a root, of whatever degree^ of a number ii
equal to the quotient arising from the division of the logarithm^ of
that number by the indeoB of the root.
Consequence. In'^ order to form any power of a number, it is
sufficient to take the logarithm of that number in the table, and
multiply it by the exponent of the power, then to seek the number
corresponding to that product. We have thus the power required.
In the same manner, in order to extract a root, it is sufficient to'
divide the logarithm of the proposed number by the index of the
root, then to find the number corresponding to the quotient, and we
have the root required. Thus, by a simple multiplication and a
simple division we find the result of the formation of a power and of
the extraction of a root^ operations in which, as we have seen, the
process is very laborious.
204. The properties which we have just demonstrated are inde-
pendent of every system of logarithms ; but the consequences
which have been deduced from them, that is, the use which we
can make of them in numerical calculations, suppose the construc-
tion of a table, containing on one side all the numbers, and on the
other the logai^thms of those numbers, calculated according to
a given base. Now in order to form this table, we must, as we
have already said, in .considering the equation a* = y, give to y,idl
possible values, . and determine the value of ^r, corresponding to
each of the values of y, according to the method of (297).
The tables which are generally used, are those whose base is
equal to 10, and their construction is reduced to resolving the
equation lO' = y. By successively making y equal to the series
of natural numbers
1 , /^, «5, 4, O, O . . . ,
9T9 Mmenis qf Algebra.
we have the equations
10* = 1, 10* = 2, 10* = 3, 10* = 4 . . .
Let us observe, moreover, that it is sufficient to calculate
directly, according to the method of article 297, the logarithms of
]J)e prime numbers
1, 2, 3, 6, 7, 11, 13, 17...;
for all the other whole numbers, resulting from the multiplication of
these different factors, their logarithms may (213) be obtained by
the addition of the logarithms of the prime numbers.
It is thus that 6 being decomposable into 2 X 3, we have
log 6 ^ log 2 + log 3 ;
in the same way
24 = 23 X 3 ; then log 24 5= 3 log 2 + log 3.
A^ain, let there be
360 = 23 X 3« X 5;
there results ' '
log 360 = 3 log 2 + 2 log 3 + log 6.
. It would be equally sufficient to place in the tables the loga-
rithms of whole numbers ; for by means of the property (203),
relating to division, we obtain the logarithm of a fractional number,
by subtracting the logarithm of the divisor from that of the dividend.
205. By supposing a first table of logarithms already construct-
ed, it is easy to construct as many others as we please after the
method of the former.
For, let a be the base of a first system already formed, b the
base of a new system to be constructed ; let us designate by JV*
any number whatever, by log JV and X, its two logarithms, calcu-
lated according to the bases a and b ; we have the equation
X
b =M
Whence by taking the logarithms of the two members in the sys-
tem of which the base is a,-
Xlog 6 = log JV.
Therefore Xr=|2i^;
log '
which proves that, by knowing the logarithm of a number in a first
system^ hi order to have the logarithm of the same number in a
second sysiemy we must divide the logarithm of the number^ ccdcur-
K
Theory of Logarithms. 277
laUd in the first system^ by the logarithm of the new Sose, ^Iso
calculated in the former system,
Heoce the logarithm of 4, in the system of which the base i& 3,
has for its value p^i log 4 and log 3 being two logarithms cal-
culated in the known system, whose base is 1 0.
Let JV, JV, JV^' ... be a series of numbers, a the base of a 8)'s-
tem already formed, b that of a system to be constructed ; we have
the series of equations,
t
whence we see, that a first table being already formed, if we wisb
to construct a new one, we have only to multiply the logarithms of
the first system by the constant quantity | — -.. This constant quan-
tity, which enables us to pass from one table to another, is called
the modulus of the new table with reference to the old table..
ni. Application of the Theory of Logarithms.
206. Multiplication and Division. Let it be required to find
the approximate value of the product 4^ x If X H-
Calling this product a?, we have (216),
log X = log 31 — log 75 + log 13 — log 12 + log 47 — log 48.
log 31 = 1,49136169
log 13 = 1,11394335,
log 47 = 1,67209786,
arith. log 75 = 8,12493874,
arilh. log 12 = 8,92081876,
jarith. log 48 = 8,31875876,
— 1,64191915 = 29,64191915 — 3Q;
adding .... 5
we obtain , • . 4,6419191
4,6419120 = log 43844.
I = «.»•
Difference ... 89
Tabular difference 99
Then 4,6419191 = log 43844,90;
the product required is 0,4384490 to within 0000001 nearly.
278 Elements ofAlgAra.
Wormniion of the Powers. We observe, in the first place, tha;
siDce in order to obtain the result of raising a number to any power,
we must multiply the logarithm of the number by the exponent of
the power, we must take at first the logarithm of the proposed
number with more than 7 decimals, if we wish to have a product
€xact to the 7*^ decimal inclusive. Now, we find in the work of
Callet, at the end of the common tables, another table, which gives
logarithms with 20 decimals; so that we can always take these
logarithms with two or three decimals more than in the common
tables.
This being premised, let it be required to form the 5^ power of
29; we have (216)
log (29)« = 5 log 29 5
now log 29 = 1,462397998,
whence 5 log 29 = 7,3 1 1 989990 ;
subtracting 3 units 4,31 19900
4,3119868 =4og 20511.
Difference 32
Tabular difference 212
ir2 = «''*'
then 20511 150 is the number sought to within a tenth.
Let it further be proposed to find the value of (2)®*.
We have log 2 = 0,301029995d;
whence 64 log 5 = 1 9,2659 197 1 84.
Subtracting 15 units, 4,2659197
4,2659022 = log 18446.
Difference 175
Tabular difference 235
11^-074
So that 4,2659197 = log 18446,74.
Then the number sought is 18.446.740.000.000.000.000, to
within tens ofirilUons; that 13, the thirteen last figures cannot be
given by the tables ; but our only object in examples of this kind
is, to form an idea of the greatness of the number ; and it is seen
with what readiness it is obtained.
Let us seek, for a new example, the value oM q ) •
Tlie following are the calculations ; by making use of the Aiith*'
metical complements, and without them.
Application of the Theory of Logarithms,
279
By MJOimdical complements,
log 2 = 0,3010299966
c. log 3 = 9,5228787453
log I =—1.8239087409
11 log I = —2.0629961499
adding 6
Without the Jhilhmeticai eomplewients.
log 3= 0,4771212647
log 2 = 0,3010299966
log I = — 0,1760912691
11 log I =— 1,9370038610
adding -}- 6
we obtain 4,0629961 .
The remainder of the calcula*
tioit is the same as that on the
other side.
we obtain 4,0629961 .
The number corresponding to
this logarithm is 11561,02 ; then
0,01166102 is the number re-
quired to within 0,00000001.
Extraction of the Roots. It is sufficient, in this operation, Uy
take the logarithms with seven decimals.
We require the 1^ r )0t of 1 162049.
We have (216),
log V A 162049 = 4 log 1162049.
Tabular difference 374
Diff. of the Dum. 0,49
log 11620= 0662061
log 1 1^20,49 — log 1 1620 = 1 83
Then
adding 4,
log 1162049 = 0662244
log 1162049 =6,0662244
I log 1162049 = 0,8664606
4,8664606
4,8664587 = log 7352.9.
3366
.1496
183,26
Difference
Ta{)ular difference
^^'1^-032-
then
4,8664606 = log 73529,32.
So that 7,352932 is the root required to within 0,000001.
Let it be ^required to find the value of
11 11
Vi^ ; we have log ^i^ = j\ (log 13 — log 27)>
By Arithmetical complements.
1. 18 = 1,11394336
compl. 1. 27 = 8,56863624
L If = — 1.68257959 = — 11 + 10,68257959
tVI. H = — 1.97114360;
adding 5
we find 4,971 14360 s log 93571,49.
280 Elements of Algebra.
Then the root required is 0,9357149, within 0,0000001.
We shall find, in the same manner,
7
>!(
IIXS
9
\ = 1,164118; {7Sy = 11047390000000;
(0,0457)^ = 0,000000000000000082984.
507. Calculation of Mgebraic Kocpressions by Logarithms,
liBt u« suppose that we have, found, for the value of the unknown
quantity of a problem, the expression
3
__ 4/{a^ — 6^) . 3 g
, «r — ,
\^(a + 6) . vc (/
^6d that by giving to a, b, Cy d, particular values, we wish to obtain
the numerical value corresponding to this expression ; we can, by
means of logarithms, reduce the question so as to have only addi-
tions and subtractions, simple multiplications and divisions to per-
form.
We lutve, in short, according to the properties (215 and. 216).
3 '
1. a? = 1. ^{a^ — b^).^a— 1. V(a + b) ^Td.
fiut
3
L V(«* — &*)-3a = VP- (« + ^) + \'{a — b) + I. 3 + 1. o]
and
1. V(a + b) s/7d ==i[\'{a + h) + i\.c + ild];
then
1. a? = I [1. (a + 6) + 1. (a — 6) + 1. 3 + 1. a]
— iU'{a+b) + \lc + i].d], •
an expression which only requires additions, subtractions, and sim-
ple divisions to perform, when a, b, c, d, are given numerically.
For example, let
a = 60, 6 = 15, c z= 16, rf = 9 ;
the expression becomes
1. x=l\l. 75+1. 46+1. 3+1. 60]— |[l. 76+^1. 16+ Jl. 9] ;
calculating separately the sum which is between the two first paren-
theses, and the sum which is between the other two, then taking a
third of the first, and Ao^of the second, we shall find
Applkaiion of the Theory of Logarithms. 381
1. a? z= 1,92784875 — 1,47712136,
or 1. 0^ = 0,4607275 ;
then a? = 2,823108.
Let there be further the expression
^ — a3-.3aa6 + 4e^'
We can at first, by separating the factor a' in the nutnerator,
and the factor cfl in the denominator, present the expression tinder
the form '
Let us now put
4b^c 36» ft*
m = — S-, n = —5-, p = -a :
the expression becomes
*"" a — 36 + 1W '
Of applying the logarithms,
1. Of = 1. a + 1. (2 a — n + p) — 1. (a — S6 + m),
an expression easily calculated, as soon as we have found the values
of m, n, p. Now the equations
46*0 3^ 6*
give ^
l.TO=:I.4 + 21.6 + l.c — 21.a, 1- n =3 1. 3 + 31. 6 — 21. a, '
J, jl? = 4 1. 6 — 3 L a.
The an of these transformations consists inf reducing thfe frac*^
tional expression to another, the terms of which are aH of the first
degree, by calculating separately the otb^r expressions wbipi^^rei^
quire only multiplications, divisions, and formatipifs ojf powei;8» /
We shall find, in the same manner,
aS ^
1. — r-g — =l,(a+6)+ l.(a — b) + comp. 1. 6 + comp. 1. d — 30,
Bour. Alg. 36
SJ83 . Eltmenis cf Algebra.
h and hf b^iog calculated according to the forEoulas
1. A = 1. J + 2 1. — 2 I. a,
1. A' == 1. 4 + J. € + 1. ^ — 1. a.
208. Exponential Equations. We have explained (209) a
QIAtbpd'fop resolving the. equation a* =e &, and yre have deduced
4[^g9Q.it;tb^ theory of. logarithms; but ^s there are tables actually
constructed, nothing prevents the use of then) in resolving this sort
ofequatipn?* v .
Nowy\jf we'take the logarithms pf the two members. of the equa-
tion a* = i&, it becqnjes (216)
07 X 1* ^ = !• &> whence x = ^.
1. a
For example, let us take «^in tlie equation 3^ :;=: 15, which, by
the method of (200), has given x = 2,465, to within 0,001 ; we
deduce from this equation
_ 1. 15 _ 1,17609126 _
''^ - W^ 047712125 "" ^'^^^ ' • •
The equation a^ = i, is called an exponential equation of the
Jjr^t oriifir;:bvit we can have equations of the form
OP
0* r= c j a* = d . , . ;
we call them exponential equc^ions of the, second, third, . • » order.
In order to form an idea of the expression cf , we must suppose
that b is raised to a power of a degree indicated by x, and that a
is r^iised to a power of a degree indicated by ^*
4r
c
In the same way, a^ indicates that after having raised c to the
power Qf a degree indicated by x^ we have then raised h to the
fMOffrer 9f 4 «tegree shown by c^^ and finally, that a is raised to a,
power of a degree indicated by i*^ ,
According to these principles, let us take the Ic^rithms of the
two members of the equation a^ = c ; it becomes
i* X !• a =* 1. c : whence ft* = L ,— ,
La'.
ApplicaUan (fdte Tkeory of f/>gantkttu. J283
or taking agkin the logarithms • «
v> 1 r 1 1. c II 11 *u ^ . 1. Lc — * L L a
a? X 1. 6 *= 1. 1 — = 1. 1, c — - 1# 1. a : thed x = i-i; .
ha ' J. 6
p. c being a decimal fraction, we can determine its logarithiti ac-
cording to the tables, as we determine the logarithm of any other
number.]
Let it be further required to resohre the equation o^ as <ii
Taking the logarithms, we have .-.>;> .'ir
X
b^ X log a :3 log d\ .
whence
La'
taking again the logarithms - <
1. Lrf^Lf. a
c* =
1.6
and performing the same operation upon this efqiiation as upon the
preceding, '>
X X\.c^\ ^^'^~^'^' "" = \.{\A.d—\.\.a)^\.\.h}
then
l.(Ll.df—l.l.a)— LLi
X
\.c
We should resolve by a similar process the exponential eq«atioflp
of a higher order. These formulas are exact, considered aigt-
brakally ; but in the applications it is easy to see that they would
•give values which are only slight approximations^ ^nd we could not
even form a very just idea of the degree of approximatipo*.
209. Refi%ark. It may happen that in the calculation of ,algf^
braio expressions, we may be led to take the logarithm of a nega-
tive number.
Let it be required, for example, to calculate the expression
a« — 63
C '
we have '
1. a? = K (a + 6) + 1. (a — 6) — 1. c.
^284 ilanents of A^ebra.
If we suppose a < 6, 1. (a — b) becomes I. (— «). Bfow must
we k)tjei|)ret tbis< result ?
Let us begin by showing that in every system of logarithms,
n^ative numbers have no logarithms. Indeed, if ^be io^e is posi^
|ive|, there exists no power of that base which can produce a nega-
tive number ; but we cannot take a negative base ; for that quantity
raised to it's different powers, would produce numbers, some posi-
tive, j6chers negative ; now the character of a base is, tb produce
all consecutive numbers, when we rais&it tQ< powers which increase
progressively.
So that it is with the logarithm of a negative number, as with its
square root, fourth root • . . ; 1. (— m) is an absurd or imaginary
expression.
We should seek in vain to vreaken this proposition by the follow-
ing reasoning ;
We know that ( — m}^ r^^^f ^h^ni by taking the logarithms on
both sides, ^ .
. r„x 2 1 ( — w) = 2 1. m ; whence 1. ( — m) = ],. m*
This reasoning fails in this point, that we put
.,,;..., l,(-«)2,= 21/(_m.)i '
according to the property of (216); now, this property supposi^^
that we can have ihe equation r-r »» = a*, a being positive, which
is impossible.
Let us return to our object. Two circumstances may be pre-
*bted. '
' Wi^t logarithms are employed only as k more simple meithod
0^ finding the numerical value of the expression ; that is, the cal-
ibulation by logarithms is then a calculation purely auxiliary, and
its use is not indispensable. Such is the case in which is found
the expression
a2 — 69
^ = -7-'
which might be calculated directly, without the assistance of loga-
rithms. It is negative, since we have supposed a <C ^ ; but if we
wish absolutely to calculate its value by logarithms, we must com-
mence by changing its sign, which gives
— - — J whence 1. i ^ = 1. (6 + a) + 1. (6 — a) — 1. c.
Aj^licaiion of the Theory (f Logarithms. 285
an expresssion which no longer contains any but logarithms of ab'^^
solute numbers } and when we have obtained the numerical value
53 aft
of > • ... ;,., it is sufficient to take the result with the sign — ^.
c
Or^ the use of logarithms is indispensable in finding the value of
an unknown quantity, and then 1. ( — m) denotes an absurdity or
an impossibility in the question.
Let there be, for example, the equation 3* = — 9.
By applying the logarithms, we find a? 1. 3 = 1. — 9, an absurd
equation ; and indeed, to whatever power we may raise 3, it is
impossible to produce — 9.
To resume ; whenever in the use of logarithms we have for our
object to find more simply the numerical value of a proposed ex-
pression, 1. ( — m) only indicates that the quantities, which ent^r
into thei proposed expression, are to undergo one or several changes
of Sign, before we apply to it the logarithmic calculation.
But if, in order to resolve an equation, we are obliged to h»f^
recourse to the logarithms, the expression I. ( — m) is a symbol of
absurdity analogous to the even roots of negative quantities.
3 10. Proportions and progressions by quotient.
Let there be at fir^ the proportion aibiicix; we deduce
be
from it 0? = — , whence, by applying the logarithms,
' 1 • "• •
1. a? = L 6 -|- 1. c — 1. a,
or laA b:\c.\xy
which proves that, if four numbers form a proportion^ their logo-
rithms form an equidifference, of which the 4^** term is the log-
arithm of the 4*^ term of the proportion.
Funher, let there be a progression by quotient
TT a : 6 : c : d : e :/ : ^ : A : . . •
It results from the definition (199), that we can write it thus ;
a^6 c d c ^
whence taking the logarithms on both sides,
b c d e f '
or
1. a —1 1. J =: 1. J — 1. c = 1. c — 1. d = 1. d — 1. c = 1. e — - Lf,
286 ElemenU of Algebra.
or finallyt
-r- la. IJ. Ic. Irf. Ic. . • . 5
then, ^ any numbers a, b, c, d . . . are in a progression hy quo^
iienis^ their logarithms are in a progression by differences. The
converse is evident.
This proposition renders the algebraic definition of logarithms
(214) similar to that which is given in the Arithmetic ; logarithms
are numbers in a progression by difference^ corresponding term for
term to numbers in a progression by quotient.
N« B. We have already shown this analogy in our Treatise
upon Arithmetic (280).
It is especially in resolving questions relating \o progresaioiis by
f]Motients that the use of logaritlims is useful.
1. If we call u the last term of a progression by. quotient, we
have (200)
« z= a f^^^ whence 1. 1« = 1. a *f- {^ — I) '• i*
For example, let it be proposed to find the 20* term of the pro-
gression
1 .3.9.27.
The formula becomes
Ltt ±= 1. 1 + 19 (1. 3— 1. 2) = 19 (1. 3— 1. 2), [for L 1 s= 0];
and we obtain, all the calculation being performed,
I. w = 3,3457339 = 1. 2216,84 ;
whence
u = 2216,84 to within 0,01.
2. If we wish to insert between two given numbers a and 6, a
number m of mean proportionals, we haye^ to determine the ratio,
(206) the formula
o = ^ U ; whence 1. O' = , -
Let
a = 2, 6=15, m =: 50 ;
it becomes
1. 16 — 1. 2
1.6 — la
l.y =
6r
Application of the Theory of Logarithms. 367
and we obtain, all the calculation being performed,
1. y = 0,0171581 = 1. 1,040299 ;
then q = 1,0402299.
If we wish to calculate immediately the twentieth mean prdpor-^*
^tio/,' which is the tweoty-^rst term of the progression $ we have
51
a?=:2(J2-) ; whence loga?=1.2H ^-^-^j '—^',
or, aU.the calculation being performed,
1. a? = 0,6441913 = 1. 4,407489 ;
so that the twentieth mean proportional is 4,407489.
(3.) We have found (201), for the expression of the sum of the
terms,
o — =— — i — •
q^l g— 1
whence
L S = 1. a + 1. (g» — 1) — 1. (j — 1).
We see, according to this formula, that we must commence by
calculating the expression 5'*, by putting 1. g» = n 1. q, after which
we can easily determine q* — 1, and consequently, 1. (g» — 1.)
We shall soon have occasion to apply this formula.
(4.) Knowing a, g, and u in the formula m = « j*^^, we may
wish to find the value of n. Now we have
1. tt = 1. a + (n — 1) !• ? ; whence n = 1 -|- ' — f^ ' .
Let it be required to find the number of terms of the progression
of which the first term is 3, the ratio 2, and the last term 6144.
We have
_ 1.6144 — L3_, , 3,31132995 _, , ,1 _,o
»-ii 1.2 "^^ "•• 0,30102999 ~" ^ i- tt - 1^;
,. . ^331132995. ,, ,. , 6 ,
(the quotient -^^^^^ is equal to 11 + g^Hj^ggg ; but we negr
lect the fraction, as it proceeds from the use of logarithms).
211. Questions relating to Compound Interest. One of the
most iniportant applications of logarithm3 is that which wa make to
questions upon the interest of money.
298 . ISHmp^qf41g^^.
ViTst General Question. 4nAfi sum vJutt^e/n: i(i»g ip¥t at mi&fr^
^ifor a certain timet at a given rate^ and at conipound interest,
ihat isj on the supposition that the interest of each year is added to
the principal of the preceding year, it is required what the amount
VfiUlf^ at the end ofthegimen time.
Let us designate by a the sum put at interest, by- n tbe nubber
of years, and by r the interest of one franc per year, which is only
ithe hundredth part of the rate of interest for 100 francs.
Since 1 franc produces r, a sum a will produce ar^ tit the end
<of a year ; so that at the end of the first year, the principal wiU
Aave become a + a r, or a (1 + r).
Let a (1 -f* ^) = ^^ ; this new principal will become, at the end
of the second year, a^ (1 + r) ; then the original principal, or a,
will have become
<(1 +r), ora(l +r)2.
We should obtain, ia the ^ame manner, at the end of the third
year^ a(l + r)^, and in general, at the end of the n^ yev
41.(1 --I- r)*« Then expressing this value by A^ we shall have the
equation
w3 = a(l + r)*, whence 1. ^ = 1. a + n X 1. (I + r)\
^Applicati^n^ It is required what a sum of 80,000 francs put
at compound interest, at the rate of 5 per cent, will amount to at
the end of 30 years.
It is sufficient to make, in the preceding formula, .
■ o ^ 30000, tt = 30, r = tK = 0,05 ; .
which gives
1.^=1.30000 + 301.(1,05).
1.1,5 = 0,021189299;
30 1. 1 ,05 = 0,63567897
I. 30000 = 4,4*712125
1. wa = 5,11280022 m L 129658,27.
Then A = 129958^,27.
The formula .^ =: a (1 + r)* containing four quantities, a, r, n,
w3, gives the solution oifour different problems ;
(I.)* To determine A, knowing a, r, and n ; this is the question
which we have just resolved.
Applicalum of the Theory of Logarithms. 289
(2.) To determine the sum which mttst actually be put at interest j
in order to obtain^ at the end ofn years, an amount A, by svppos"
ing the principal put at compound interest , at the rate of r fir one
. franc.
New, from ihe equation ^4 = a (I + r)*, we deduce
1. a = 1. ^ — nl. (I +r);
and this new formula will give the value of a.
This second question constitutes the rule of compound discount ;
for it is reduced to finding the actual value of a sum A payable ia
n years, by having regard to the interest of the sum and to the
interest of the interest.
(3.) To determine the rate of interest at which we must put a
sum a, in order to obtain at the end of n years, at compound inter-
est, another sum A.
The formula would be
n
1 + r = J-, whence 1. (1 + r) = - — ^-^.
Knowing 1 -}- r, we should easily obtain r, and consequently, the
rate of interest for 100 francs.
(4.) Fiiialiy, to determine the time during which a sum a must be
put at compound interest, at the rate of r for 1 franc, in order to
produce an amount A.
The formula would be
If we wish that A should be double, triple, quadruple • • • of a,
the formula would be simplified.
Indeed leiAi=ika; the formula A zz a(l'\* r)*, is reduced
io ka =: a {I + r)*, whence
Ik
that is, the value of » t* independent of the principal originally put
at intereet.
Second General Question. To determine what sum- must be
actually put at interest in order to receim at ihe end of each year, a
given amount b, so as to be entirely reimbursed for the principal,
the interest of the principal, and the interest of the interest after a
Bour. Alg. 37
290 Elements of Algebra.
number n of years ^ the intereet being at the rate oft for one franc
per year*
hefa be the sum required ; this priocipal would become, at the
end of n years,
a (1 + r)».
It is necessary then in determining what the sums paid each year
will become at the end of the n^ year, that the amount of the
results should be equal to a (1 -{^ r)".
Now, b given at the end of the first year, or at the commence-
ment of the second, becomes at the end of the n^ year
In the same way, b given at the end of the second, or at the
commencement of the third year, becomes at the end of the n^
year
6(1 +r)»^«.
We should find, in a similar manner,
6(1+ r)»-3, b (J + r)»-S . • . 6 (1 + r), 6,
for the values of the other sums 6, at the end of the n^ year. We
have then the equation
a {I +r)« =
b{l+r)*-^+b{l+r)*^+b{l+r)^+...+b{l+r)+bi
but the second member of this equation, considered in an inverse
order, is evidently the sum of the terms of a progression by quo-
tient, the first term of which is 6, the ratio 1 + r, and the number
of terms n.
So that this sum has for its expression (201)
b{l+r)-^b b[{l+r)*-l] ^
then finally we have the equation
or, applying the logarithms,
1. fl = Li + l.[(l + r)«— l]_Lr— iil.(l+r).
This new formula, containing four quantities, a, 6, r, », also gives
rise to four different problems.
Application of the Theory of Logarithms.
291
The following are eounciations of several questions which are
connected with the preceding.
Required the number of years for which we must put a^um a,
at compound interest, at 5 and \Q per cent, in order to double that
sum.
(Ans, At 5 per f, 14 years 2 months; at 10 per |, 7 years 3
months.)
Required the sum which must be put at interest at the present
time, in order to produce for 12 years, at the end of each year, a
sum of 1500 francs, so that the whole principal and interest may be
repaid at the end of the twelve years, the interest being seven and a
half per cent per year.
{Ans. 11602,91.)
A person has bought a property of 100,000 francs, which is to be
paid for in fifteen equal payments at compound interest ; the rate
for each interval of payment is 5 per cent. Required what will be
the amount or the quota of each payment.
{Ans. 9634,22.)
A certain number of men a increases every year by the hun-
dredth part of what it was the preceding year ; required in how
many years the number will become ten times greater.
{Ans. 231 years nearly.)
Suppose that from a barrel of 100 pints of wine, we draw each
day a pint, which we replace by a pint of water ; required, 1. how
much vnne will remain in the barrel, when we have replaced the
fiftieth pint; 2. in how many days the wine wUl be reduced to one
half, one third, or one fourth.
Ans. to the first part of the question, 60| pints.
Ans. to the second part, 69 days for the half, 109 days for the
third, and 138 days for the fourth.
NOTE.
IV. Of Continued Fractions*
166. Continued fractions originate in the approximate value
of fractions whose *erms are considerable, and prime to each other.
In order to be better understood, let there be proposed the frac-
tion ^{|, of which it is easy to show that the two terms are prime
to each other, and which for that reason is (157) irreducif^lc.
By leaving this fraction under this form, it becomes difficult to
obtain a just idea of it ; but if,, by means of a known principle, we
divide its two terms by 159, which does not change its value, it
becomes /^^tv, or by performing the division indicated in the de-
nominator,
1
3 + V.V
This being premised, let us neglect for ihe moment the fraction
T. V y ^'^6 fraction ^, which results, is a liltle greater than the pro-
posed fraction, since we have diminished the denominators.
On the other hand, if, instead of neglecting j^^, we replace this
fraction by 1, which gives .-y-T—i- or 7, this new fraction is, in its
o -|- 1 4
turn, smaller than the proposed fraction, since we have increased
tlie denominator.
Whence we may conclude that the fraction }{| is comprehend-
ed between ^ and \. This gives already a sufficiently exact idea
of the fraction.
If we wish a greater degree of approximation, we have only to
perform the operation with jV*, , as we have done with J||, and the
proposed fractio;) beco.nes
_1
3+ '
9 + H-
I
If we neglect |^, ^ is greater than ^-^^ whence it follows that
h%y '* smaller than \^\. But . is reduced to -j^ or ^ ;
294 Jfote.
so that the proposed fraction is still comprehended between \
and /j.
The difference of these two last fractions, ^duced to the same
denominator, is
% — 27 1
84 ^"^gJ-
Then the error we may commit by taking \ for the value of the
proposed fraction is less than j'^.
By performing the operation upon ^f , as we have done with the
preceding, we have
and the proposed fraction may be put under the form
1
1
3 +
9 + i
1+*
15'
Let us neglect ^V i the number |, or 1, is greater than j-f, then
I 1
9+1' *" lO'
IS smaller than ^y^.
Then
1 1 10 . u 150
i ' o' a + ^r. or 31. 's greater than ^.
Whence we see that jf | is comprehended between /^ and ^f •
The first is too small and the second too large.
Now the difference of these two fractions is |^f — /y, or j^ j ;
so that the ferror which we commit, by taking either ^y, or ^f, for
the value of the proposed fraction, is less than j|j.
We see how, by this series of operations, we succeed in finding,
in more simple terms, fractions which give the approximate values
of another fraction of which the terms are very considerable.
The expression
1
3 + ^-
9+i
1 + i
15
is what we call a continued fraction.
Kote. 295
In general, we understand by continued fraction, a fraction
\yhich has for its numerator unity, and for its denominator » whole
number, plus a fraption which has also unity for its numerator, and
for its denominator a whole number, plus a fraction, and so on.
Frequently the proposed fractional number is greater than unity.
So that in order to make the definition of a continued fraction more
general, we must say : A continued fraction is an expression com-
posed of a whole number, plus a fraction which has for its numera-
tor unity, and for its denominator, &z;c.
Such is the expression
(a, 6, c, J, . . . Being whole numbers).
167. By reflecting on the method which has just been pursued-
in order to reduce ^f f to a continued fraction, we see that we first
divided 493 by 159, which gave 3 for a quotient, and for a re-
mainder 16. We then divided J 59 by 16, which gave for a quo-
tient 9, and for a remainder 15; we then divided 16 by 15, which
gave 1 for a quotient, and 1 for a remainder. Thence it is easy to
deterrfiine the following process, in order to reduce a fraction or a
fractional number to a continued fraction.
Proceed with the two terms of the proposed fraction, as if to
find their greatest common divisor (49). Continue the operation
till there is obtained a remainder equal to zero, and the successive
quotients at which we shall arrive, will be the determinators of the
fractions which constitute the continued fraction.
In the hypothesis in which the proposed number is greater than
unity, the first quotient represents the entire portion which enters
into the expression of the continued fraction.
We may, according to this method, reduce to continued frac-'
tions the two numbers tVV ^"^ IH*
The following is the fornqi of operations
(I ^ 149 ^ ^ 1 1®I?I?
(1.) \^\ ¥|2|3
1
2'
396
^ote.
Then
65
149
1
2 +
3 +
2 +
2 +
1
P-)
whence
829
847135
2 2
77
1
5819
1 3
1
15'
629
347
^=2 +
1
2 +
i +
1 +
1
3 + —
^19
Continued frnctions possess a greater number of properties, the
discovery ol whicli has been the object of the labotirs of the most
celfbnited geometricians. We will here me jtion the elementary
properties only ; those of which we make frequent use, and whose
denu>nstrations are founded on the first principles of Algebra.
We refer, for more ample details, to the Additions of Lagrange to
Euler's Algebra.
QUESTIONS FOR PRACTICE.
DIVISION.
Cases in which the Divisor is not an aliquot part of the Bf^vidend*
1. |--7— = a — ax + aar^ — oic^ + aa?* — .....
q a a a j^ a a ,
. a a ^^ a •, g , g ,
*• j:ri -; + P + ^ + ?'T"
6- r:ri = 4 + -p-* + -p-* +-64-** +
POWERS OP POWERS.
1. [((o")»)']*= o"w.
-.n-«
3. [((a—)—)"'] =«"*»
4. r((«")"")~'l sso-"^
^our. ^. 38
#
10.
298 Questions for Practice.
7. (o»6-«c'd)' = a"'6-*c''d'.
9. [o8 (a + i)2]» = a*«(a + 6)*».
(a" 6" c' d-«V-' _ a-*' J-"' c-*' <??■
13. (_o»)« = — a".
14. ( — 6-3)4 = J-M,
15. [(( — a)')4]5 = o".
16. K — a)-3]^ = — «w.
17. [( — o)-*]-6 = a«.
18. ( — o)2» = a"".
19. ( — a)*»+i = — o*H-i.
HOOTS OP LITERAL EXPRESSIONS.
Roots of Simple Lateral Expressions.
m
li j^a^ = a*.
m
«
«
««J«^ a* ftp
4.
d*'/" — (fiefs'
r
6. ViT«®«***^y*~®* = Jc^a?'^^*-*.
Questions for Practice.
399
27 «9 aJa (o» + a^)-«" _ 3b*t^a*
8 A-** A*
2 A' {a* + a»)^ '
' ' ^ 32 c« d-15 — 2ac(2 + a?)«'
10. l^(2^a^b^x^^^^^^^^
1 1 f/^J^ 1 ^\ _
63^2^-
THE ARITHMETIC OF ROOTS.
Addition and Svbtraction.
1. i y'a + c ^a — d t^a = (6 + c — d) y'a.
fi A R A fi
2. 3 V5 + 17 V5 — 12 V^ — 7 V5 = \/5.
3. 6 V2 — 5 -v/2 + f v2 — I y2 = f V2.
. .. * *^ * 26 * • 2ft\*
4. 6v|-2VH-av| — '^Vl = (4 + a--~)vf
6. 5v9—2v^l4+v^— 5^14—2^9=3^9— TVU+V^'
6.
7.
.
r 7 m 3
10 V2 + 5v« — 7v/5 + 2Vfl
■sj
•<
7 m 3
6 v^2 + v8 + 4 V^ — 3 Vfl
7 m 3 ^g
— 3 v^2 — 9 v8 — 3 V^ + Vfl + V«*
J2V2 — 3^8—6^5+ Va6
r 5 m «
13v/12a%c + 17^3 — 5^6
•
7vl2a26c+ 2^6 + 3^)3— 2av/c + 4y9a
6 5 7
^— 20vl2a^ic + 9v/12a26c + \/c — Sv^o
20v3 — 3^6 + ^s/\%c?bc — (2a— l)\/c— f \/9«-
SCM) Questions for Practice.
3
12 V7 — 3 V6 +IV11 — 5 Vi3
16 v6 a6 — V 9 c^ + 3^7 a — vlO
5 m 4 4
8^/9 (?— 6 v^7a + 3v6ai + 2v/10
13v^a6 + 7v9c3 + 8v7fl — VlO— 2vl0,
Division*
m
1. Vo:V6 = Jj-
3. a:C^J = Jy.
4. a : \/a =: Va.
3
S 5
6. V«i-^c« : J^-s^ = J— ^r-.
7
15 ,
8. ^<^be:A/ab^<?= ^^.
3 6
10. 4-V/12 :2V3 = 2-v/V-
11. V6'* = 2=V2.
Questions for Practice* 301
Sn 3» 6n
13. c V(a^ — a^) : V(« + <») = c V(a — a?).
14. v(a2 6 _ J9c) : s/{a — c) = b.
15. v(a^_^S):(a-;r)J^.
• EXPRESSIONS.
m Z
1 --
m "L £ 1 1
3. Vfl~iPc« = a" i~ C" = (a* b^ c«)".
m
n p Jll_iL *
4 1^ 3 _a
6. cy^a^ + 'T — :=ica* + da ^.
6. ^a^b c=zJbK^=z {c? b cy.
8. VXi±^) = (e + d)^-i
3QS Questions for Practice.
ARITHMETIC OP FRACTIONAL POWERa
Multiplication.
m p m J p mq-^np
1. a» X a^ = a*^ = a «« .
m p m p mg— up
2. a* X fl « = a» i^ zz: a "« .
3. a «Xo «=o " "«=a ^ »~ ' = a "« .
4. a^ X o^ = fl^* = a^ l/a^^
.._l 7 1 3 1 SO
6. a * X a* X o * = a" = a s/a-
6. a *Xa *=a ^ = -^ —
a^a^
__3 ^51 1 3 C ' 3 fl
8.
8
4.
ff 7 «^ issa 867 105
fi 3 A 4 9 3 190 190
10. V V«^ X VV«® = «^^' «^ = V»^ == Vfl • V«-
»• j^x, j'4-T-?-*=<^-«"(«+')-»
(a + «)8 V(e^ - ff
Division.
I.. o»:a'=: a* « = a "« .
2 y ^ I P mq+np
3. a *:a« = a " « = a V^rn/ =: « «* ,
Questions for Practice. 303
1
. i J i ca ^^ c
3
3 1 .„^7 1 /i^Ti* /»2 4
c c
i 1
7. A:_^ = __3^.
a '6^ g T d ^ _ o « 6T t c _ a' ^ c ". i^^ d"
IS
4 6 13 3
9. {o? — 2Va^ 63 — aVa^ 6^ + 2 6 ^6) : ( V^ — V*)
= (a»— 2a^ 6^ — a* 6^ + 2 b^^) : (a^ — 6^)
=: o^ — 2 i* = aVa — ^s/h^-
APPLICATION TO THE FINDING LOGARITHMS OP PRO-
DUCTS, QUOTIENTS, POWERS, AND ROOTS.
For General or Literal Expressions,
1. log a 6 c d = log a + log b -\- log c -{- log d,
fs
^' ^^S^ = log/ + log g- — log c — log d.
3. log a^b*(P = wlog a + n log 6 + 2; log c.
^- '^ 77:^ = »i log a — nlog J —p log c — j log £?.
m p
m . p
6. log a* 6 « c = — log a — t- log 6 + log c.
6. log vftf»6"^c« = — log a — log J + — log c.
304 ^ttesHonafor Practice.
7. log j^ = log a + ^ log c — log 6 — J log d.
8. log ^ — — — ^^— =nlog(o4-ft)+'nIogc — hg(c-{-dy—^logd.
(c -)- d) ^tP
*• '°S J^TfT)' = - «» Jog (« + *")•
10. log -5p-i = - i log (a + b).
11. log v(a»— a^) = ^ log (a« — a:») = ^ log (a + *)
1
+ -log.(«t — a;).
12. a; log a = log a*.
13
. tt log a + «i log 6 — p log c = log — J-
14. »log(a + y)+logc — mlog(a — y)==log-^^^f^
n
15. -log(2a + 36) — flogc = log ^(^3'''^^l
Vc*
1
THE END.
I
r.
r
I
§