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ELEMENTARY ALGEBRA 



THE MACMILLAN COMPANY 

NBW YORK • BOSTON • CHICAGO • DALLAS 
ATLANTA • SAN FRANCISCO 

MACMILLAN & CO., Limitbd 

LONDON • BOMBAY • CALCITITA 
MBLBOURNB 

THE MACMILLAN CO. OF CANADA, Ltd. 

TORONTO 



^ ■J 



JOHN WALUS (1616-1703) 

Professor at the UniverBlQr of Oxfoi^ In 1686 he published it 
Treatise of Algebra which was the means of making the science of 
algebm more widely known in England. 



ELEMENTARY ALGEBRA 






^econD i^ear Course 



BY 

FLORIAN CAJORI 

OOLORADO COLLJBOE 
AND 

LETITIA R. ODELL 

NOBTH 8IDB HIGH 80HOOL, DBNYBB 



THE MACMILLAN COMPANY 

1916 



• •• -•, • •••-•• • 

• ' • • •••••••• 







GOPTBIOHT, 1916, 

bt the magmillan company. 



Set up and electrotjrped. Published July, 19x6. 



CAJORI 



J. 8. Cashing Co. — Berwick A Smith Co. 
Norwood, Mass., U.S.A. 



PREFACE 

This book contains a brief review of the fundamental opera- 
tions of algebra followed by a thorough presentation of the 
topics usually included in the work of the third half year. 
The material is so arranged that the choice of topics for review 
or advance study may be easily made. 

The book contains not only a large number of practical 
problems but also practical applications of graphs. Material 
for drill in the manipulation of exponents and radicals will be 
found in Chapter VII. Great pains have been taken to make 
the subject of logarithms accessible to beginners. The most 
difficult part in learning to compute with logarithms is the 
process of " interpolation." Of the various arrangements of 
logarithmic tables which have been suggested, we have selected 
the one which renders interpolation easiest. The chapter on 
logarithms is introduced earlier than usual, on account of the 
great practical importance of the subject. If desired, it can 
be taken up later, after Chapter VI. The concept of disfunction 
does not receive isolated and abstract treatment; it is pre- 
sented as a fundamental idea in proportion, variation, and 
graphics. Its connection with problems of everyday life is 
firmly established. 

The aims of the First Year Book have been kept in view in 
the preparation of this text. Emphasis is thrown upon clear- 
ness of exposition and the use of expressions which recall the 
axiomatic processes involved. Continued stress has been laid 
upon oral exercises. 



911236 



vi PREFACE 

Several of the practical problems given in the text were 
suggested by the perusal of an English book, T. Percy Nunn's 
Exercises in Algebra (including Trigonometry), Part 1, 1913. 

The authors have received help from several teachers. Spe- 
cial mention should be made of Mr. E. A. Cummings of the 
North Side High School in Denver, who has worked most of 
the exercises and offered valuable suggestions. 

FLORIAN CAJORI. 
LETITIA R. ODELL. 



TABLE OF CONTENTS 



OHAPraB PAGB 

L Elembntabt Definitions and Operations .... 1 

Fundamental Definitions 1 

Addition and Subtraction 2 

Order of Fundamental Operations 3 

Multiplication and Division 6 

Parentheses 7 

Review Exercises 9 

Equations 11 

Simultaneous Linear Equations 13 

Positive Integral Exponents .20 

Fractional and Negative Exponents 21 

Powers and Boots 25 

Factoring 32 

Fractions, their Multiplication and Division ... 37 

Highest Common Factor and Least Common Multiple 43 

Addition and Subtraction of Fractions .... 44 

Quadratic Equations 48 

Problems Involving Quadratics 52 

n. More Advanced Theory and Operations .... 56 

Fundamental Laws of Algebra 56 

Remainder Theorem and Factor Theorem ... 59 

Binomial Theorem (preliminary treatment) ... 65 

Systems of Linear Equations. Determinants ... 67 

Problems Involving Linear Equations .... 74 

Division by Zero Impossible ...... 76 

Equations with Fractions 77 

Miscellaneous Practical Problems 80 

vu 



Viii TABLE OF CONTENTS 

CEAVm PAGK 

III. Proportion, Variation, Function 86 

Proportion 86 

Functions 87 

Different Modes of Variation 88 

Variation Shown in Graphs 90 

Problems in Proportion and Variation . ... 02 

Graphs Exhibiting Empirical Data .... 101 

IV. Logarithms 102 

Logarithmic Curve 103 

Fundamental Theorem 106 

Finding Logarithms 108 

Finding Antilogarithms Ill 

I*roblems 115 

Exponential Equations 116 

V. Quadratic Equations and their Propbrtils . . 119 

Equations Quadratic in Foi^m 119 

Relations between Roots and Coefficients . . . 120 

Nature of the Roots 122 

Graph of the Quadratic Equation ax^ -\-bz-\-c = y . 128 

VI. Systems of Equations Solvable bt Quadratics . . 127 

A System of Two Equations, One Linear . . 127 

A System of Two Equations, Both Quadratic . . 134 

Possibility of Solution by Quadratics .... 135 

Problems 137 

VII. Exponents, Radicals, Imaoinaries 139 

Meanings of Different Kinds of Exponents . . . 139 

Different Kinds of Numbrrs 140 

Simplifying Radicals 141 

Operations with Radicals 144 

Square Root of a ±2v7) 148 

Irrational Equations 149 

Irrational Equations Quadratic in Form . . .152 

Graphic Representation of Complex Numbers . . 154 

VIII. Series and Limits 168 

Arithmetical Series 168 

Geometrical Series 163 

Infinite Geometrical Series 170 

Theory of Limits 173 



TABLE OF CONTENTS ix 

PA6B 

SUPPLBMENT 177 

The Highest Common Factor by the Method of Division . . 177 

Solutions of Quadratic Equations 180 

Mathematical Induction and Proof of the Binomial Theorem . 181 

Applications of the Binomial Theorem 185 

Table of Squares and Cubes, Square Roots and Cube Roots . 188 

Review Exercises Selected from College Entrance Examinations 190 



* 



^ , J »* > 



ELEMENTARY ALGEBRA 



SECOND YEAR COURSE 



CHAPTER I 

ELSMSNTARY DEFINITIONS AND OPERATIONS 
FUNDAMENTAL DEFINITIONS 

1. In arithmetic, numbers are commonly represented by 
Hindu-Arabic numerals. In algebra, numbers are represented 
also by letters. 

Any combination of numerals, letters and symbols of 
operation, which stands for a number, is called an algebraic 
expression. 

An algebraic expression may consist of parts which are 
separated by the -f- or — signs ; these parts, with the signs 
immediately preceding them, are called terms. 

Thus, in + a^ - 2 a& + 3 &3, there are three terms, + a^ — 2 a&, +3 b^. 

Each of the numbers which multiplied together form a 
product is called a factor of the product. 

A factor consisting of one or more letters is called a literal 
factor. 

Terms which have the same literal factors are called similar; 
terms which do not have the same literal factors are called dis- 
similar. 

In 4 a3& - 3 ab^ - 10 a% the terms + 4 a^b and > 10 a^b are siinilar ; 
the terms —Sdb^ and - 10 a^b are dissimilar. 
B 1 







2 ' ELE^BmARY ALGEBRA 

. • • .• ••: .••••:: : •• : 

• All algebraic expression of one term is called a monomial, of 
two terms a binomial, of three terms a trinomial, and of several 
terms a polynomial. 

When a number is the product of two factors, either factor 
may be called the coefficient of the other factor. 

Often the word coefficient is applied only to the factor which 
is expressed in numerals. 

Thus, in 6 xyz^, the numeral 6 is called the coefBcient of xyz^. 

An exponent is a number placed at the right and a little 
above another number, called the base. 

When the exponent is a positive integer, it indicates how 
many times the base is taken as a factor. 

The exponent expresses the power to which the number is 
raised. 

Thus, the 4 in a* expresses the fourth power of a. 

Find a* when o = 2. Has o* the same value as 4 a, when a = 2 ? 

The absolute value of a number is its value regardless of its 
sign. 

Thus + 5 and — 5 have the same absolute value, 6. 

ADDITION AND SUBTRACTION 

2. Addition : If similar terms have like signs, find the sum 
of the absolute values and prefix the common sign. 

If similar terms have unlike signs, find the difference of the 
absolute values and prefix the sign of the one which has the 
greater absolute value. 

If the terms are dissimilar, the addition is indicated in the 
usual way. 

Thus, the sum of 3 a and 5 & is 3 a + 6 &. 

To check an example in addition : 

Substitute numerals for the letters and find the sum. This 
sum must equal the result of substituting the numerals in the 
answer. 



ELEMENTARY DEFINITIONS AND OPERATIONS 3 

3. Subtraction: Conceive the sign of the subtrahend to be 
changed, and then proceed as in addition. Check by substituting 
numerals for the letters. 

In what other way may subtraction be checked ? 

WBITTBN BXEB0ISB8 

4. Find the sum of : 

1. 7a-36-f 6c; 4a + 56-12c; -6a + 2b + Sc. 

2. 4aj* — 3aj*y — 4a^*; 3a^ — 8a?y« — Saj*; 3aj* — 6a5«y. 

3. 5a» — 6a"6" — 7c»; 2a" 4-3a»6» + 8c»; — 5a"6» — 2c". 

4. 3(a - b)+4:{a + b) ; 7(a - 6)- 6(a + 6) ; - 6(a - b) 

-hll(a + 6). 

6. a4-3a«4-9 a*-4a4-6; 7 a^ -}• 3a^ -^2 a^ + 10a + 7 -, 

-4a*-8a»-5. 

6. From 2 m — 3 m« + 4 n take m — 3 mw — 9 n. 

7. From 2a^'-2ab + Sb^ take a^ — ab — b^. 

8. From 4 4- 3c -8d-9e take 7c 4- 5c -10 -2d. 

9. From a* 4- 2 a*^*' — 6b^ take 2 a' — 5 a'ft*' + Sfe*'. 

10. From 3(aj + y)—7(x — y) take 4(a: — y) — 6(a? 4- 3^). 

11. From (a? — y)a — (« 4- y)b take (a? 4- y)a + (a — y)6. 

12. What number added to 7 aj^ — 5 a?y* — 6 a?y will give 
2a^2 — 3aj*y4-4ajy? 

ORDER OF FUNDAMENTAL OPERATIONS 

5. The values of many algebraic expressions depend upon 
the order in which the operations of addition, subtraction, mul- 
tiplication, and division are performed. To avoid confusion it 
has been found necessary to adopt certain rules, so that an ex- 
pression shall always be interpreted in the same way and 



4 ELEMENTARY ALGEBRA 

^ represent the same value. The following rules are agreed 
upon: 

I. A succession of multiplications and divisions shall be 
performed in the order in which they occur. 

Thus 6. 8 + 4. 8 = 48 +4. 3 = 12. 3 = 36. 

II. A succession of additions and subtractions shall be 
performed in the order in which they occur. 

III. A succession of multiplications, divisions, additions, 
and subtractions shall be performed in accordance with I and 
II ; the multiplications and divisions being performed before 
any additions and subtractions. 

Thus, 20 + 6. 7-4-3. 6 = 20 + 42 -4- 16 = 43; 
17 + 15 + 8-8. 2-*.4 = 17 4-6- 16 + 4 = 17 + 6-4 = 18. 

6. If there are parentheses, apply the preceding rules to the 
expressions within the parentheses first ; then to the resulting 
expression as a whole. 

The forms ( ), [ ], { } go by the general name of " paren- 
theses" but they are designated by special names when it is 
necessary to distinguish one from the other. 

Thus, [ ] is called a " bracket," { } a " brace," but ( ) is 
always called a " parenthesis." 

The " vinculum " is also used to denote aggregation, thus 

a + &-*■ fit — ^« 

ORAL BXEB0I8B8 

7. Simplify: 

1. 16-2 4-14. 7. 28-5-7.2 4-5. 

2. 3a?-42^4-8aj4-3y. 8. ^x-iy-^-^x + ^y. 

3. 2a-7a-3a4-8a. 9. 15 • 3 -4-5 4- 10 -4. 

4. 10 -5m -15 4- 20m. 10. 24 4- 5.5-10 - 7-(15-^2). 
6. 5-(7_2)4-[8-i-3]. 11. 48 4-3(18-^-9.3)^9. 

6. 25 -^ 5 4- 15 4- 40. 12. (12a-7a)-(13a-20a): 



ELEMENTARY DEFINITIONS AND OPERATIONS 5 

MULTIPLICATION 

8. To muUiply one term by another : Multiply the numerical 
coefficients ; annex the letters, giving to each letter in the product 
an exponent equal to the sum of its exponents in the two factors. 

Law of signs: Like signs give plus, unlike signs give minus. 

The degree of a term is the sum of the exponents of the 
literal factors. 

Thus 4 Qcy^afi is of the sixth degree. 

An expression is homogeneottSy if all its terms are of the 
same degree. 

In a^ + 2 a^ — 6 x^ every term is of the fourth degree ; the expression 
is homogeneous. 

To multiply a polynomial by a polynomial: Arrange the 
terms in both polynomials according to the Ascending or 
descending powers of some letter; multiply each term of 
the multiplicand by each term of the multiplier and add the 
partial products. 

Multiplication may be performed by detached coefficients. 

For example, (2ic*-7aj»-5aj*4-6aj- 3)(6 a^ - 4 a? 4- 6). 

2-7-5+6-3 

6-4+5 

12-42-30 + 36-18 
_ 8+28 + 20-24 + 12 

10 - 35 - 25 +30-15 

12-50+ 8 + 21-67 + 42-15 

The product is 12 aj«- 50 a* + 8 aj* + 21 aj» - 67 a^ + 42 a? - 15. 
If any powers of x are lacking in either polynomial, the 
terms in question must be represented by zeros. 

Thus, in multiplying ac" + 2 05 — 4 by 05* + 05* + 1, we must write 
1+0 + 2- 4and 1+0+1 + + 1. 

Check an example in multiplication by substituting numerals 
for the letters. 



6 ELEMENTARY ALGEBRA 

JDIVISIOH . 

* ' • 

9. To divide oiie term by another : Divide the numerical 
coefficients ; annex the letters, giving to each letter in the 
quotient an exponent equal to its exponent in the dividend 
minus its exponent in the divisor. Observe the law of signs : 
Like signs give plus, unlike signs give minus. 

To divide one polynomial by anothsr : 

I. Arrange the terms. 

II. Divide the first term of the dividend by the first term of 
the divisor and obtain the first term in the quotient. 

III. Multiply the divisor by this term in the quotient. 

IV. Subtract the product from the dividend. 

V. Treat the remainder as a new dividend and proceed 
as before. 

YI. Keep each new dividend arranged in the same order as 
the first dividend. 

Divide 14 a:* - 27 ax* + 21 aW - 32 a* + 12a8a; by 2x8 - 8 a« + 4 a«. 

2a;2-3ax + 4a« 



14jr* - 27 oa* + 21 a'^a + 12 a»x- 32(1* 
14 re* - 21 ax» + 28 a%c2 



7a:2-3ax-8a« 



- 60*8- 7a2a;2 + i2a»a5-32(i* 

- 6aofi+ 9 qagg -. 12 flgg 

- 16 a2a;2 -I- 24 a'te - 32 a* 
-16a2aja + 24a8'x-32a* 



By detached coefficients this division may be performed as follows : 



14 _ 27 + 21 + 12 - 32 
14 - 21 + 28 



2-3+4 



7-3-8 



- 6- 

- 6 + 


7 + 12- 
9-12 


-32 


^_ 


16 + 24- 
16 + 24 - 


-32 
-32 



The quotient is 7 a;^ - 3 ox — 8 a^. 

Division may be checked by substituting numerals for the 
letters, care being taken to avoid a zero divisor. 



ELEMENTARY DEFINITIONS AND OPERATIONS 7 

ORAL BXBBOI8B8 

iricaJ 10. 1. In what other way may examples in multiplication 

the and division be checked ? 
dend 2. If there is a remainder in a division, what must be done 

gns:' with it in checking? 

3. If the multiplicand and the multiplier are arranged 
according to the descending powers of some letter, how will 
the product be arranged ? 

11 of 4. If the multiplicand and the multiplier are homogeneous, 

what will be true of the product ? Of what degree will it be ? 



I 



eed 



as 



2 I 

• I 

I 
I 



WRITTBN BXBRCISB8 

11. Expand and check : 

1. (7a-h2a2-44-2a»)(3a-3 + 4a«). 

2. (aj*« — 2 aj«3/* -f y»)(af — ^). 

Divide and check : 

3. a^-Sa^b + Sab^-b^hya-b. 

4. ixfi — ]^ hy X — y, and by a? 4- y. 

6. 12 aj-+< + 20 af+» - 107 ic»+2 ^ 37 0^+1 -f 33 af by 6 aj» - 11 a^. 

6. m* -f w' 4- i>' — 3 mnp hy m-^n+p. 

7. a»-&' + c» + 3a6cby a^+V + c^-f aft-ac + ^w. 

8. 2,*-|2^-h||2/»-||2/«H.5y_^byy-|. 

PARENTHESES 

12. I. A parenthesis preceded by a -j- sign may be removed 
without changing the signs of the terms within the parenthesis. 

II. A parenthesis preceded by a — sign may be removed, 
le provided that the sign of each term within the parenthesis be 
changed. 



L 



8 ELEMENTARY ALGEBRA 

WRITTEN BXEBOI8BS 

13. Simplify: 

1. 18 -(7-2) -3. 

2. 5a4-(6a-4a)-(9a-2a). 



3. 8 a: - [2 aj 4- (3 aj - y) -f 7] + [5 « - 2 y + 3]. 

4. -7m4-[2n- j4 m- (3 m-5n)- 8 ii{ -f 9m]. 

6. 10c- {4d-(9c-5d)| - {(2c -d)- (4c-f 7d)}. 

6. (aj-f 2)(aj-3)-5(a^-3« + 2)-10(aj + 3). 

7. {5a«-3[-5+(a-f 2)(a-5)-7]| -a(a-l)\ 

9. (a + 6)(a - 6) - (a - 6)2 4- (a 4- 2^)'. 

10. 2{ -4-(7 -3) -f (-8- 2) -5(9-3 + 5)1 4-11. 

11. m-{-[— (1— m)— l]-m|— {m— (5-4m)-(44-m)}. 

14. Terms may be inclosed in a parenthesis by reversing 
the rules for removing parentheses. 

When terms are inclosed in a parenthesis preceded by a 4- 
sign, the signs of the terms are not changed. 

When terms are inclosed in a parenthesis preceded by a — 
sigriy tjiersigns of all the terms are changed, 

WRITTEN EXERCISES 

15. Collect the coefficients of a? in a parenthesis preceded 
by a 4- sign, and the coefficients of ^ in a parenthesis pre- 
ceded by a — sign : 

1. a>x — ay '\- bx — by, 4. 2x ^3y —- dx -{■ dy — fx. 

2. 7nx + na; 4- f^/y — ^y. 5- py — 9^ —px-{- qy. 

3. abx + cdy — cdx— aby, 6.-7 aaj+3 6y 4-4 bx— 5 ay. ■ 



ELEMENTARY DEFINITIONS AND OPERATIONS 9 

REVIEW EXERCISES 

16. liA = 2a^-^ab+7b', 5 = - 5a*+ 2a6-46«, 

C == S a" + 10 ab-- 6 b% D = 4a«-9 6« + 3a5, 

find the values of : 

1. ^ + 5 + C4-A 4. B- A + G-D. 

2. A-B-^O-D. 6, G-B + A-D. 

3. ^A+B-C + D, 6. D-^A-G-B. 

If a = 1, 6 = 3, c = — 2, d = 4, n = 2, find the numerical 
values ' of : 

2a-3c + 4d^ 3a^-2a--26-. 

7c- 2d 

8. ?^ + ^-. 13. (a+6)«-(c-d)». 

Q a^ + &^ 14. (c + d)(c-cr>+5^- 

• c2-hcP' ^"^ 

, ^ a" — 6* X6. 2 d -^ 3 c • dn — &c ^ d. 

10. — • 

11. 5a-^(6-c)H-d-n. 16. Vfe^+VS^. 

Find the sum of : 

17. (a 4- b)x - (a - 6)2/ and (a - 6)aj + (a + ^)y. 

18. |a-|6 + ic; ia + i6-^c;andfa + i6-Hc 

19. |aj-fy + i2;; |a? + |y-i2;; and|aj-|y + |2. 

20. [(a + c)aj 4- (^ + c)y] - [(a - c)a; - (& - c)y]. 

21. [5(.'r4-2/)-7(aj-y)]-[-2(aj + y)-3(«-y)]. 

Simplify : 

22. aj2-(?/2-2^)-[y^-(»''-«')] + [25'- (y^-aj«)]. 

23. aj-22;-f3y-M?-[3a?-(5«-y-7w?)]-5« + 4a?{ 

, 24. (iB« + 2iC-^- 3af-2-l)(aj-l). 



10 ELEMENTARY ALGEBRA 

26. (a"+* — 4 a« + 5 a-"* + a'^^a + 1). 

26. (a"4-&")(a"— 6»). 

27. (|a'-|a6-hi6»)(|a-i6). 

28. (a«6'»+^ — 6 a'6"* + 4 a'»6"»-i) -«- 2 a-ft*"*"*. 

29. (aj»2r — af +y+^ 4- aj*--y*-*) -5- af -y-*. 

30. (a** 4- a** + 1) -5- (a** - a* -f 1). 

31. (9 a*» - 26 6^) -s- (3 a« + 6 fe'-j 

32. (iaj»-|xV + M^-ty')-^(iaJ-iy). 

TYPE FORMS IN MULTIPLICATION 

17. 1. (a ± 6)« = a* ± 2 a6 + &». 

Take the upper signs together, and the lower signs together. 

2. (a-f 6--c)«=a*4-&*4-c2 + 2a6-2ac-26c. 

3. (a + 6)(a — 6) = a2 - 6». 

4. (aj + a)(aj 4- 6) = a.** 4- (a + ^)aj 4- a5. 

6. (ax 4- 6)(ca5 4- cf) = ocaj* 4- {ad -\- bc)x + hd. 
6. (a± 6)« = a«±3a*64-3a6»±6«. 

ORAL BXBROISB8 

18. Write, by inspection, the va4ue8 of : 

1. (a4-2)«. 6. (a?-^h)\ 9. (a»-6»)«. 

2. {b-Sy. 6. (c'-d')*. 10. (aj»4-6')*. 

3. (2 a: -1)2. 7. (c2 4-7(P)«. 11. (m»-2n2)«. 
4 (3 a -4)*. 8. (5«-3y2)«. 12. (8a-6a?«)2. 

13. (aj — y4-«)'. 18. (aj4-4)(aj4-9). 

14. {2x- y- z)\ 19. (a? 4- 20)(aj - 15). 
16. (5aj — 2 2/4- 25)*. 20. (aj - 8)(aj - 6). 

16. (a? -33/ -4)*. 21. (a4-4c)(a — 3c). 

17. (a?-7)(aj + 3). 22. (2a4- c)(3a - d). 



^ 



ELEMENTARY DEFINITIONS AND OPERATIONS 11 

23. (3aj4-4)(2a:-7). 31. (a - 2c)(a- 7 c). 

24. (56-6c)(46+3c). 32. (a«4- 26)(a«4-36). 
26. (x^y)(a — b). 33. (a» — 9 c)(a» + 8 c). 

26. (3y-4)(3y4-8). 34. (6a-4 6)«. 

27. (a« + c)(a« — c). 36. (a? + yy. 

28. (2a + 6)(2a-6). 36. (3aj-4)«. 

29. (5aj« — y»)(3/»-f 5aj«). 37. (x-yf. 

30. («• 4- 3/*)(»' - 3/*). 38. (2 a 4- 1)'. 

Complete the squares : 

39. 4aj*±12aj + ? 42. 4m«±?-f9w*. 

40. 9a«±24a4-? 43. a*'±?+366»». 

41. ?±18 6c + 81c». 44. 25aj*-20 4-? 

EQUATIONS 

19. An equation expresses an equality. 

In other words^ an equation is a statement that two expres- 
sions stand for the same number. 

2»-|-6 = 7a? — 4, and (a + 6)* = a* -|- 2 a6 -f 6* are equations. 
The first is an equation of condition^ because it is true only 
under the condition that a? = 2. The second is an equation 
of identity^ because it is true whatever values a and b may 
have. 

In this chapter we deal with equations of condition. The 
root of an equation, containing one unknown, is a number 
which, when substituted for the unknown, "satisfies" the 
equation by reducing both sides to identical numbers. 

An equation of the form 

ooj* + 6aj»-^ 4- ... 4-^ = 0, 
where n is a positive integer, is said to be of the nth degree. 



12 ELEMENTARY ALGEBRA 

The degree of any equation is indicated by the exponent of 
the highest power of the unknown. 

Thus, a; -> 7 = 2, is an equation of the 1st degree, called linear. 

a;3 — 3 X = — 2, is an equation of the 2d degree, called quadratic. 

x8.62(^ + lla; = 6, is an equation of the 3d degree, called cubic. 

x« — 1 = 0, is an equation of the 4th degree, called quartic. 
etc. 

20. The equilibrium of a balance is not disturbed so long as 
like changes in the weights are made simultaneously on bpth 
sides. So in equations, we may add the same number to both 
sides, or subtract the same number from both sides, or we may 
multiply or divide both sides by the same number (except divi- 
sion by zero). 

The equality is maintained during all these changes. 

Solve »(« + 4)=a^ — 3a? + 5. 

Remove the parenthesis, x^ -\-Ax = x^ — Sx-\- 6, 

Subtract x^ from both sides, 4x = — Sx + b. 

Add 3 x to both sides, 7 a; = 5. 

Divide both sides by 7, x = f . 

Check : In the given equation, substitute ^ for x, 

WBITTBN BXEBOI8E8 

21. Solve and check : 

1. (aj-2)(a?-3)=a:(« + 4). 

3. (m — 4)*4-(m4-4:)2 = m(2m-f-l). 

4. (r 4- 1)' - r(r — 1)= r2(r + 2)-f 5 r - 1. 

6. y(t/ + l)+(y-hl)(2^ + 2)=(t/ + 2)(t/+3)+y(t/-f-4)-9. 

6. «-.[4 + {4-(4 + 01]=0. 

7. 6«-(4«-8)-{5-3«~(7aj-4)| = ll. 



ELEMENTARY DEFINITIONS AND OPERATIONS 13 

8. s-|-3-[s-9-3{9-4(6-«)-sn = 3. 

9. 20(1 - X)- 3(x - 6)- S[x + 8 -458 - 3(1 - «)}]==- 6. 

10. (x 4- 8)(aj - 5)(x 4- 6) = (» + 2)(x 4- S)(x + 4). 

Solve each of the following equations for each letter in terms 
of the others : 

11. A = iab. j3 ;S = -(a + Z). ^** ^i^i=^^2- 

12. S = ia(b-^by ' ^ 16. = 1(27' -32). 



I 



SIMULTANEOUS LINEAR EQUATIONS 

22. If a liQear equation contains but one unknown, one 
value may be found for that unknown. 

If a linear equation contains two unknowns, an unlimited 
number of simultaneous values may be found for them. For 
that reason the equation is called indeterminate. 

In « + y = 3, we have x = l, y = 2; a; = 2, y = l;aj=— 2, y = 5j 
a: = 0, y = 3 ; a; = 3, y = ; etc. 

If these pairs of values are 
plotted, the points thus obtained 
will be seen to lie on a straight 
line, as in Fig. 1 ; hence the name 
linear equation. 

Two points determine the posi- 
tion of a straight line. The graph 
of a linear equation in two un- 
knowns is most easily made by 
locating the points at which the 
line crosses the coordinate axes. ^^' ^• 

It was seen above that in the equation aj-|-y = 8, 05 = 8 when y = 0, 
and x = when y = 3. In Fig. 1, 

x = S, y = locates the point A. 
35 = 0, y = 3 locates the point B, 
The two points A and B determine the line. 



I 1 t I I III l^*T^T— T— ' 
IIIIIIIIIII 

zz_sszzzzzz 

Q ^ ZX 



14 ELEMENTARY ALGEBRA 

BXEBOI8B8 

23. Make graphs of the following equations : 

1. aj4-y=7. 6. 4a;~9y = 36. 11. y = 0. 

2. aj — y = 2. 7. 5a? = 10 — 2y. 12. a: = 0. 

3. re — 15 = 3 y. 8. aj = y. 13. x = 3. 

4. 3aJ— 4y=*12. 9. aj — 3y = 0. 14. y = — 4. 
6. 6-2y — a;=0. 10. 3aj — y = 6. 16.a; = — 2. 

24. If the graphs of two linear equations in x and y inter- 
sect, the values of x and y at the point of intersection satisfy 
both equations ; this value of x and of y, called the coordinates 
of the point, constitute one solution. 

Since two straight lines cannot intersect in more than one 
point, two linear equations in x and y cannot have more than 
one solution* 

Two linear equations in x and t/, representing lines which 
intersect in one point, are called independent. 

WHi'ri'BN BXEB0I8BS 

25. Make graphs of the following equations and determine, 
from the graph, the coordinates of the point of intersection. 
Check, by substituting the coordinates in the equations. 

We are not able to draw figures that are absolutely accurate. Hence, 
solutions obtained from graphs are usually only approximations to the 
true values. 

1. a; + y = 6, 4. a: = 3, 7. 3« = 2y, 

re — y = l. 2a:— 3y = 9. a?-f4 = 7y. 

2. 3aj — y = 7, 6. y = — 2, 8. a? — 3^ = 0, 
2a;-y = 5. 6a: — 3y=16. 3a:-f23^ = 6. 

3. a? 4- 3 2/ = 8, 6. a? = y — 6, 9. a? -f 1 = 0, 
2aj-f-y = l. 6y = a: + 10. y — a?=6. 



ELEMENTARY DEFINITIONS AND OPERATIONS 15 

26. The two linear equations, 

2x-h2y = 10, 

can be satisfied by the same values of x and y ; for example, 
jc = 1 and y = 4, or a? = 3 and y = 2, or x = 5 and y = 0, etc. 
Moreover, by dividing both sides of the second equation by 2, 
the second equation can be reduced to the first. The two equar 
tions are therefore not independent ; they are really different 
forms of one and the same equation and represent one and the 
same straight line. They are called equivalent equations. 

27. The two linear equations, 

x — y=z5, 
x-y=l, 

are called inconsistent^ for the reason that no finite value of x 
and of y can be found, such that x— y is equal to 5 and also 
equal to 7 ; no pair of finite values of x and y satisfy both V 
equations. This is geometrically evident from the fact that 
the graphs of the two equations are parallel lines. 

Two linear equations in two variables x and y belong there- 
fore to one of three groups : 

1. The two equations are independent and represent two 
lines which intersect in one point, or 

2. The two equations are inconsistent and represent two 
lines which do not intersect (are parallel), or 

3. The two equations are equivalent and represent one and the 
same straight line. 

WBITTBN BXBBGISBS 

28. Make graphs of the following equations : 

1. 3aj-6y=8, 3. 2yH-3aj = 0, 
oj— 2y = 5. 9aj-h6y = 15. 

2. 5y-2x = 4:, 4. 2a:-4y = -10, 
4a:-10y=:-8. x-2y = -5. 



16 ELEMENTARY ALGEBRA 

6. a— 3y = 2aj-7, 7. aj = 3, 9. a;-h4 = 0, 

3ajH-2y=4aj + 5y-7. y = -2. y-l = 0. 

6. 5aj — 7y = ll, 8. aj-hy = 0, 10. aj = 0, 

10aj-14y = 20. aj-y=sO. 3^ = 0. 

11. Can you tell, before making the graphs, whether two 
equations are equivalent, or not? Inconsistent, or not? 

12. Can you tell whether the graphs will be parallel, or not? 

13. Can you tell whether the graphs will go through the 
origin, or not ? 

29. Systems of independent linear equations may be solved 
by a process called elimination. 

The elimination may be accomplished in one of three ways : 

I. Addition or subtraction. — Make the coeflB.cients of one 
variable alike in both equations and then either add or sub- 
tract the sides of the equations. 

II. Substitution. — Find the value of one variable in terms 
of the other in one equation, and substitute that value in the 
other equation. 

III. Comparison. — Find the value of one variable in terms 
of the other in both equations, then equate these values and 
solve the resulting equation. 

EXBBCISBS 

30. 1. Solve by addition or subtraction : 

5aj-h3y = 69, (1) 

4aj-7y=-20. (2) 

Solution. Multiply (1) by 4, (2) by 5, 20 x + 12 y = 276, (8) 

20a;-36y=-100. (4) 

Subtract (4) from (3) , 47 y = 376, 

y = 8. 
Substitute in (1), a; = 9. 

CTieck : Substitute in (1) and (2), 46 + 24 = 69, 

36 -66 =-20, 



ELEMENTARY DEFINITIONS AND OPERATIONS 17 



2. Solve by substitution, 


5 + 6 = *' 


(1) 




7 8 14' 


(2) 


Solution. From (1), 






Substitute in (2), 


6 y 1 
7 8 14' 

160 201, 7y= 4. 
111^=164. 

y = 12. 




Substitute in (1), 


f+2 = 4. 
a; = 10. 




3. Solve by comparison, 


3 a - 13 y = 1, 


(1) 
(2) 


Solution, From (1), 


^ 13 


(3) 


From (2), 


y = 16a;-9H. 


(4) 


Compare (3) and (4), 


8^=15x-9H, 

3x-l = 195x-129, 
192 a; = 128, 
x = f 


1 


Substitute in (3), 


y = A- 




Solve a,nd check : 






4. 12 « + 11^ = 12, 


•• !+"-¥' 




42 a -h 22 y = 4(H^. 


X 

• 


% 



18 ELEMENTARY ALGEBRA 

3 9 3 

^2 4 20 

PROBLEMS 

31. 1. The sum of two numbers is 180 and their difference 
is 12. Find the numbers. 

2. One number is 27 more than another. Their sum is 
143. What are the numbers ? 

3. Of two numbers, twice the greater exceeds three times 
the smaller by 4 ; but the sum of the greater and twice the 
smaller is 44. Find the numbers. 

4. Two pounds of sugar and three pounds of flour cost 19^, 
and three pounds of sugar and five pounds of flour cost 30 ^. 
What is the price of each per pound ? 

5. Three yards of velvet and five yards of silk cost me 
$ 15.75. If I had bought two yards more of velvet and two 
yards less of silk, my bill would have been $ 18.25. What 
is the cost of the velvet and silk per yard ? 

6. In five hours A can walk 1 mi. more than B can walk 
in 6 hours ; in seven hours A can walk 3 mi. more than B can 
walk in 8 hours. How many miles an hour can each walk ? 

7. A, B, C, D, together have $ 300. A has $ 10 more than 
C ; B has $ 5 less than half as much as D ; and A and B have 
together $ 5 less than twice as much as C. How much has each? 

8. A number consists of two digits whose sum is 12. If 
three times the sum of the digits be subtracted from the 
number, the digits will be reversed. What is the number? 

9. The digits of a three-digit number will be reversed, if 
396 be added to the. number. The units' digit is 3 times the 
hundreds', and if double the tens' digit be increased by 6, the 
result will equal the units' digit. Find the number. 



^ELEMENTARY DEFINITIONS AND OPERATIONS 19 

10. In a purse there are 3 times as many quarters as nickels, 
and twice as many half dollars as nickels. The total value of 
the coins is $ 3.60. How many of each kind are there ? 

11. Some boys bought a boat and found upon paying for it 
that if there had been 2 more of them, each would have paid 
a dollar less ; but if there had been 2 fewer, each would have 
had to pay $ 1^ more. How much did the boat cost ? 

12. A rectangle is four times as long as it is wide. If it 
were 3 in. shorter and 2 in. wider, its area would be increased 
15 sq. in. Find its dimensions. 

13. It costs as much to sod a square piece of ground at 25 ff 
a square yard as it does to fence it at $ 1 a yard. How long is 
the side of the square ? 

14. A walk is laid 3 ft. wide around a rectangular court, 
which is 15 ft. longer than it is wide. The area of the walk is 
960 sq. ft. Find the dimensions of the court. 

15. A man invests part of $ 5480 at 5 % and the remainder 
at 4 %. The total annual income is $244. How many dollars 
has he in each investment ? 

16. The yearly income from a 5 % investment is $ 97.70 
more than that from a 6 % investment. The sum of the two 
investments is $ 6420. How much is invested at 6 % ? 

17. A boy weighing 100 lb. is 6 ft. from the fulcrum of a 
seesaw. He balances a boy who is 8 ft. from the fulcrum. 
What is the weight of the second boy ? 

By careful measurement it has been ascer- i—r 
tained that, to maintain a balance, the lengths 
of the arms of a lever and the corresponding 
weights must conform to the law, hwi = hv32 ; 

that is, the product of one weight and its distance from the fulcrum is 
equal to the product of the other weight and its distance from the fulcrum. 

18. Two boys together weigh 170 lb. They balance when 
one is 8 ft. and the other 9 ft. from the fulcrum. How much 
does each weigh ? 




20 ELEMENTARY ALGEBRA 

19. If the lever is 8 ft. long, how far from the fulcrum 
will two weights, 30 lb. and 50 lb., have to be in order to 
balance? 

20. A weight of 180 lb. is carried between two men by 
means of a pole. One man is 5 ft. from the weight, the other 
is 4 ft. How many lb, does each man lift ? 

POSITIVE INTEGRAL EXPONENTS 

82. In the expression a* we say a is raised to the nth power. 
When n is a positive integer : 

I. The nth power of a is the product obtained by using a 
n times as a factor. 

a • a • a • • • to n factors = a*. 

II. The nth power of a* is equal to a* • a* • a* • • • to n 
factors = a*". 

Hence (a*")" = a"". 

III. The nth power of a^ft** is equal to (a* • a* • a* • • • to n 
factors) (6' • 6" . 6' • • • to n factors) = a"*" • h^ = a'"*6'*. 

Hence {oFbpy = af^'^hf*'*. 

IV. The nth power of — is equal to 

a* • a* • a* • • • to n factors a*** 
ftp . 5j» . 5p . . . to n factors 6** 



Hence, ( — ) = — • 

' \bpj bP'* 



33. From the law of the signs in multiplication : 

1. An even power of any real number is positive. 

2. An odd power of any real number has the same sign as 
the number itself. 



ELEMENTARY DEFINITIONS AND OPERATIONS 21 

BXBBCISBS 

34. Write the values of : 

1. (2a'6)'. jj /_^V 

2. i-lahf)*. ^ ^^ 

4. /'3<«f\*. 13. (2a»6c»)'. 

^'^'^'^ 14. -(_a*6'c«)». 

7. (0^2'')". ^^ f^^_f\\ 

8. (a^+'ft*"")'. A ^^^*^ / 

19. (-|a*6")»». 

20. (6a^yh)^. 



10. 



(23^)' 



MEANING OF FRACTIONAL EXPONENTS 

35. When an exponent n is a positive integer, we know that 
a* = a • a • a • • • to w factors. That is, the exponent n indicates 
that a is taken n times as a factor. 

What is the meaning of a* ? It would be absurd to say that 
a* signifies a taken ^ times Vs a factor. A number can be 
taken ^s a factor only a whole number of times. 

For the purpose of attaching a meaning to a* we stipulate 
that we shall be able to multiply a' by a* according to the 
same rule by which a* is multiplied by a^ The product of a* 
and a* is found by adding the exponents; that is, a^>a^ = a*. 

If we add the exponents in the multiplication of a' by a% 
we obtain, 1 1 1 1 1 



/ 



22 ELEMENTARY ALGEBRA 

It is seen that a' is one of the two equal factors of a, or the 
square root of a. 

Hence a^ is another way of writing the square root of a. 
Likewise, the meaning of a* is found by taking a* four times 

as a factor, thus, i s s « 

a* • a* • a* • a* = a*. 

Hence a* means one of the four equal factors of a' or the 
fourth root of aK 

In general, to find a meaning for a«, where p and g are pos- 

itive integers, we take a« as a factor q times, and add the 
exponents. We obtain, 

- - - . j» J -+ — r-+ ••• to c terms — _ 

a« . a« • a* • • • to g factors = a« « « = a« = a'. 

It is seen that a« is one of the q equal factors of a'. Hence 
t 
a« means the gth root of a^ Thus, 

The numerator of a positive fra^ional exponent indicates the 
power of the base and the denominator indicates the root of that 
power. 

In finding the value of 27*, we may square 27, which gives 
729, and then take the cube root of 729, which is 9. Or we 
may take the cube root of 27, which is 3, and square 3, getting 
9 as the result, that is, 

27* = a/272 = {V27y. 
In general, we have. 

All irrational numbers which are expressed by the use of 
radical signs can be expressed by the use of fractional ex- 
ponents. In fact, the simplification of expressions usually can 
be effected more easily by the latter. Thus, 

■v/16 a^h^d^ = 16*a*6^c* = 2 a^l^c. 



X 



X 



ELEMENTARY DEFINITIONS AND OPERATIONS 23 
MEANING OF A ZERO EXPONENT 

36. We proceed to assign a meaning to aP, where a is not 
itself zero. For simplicity we stipulate, as before, that in the 
multiplication of two powers having equal bases, we find the 
product by adding the exponents. 

Accordingly, a"* >a^=z a""*"® = a"*. 

Divide both sides by a*, 2!!j_?L — ?5!!. 

a" a"* 

Simplify, a* = 1. 

Therefore, any number y except 0, vnth a zero exponent is equal 

tO'l, 

MEANING OF NEGATIVE EXPONENTS 

37. Let m represent a positive integer or a positive fraction. 
What is the meaning of a"^, where a is not zero ? Working on 
the assumption (§ 35 and § 36) that in multiplication we are 
permitted to add the exponents^ we obtain, 

a"* • a~*" = a'*""' = a\ 
Since a° = 1, a* • a"~ = 1. 

Divide by a**, a~** = — • 

a"* 

Therefore, any number with a negative exponent is equal to 1 
divided by that number with a positive exponent. 



Thus, 



a-»6* 


1 


.6* 


6* 

c 


6*. 


• c 


a6« 


a-^b-^c 


" 1 
a* 


1 ~ 

— • c 

6« 


c 



Therefore, a factor may be moved from the numerator 
to the denominator of a fraction, or from the denomi- 
nator to the numerator, provided the sign of its exponent be 
changed. 



24 ELEMENTARY ALGEBRA 



-2 



+ 6 _5 



Care must be taken to transfer only /actors. In ^ — ^^, ar^ is not a 

c 

factor of the numerator. "*" is not equal to , • Since a~^ = ^ « 

i + 6 

it follow that ?:^±^ = ?l_ = li^. 

c c a^ 



PRINCIPAL ROOTS 

88. Since 6 • 6 = -h 36, and ( - 6)(- 6) = + 36, it follows that 
both -h 6 and — 6 are square roots of 36. The two square 
roots of 36 are usually written in the form ± 6. Similarly the 
two square roots of a^ are written ± a. 

The positive square root of a number is called the principal 
square root Unless otherwise stated, we shall consider only 
the principal square root, and write V36 = 6. When V has 
no sign before it, or has the + sign before it, the principal 
square root is always understood. 

When we write — V we mean the negative square root. 

Thus, \/36 or (86)^ stands for -h 6, - V86 or -(36)* stands for — 6, 
±V86 = ±(36)1=±6. 

It can be shown that there are 3 different cube roots of a 
number, four different fourth roots of a number, and, in general, 
n different nth roots 'of a number. 

For example, there are three different cube roots of &' ; namely, 6, 

^^ "*" &, "" ^ ""^^^ 6. The fractional coeflacients in the last two 

2 2 

roots involve the imaginary number V— 3. Only one of the three roots is 

real, namely, the root b. We call h ihe principal root. 

In elementary algebra the principal root is the only root 
usually considered. 

I. .If a is a positive number, then the principal nth root of a 

is its positive value ; we designate it by a" or ^a. When n is 
an odd number, then this principal root is the only real root. 



ELEMENTARY DEFINITIONS AND OPERATIONS 25 

1 
When n is even there is still another real root, namely, — a* or 

Thus, the principal cube root of 64 is (64) t or v^ = 4 ; this is the only 
cube root of 64 that is real. 

The principal fourth root of 81 is (81) i or \^ = 8. There is another 
real fourth root of 81 ; namely, —3, for we see that ( — 3) ( — 3) ( — 3) (— 3) 
= 81. 

II. If a is a negative number, and n is an odd integer, there 
is no positive root ; there is a negative root and that is taken 
as the principal nth root. 

Thus, the principal cube root of — 27 is C— 27)^ or -^—27 = - 3. 

III. If a is a negative number, and n is an even integer, then 
all the roots are imaginary. Imaginary numbers will be dis- 
cussed more fully later. 

Thus, v^— 64 or ( — 64) • represents imaginary roots. 



POWERS AND ROOTS 

39. It is readily seen that 

(a^y^^c^.a^^a^'C^^a."^ 

This result may be obtained at once by multiplying the 
exponents 3 and 4. In this process algebraic expressions are 
raised to powers ; it is called involution. 

In general, (a*)* = a"*" 

((aryy = a~»» 

11 
It is agreed that each of the symbols a% Va, 6*, -Vb, (a6)n, 
^^/ab shall represent only one nth root, namely, the principal 






26 ELEMENTARY ALGEBRA 

nth root. For the principal real roots the following formulas 
can be shown to be true : 



(a"")* = a"* or Va"*" = a"*. 

Ill 

a* • 6" = {ahy or Va • Vh = Va6. 



? = r?Yor^ 



y/a _ */a ^ 



5^ W V6 - 
Express each of these formulas in words. 

• 

The square root of 2 cannot be exactly expressed by the 
Hindu-Arabic numerals. One can approximate its value by 
extracting the square root to two, three, or more decimal 
places, thus: V2 = 1.41+, V2 = 1.414+, V2 = 1.4147+, and so 
on. The radical V2 or 2*, and other radicals of the same 
kind, V3 or 3i, ^5 or 5^^, etc., whose values can be found ap- 
proximately, but not exactly, represent numbers called irra- 
tional numbers. 

ORAL BXBBCISBS 

40. Express with fractional exponents and simplify : 



1. V3 win*. 5. 2V2a^. 9. \^5 . Vo^. 

2. A/2a^h\ ^ 10. -v/5.^35. 

7. 3a/^. 

3. V-21a'c\ ^^'_ 11. ^.■^. 

, 8. -7\/-. 7, 

4. ay/^M. ^y 12. 6V2^^*. 

Express with radical signs : 

13. 2(7 y)l 15. Sh^yl 17. 5(x + 2/)*. 

14. Sajij/*- ^®* "1* 18- 4(aj*-y2)*. 



4.27*. 



ELEMENTARY DEFINITIONS AND OPERATIONS 27 

Simplify : 

19. 49-i. 26 2^ *'■ (-3)"'-(-^)'- 

20. 81*. ■ 8* ' 82- (M)-*. 

21. 50 . 16i. ''' ^f!„r- ^'' <^>"'-_, 

, , 27. -^-^- 84. a».(|)-*. 

22. 16-i.26i. 4-'a-> 

23- iis) • 29. a-»(a^ + 6). ^_,^_, 

24. (^)*. 80. 36-^-343*. **• '^i;=r* 

37. £l^. 38. «1±^. 

m~* + b~* Multiply numerator and denomi- 

' mr* — 6~*' nator by tfibf. 

Find the principal roots of : 

40. Vxy. .g 1 256 a^b* »/ ->^a" 

41. V4^. >'289m"- >/««««• 

8. 



42. V9^. 60. </256^. 56. Vl25a!>Y. 

43. V26^. ^sr^^ "• A^ 

44. VIOO ai6. ^^' XUoO'riOwie' 



h^^ 



si- 512 a^ 



400 a^y « 

45. ^-8a36i2. ^ 58. \/— ^ 

46. ■\/64.afifz'^. 



x^ 



47. ^ -343a36i« . 53. V/^?^. ^^' ^^^^' 

48. V^/1^. e,.^ 60. #^'. 

\ 729 2/3 54. ^J/729^y%24; \ 76r 

Find the ^loo reaZ fourth roots of : 

61. 10,000 a2&3c*(a + ^)*- «3. j\ x-*y'^^(x - yy. 

62. 625 7/i8n«p3(a2 - 52)2. e4. .OOSl(x - yyzhi}-^. 



28 ELEMENTARY ALGEBRA 

Find the real fifth root of : 

65. -32iB6(m+7i-j9)-«y»24. 66. 243 a»(6 - c)i«ar«2*. 

67. What kind of roots of positive numbers have two real 
values ? Which of those two real roots is the principal root ? 

68. What kind of roots of positive or negative numbers 
have only one real value ? 

69. What kind of roots of negative numbers have no real 
value? 

SQUARE ROOT 

41. Since (a -h 6)2 = a^ + 2 a6 + d«, 

Va2-h2adH-62 = a + 6. 

Thus, by inspection, we can extract the square root of a tri- 
nomial which is a perfect square. 

Or we may find the square root in this way : 

a2-h2a6-h6* |a-h^ 
a* 



2a6-h&* 



Trial divisor, 2 a 
Complete divisor, 2 a + 6 

Thus the square root is a + 6. 

This simple case will enable us to devise a rule which is 
applicable to more complicated cases. The procedure is as 
follows : 

I. Extract the square root of the first term and subtract its 
square from the polynomial, leaving 2 a6 H- bK 

II. In 2aby we see the factor b which we know is the 
second term of the root. This factor b may be obtained by 
dividing 2ab by 2 a. Thus we call 2 a the trial divisor. 
Since a is the part of the root already found, we see that 
the trial divisor is double the root already found. After b h^s 
been found, add it to the trial divisor and we have 2 aft + 6, 
the complete divisor. 



ELEMENTARY DEFINITIONS AND OPERATIONS 29 

III. Multiply 2ab + bhj by and subtract. 
This process may be extended to finding the square root of 
any polynomial. 

Find the square root of x* + 21xa-86a:-f 86 — Oa*. Arrange the 
terms according to the ascending or descending powers of some letter, 



x*-6a« + 21ic2-36a; + 36 \x^-Sx + e 



— 6ar« + 21x3 — 36x + 86, 1st remainder 
-6x8+ 9x2 



12 x^ - 86 X + 86, 2d remainder 
12x2-86x + 86 



Ist trial divisor, 2 (x^) = 2 x^ 

Ist complete divisor, 2 x^- 3 x 

2d trial divisor, 2(xa - 3x)= 2x2- 6x 

2d complete divisor, 2 x^ — 6 x + 6 

* 8d remainder 

The procedure is as follows : 

I. The square root of x* is x'. Here a = x^ ; 2 a = 2x2, the first trial 
divisor. 

II. 2 a6 = — 6x8 ; divide — 6x8 by 2x2 and we obtain — 8x, the value 
of &. Thus the first complete divisor is 2 x2 — 3 x. 

III. Multiply 2x2- 8x by —8x and we obtain —6x8 + 9x2. 
Subtract this product from the 1st remainder and we obtain 

12x2-86x+86. 

IV. Treat this remainder as we did the first remainder and proceed 
as before. 

Since the third remainder isO, x2 — 3x + 6is the required square root. 

BXBBCI8BS 

42. Find the square root of : 

1. 4 a« - 28 a»6« -h 49 6". 

2. 4a«-12a6 + 4ac + 962-6&c-hc*. 

3. ai2 — 2icy-h6a:H-y2 — 6y-h9. 

4. 4ic*-20ic»-30aj + 9H-37a^. 

6. 9aj«-2ic*-3aj* + l-h 10 aj'- 12 a^-2 a. 

6. 9 a« -h 9 d« -h 24 a'^ft -h 24 ab^ - 8 a^ft* - 8 a«6* -50 a»6». 

7. ^-5aj + 25. 8. ^'-2 + i. 
4 9 a* 



30 ELEMENTARY ALGEBRA 

^ 4 4 , , 187 , 15 ^25 

9. -m* — rnr -\- ——nr :-wi4--r' 

9 48 4 4 

tix A 4a , a* , 2a' 4a' , a* 

11. a' -H a5, to three terms. 

12. 1 — a, to four terms. 



SQUARE ROOT OF ARITHMETICAL NUMBERS 

43. 1« = 1, 2* = 4, ..., 9* =81, 10« = 100, ..., 90« = 8100, 
100* = 10000, 1000» = 1,000,000. 

This is sufficient to show that a perfect square consisting of 
one or two digits has one digit in the root ; one consisting of 
three or four digits has two digits in the root ; one consisting 
oijive or six digits has three digits in the root ; and so on. In 
other words, the square of a number has twice as many inte- 
gral digits, or one less than twice as many, as the number 
itself. 

Hence, to find the square root we must separate the number 
into periods of two digits each, towards the left and right from 
the decimal point The period farthest to the left may have 
one or two digits, whereas the one farthest to the right must 
have two, a cipher being annexed, if necessary, to complete it. 

The square root has therefore one digit corre9ponding to every 
period in its square. Separating the square number into 
periods enables one to find one digit in the root at a time. 

This is seen more clearly if we consider that the square of 
any number of tens, say 4 tens (40), ends in two ciphers (1600) ; 
hence the two digits on the right are not needed to find the 
tens' digit (4), and are set aside until the unit's digit is to be 
found. 

Likewise, the square of any number of hundreds (400), ends 
in four ciphers (160,000), and all of these may be set aside until 
the hundreds' digit is found and they are needed for the tens' 
and units' digits. 



ELEMENTARY DEFINITIONS AND OPERATIONS 31 

After pointing off the number into periods, the method is 
similar to the one used for polynomials. The procedure is 
based on the formula d^-{-2ab -^b^ = (a-\- by. 

When dividing by the .trial divisor, exclude, for brevity, the 
right-hand digit in the remainder. 

The reason for this is seen in the example which follows : 
the trial divisor 4 is equal to 40 of the next lower units. Now 
14 divided by 4 gives the same digit as 148 -s- 40. 

Point off one decimal place in the root for each period in 
the decimal. 

Find the square root of 648.643241. 

648.643241 1 23.421 
4 



148 1st remainder. 

129 



1964 2d remainder. 

1866 



9832 3d remainder. 

9364 



46841 4th remainder. 
46841 



1st trial divisor, 2 (2) = 4 
1st complete divisor, 43 
2d trial divisor, 2(23) = 46 
2d complete divisor, 464 
3d trial divisor, 2(234) = 468 
3d complete divisor, 4682 

4th trial divisor, 2(2342)= 4684 

4th complete divisor, 46841 

6th remainder. 
Point off three decimal places in the root. 
Thos, the square root is 23.421. 

To find the square root of a fraction, which is not a perfect 
square : Reduce the fraction to a decimal and find the square 
root of the decimal. 

EXERCISES 

44. Find the square root of : 

1. 6889. 6. 567,009. 

2. 2025. 7. 2777.29. 

3. 161,604. 8. 88.454025. 

4. 337,561. 9. .00698896. 
6. 7.3984. 10. .059049. 



32 ELEMENTARY ALGEBRA 

Find the square root, to 3 decimal places, of : 

11. 2. 13. 5. 15. 3|. 

12. 3. 14. 1.005. 16. f 

FACTORING 

45. Factoring is the process of finding two or more expres- 
sions whose product is a given expression. 

An integral expression is one that contains no fractions. 
A rational integral algebraic expression is one containing no 
fractions, no negative exponents, and no indicated roots. 

2 + 3 aaj + aj'* is a rational integral expression ; J x* + J x + 1, a;-* + 6, 
y/a + x + y, X* + 10, are not rational integral expressions. 

A prime factor of an integer is an integral factor which is 
exactly divisible only by itself or one. 

Thus, and 13 are prime factors of 65. 

A prime factor of a rational integral expression is a rational 
integral factor which cannot itself be resolved into rational 
integral factors other than itself ojid one. 

Thus, a — 6 and a^ + aft + 6* are prime factors of a^ — 6«. 

In the following exercises we shall confine ourselves to find- 
ing 2>rime factors of rational integral expressions. In § 83 
irrational factors will be required. 

I. Type form : aft + ac — ad. 

ab + a4i —ad=: a(b 4- c — d). 

46. Factor.: 

1. y^-2y' + Sy*-y. 3. a(x + y)-b(x + y). 

2. 80x»y*-160 ajy -240 o^, 4. (a? - y)c -(« + y)c. 
6. m{c — d)— n{c — 6i)-\-p{c — (Z). 

6. a{m — n) + b{m — n). 

7. m(a — &) — w(6 — a). 



ELEMENTARY DEFINITIONS AND OPERATIONS 33 

8. a(x-Sy)+b(Sy'-'X), 

®- (a? — y)— 2a(y — aj)-H36(y — a). 

10. c(a — & — c)+d(6+ c — a). 

11. Type form: ax -\- bx -\- ay + by. 

ax -{- bx +ay + by == a{x -^ y)'\' b(x '\' y)=^ix+y)(a + b). 

47. Factor: 

1. Sx + Sy-{-ax + ay. 6. 2ac — 36c — 2ad4-36d. 

2. 005 — ay — &aj + 6y. 6. a"c" — &"c" -f- a"d" — 6*d*. 

3. 2aaj — 2 6a5 + & — a. 7. a'— a6-Hac— ac+6c— c*. 

4. 4a— 46 + ac — &c. 8. aV-h6V — aV — ^V* 

III. Type form : 

tf ± 2a6-H 6* and fl^-hft* + c* + 2a6 - 2ac- 26c 

a*±2a6-h6«=(a±6)*. 
a* -h 6* + c* 4- 2 a6 - 2 oo - 2 6c = (a + 6 - c )*. 

48. Factor: 

1. 9r*-24r« + 16a». 3. a* + a + i. 

2. a*»-14a»-h49. 4. (aj-y)*-2(aj-3^)+l. 
6. 4(c - 3)2 - 12(c2 - 9) + 9(c + Sy. 

6. sfi-\'f/^'{-z* — 2xy^23DZ + 2yz. 

7. a* + 6*4-c'--2a6 + 2ac — 26a 

8. l + a* + 6* + 2a-26-2a6. 

9. 4a* + 6* + c^-4a6 + 4ac-26c. 
10. m*4-n»-h9 — 2mw- 6m + 6w. 

IV. Type form: tf-6*. 

49. a«-6* = (a + 6)(a-6). 

a*-2a6 + 6«-c*=(a-6)«-c* = (a-6-hc)(a-6-c). 
a* — c* — 2cd— cP = a*— (c + d)* = (a-hc + d)(a — c — d). 
a«-2a6 + 6*-c» + 2cd-(f» = (a-6)«-(c-d)« 

=(a — 6 + c — d)(a — 6 — c -h d). 



34 ELEMENTARY ALGEBRA 

50. Factor: 

1, c8~64. 6. 100aW-a*-hl2a6-366». 

2, 121c«-144d*. 6. 36y» + 84y + 49-92!«. 

3. aj* — m* — 2mn — w*. 7. m« + 10m» + 25 — 81n*. 

4. 49a«-64 + 166-&«. 8. (a - 3)« - (6 - 2)*. 
9. 46» — a?* + 4a^-ha*4-4a& — 4y«. 

10. 26(f*-l-9c*d«-10ad + a2 + 6cd. 

V. Type form : jfi + bx + c, 

51. Factor: 

1. aj*-h5a; + 6. 6. r» — 4 r« — 77 ««. 

2. a;2-5a? + 6. 7. b^c^ -^ 7 bey -{- 10 yK 

3. yj- 7^4-6. 8. c^d* H- 7 cc«« - 120 (f*. 

4. 2/2 + 72/-98. 9. <«-23«*-60. 

6. 2!2 + 102 + 9. 10. a*2/® ■- 16 ic'y^ + 39 2^ 

VI. Type form : or^ + 6x + c. 

52. ao?* + &« -h c = (pa? + g)(ra? 4- «), wherein pr =: a, ps -{- qr 
= 6, g« = c. 

By this method p, g, r, « must be found by trial, so that the 
three conditions just named will be satisfied. 

Factor 6a^ — 5x—6. 

Here a = 6f 6 = — 6, c = ^6. 

If possible, we must find numbers p, q, r, 8, such that pr = 6^ gs = — 6, 
p« + gr = — 5. 

pr = 6 suggests the values jp = 3, r = 2, orjp = 1, r = 6, etc. 

g« = — 6 suggests the values g = 3, « = — 2, or g = 2, s = — 3, etc. 

Try the different sets of values of p and r, whose product is 6, with 
each pair of values of g and s, whose product is — 6, until a combination 
is found which satisfies ps -{- qr= —6. 



ELEMENTARY DEFINITIONS AND OPERATIONS 35 
Try the sets of values p = 3, r = 2, g = 2, a = — 8. 

Examine (3x 4-2)(2x-8) 

I I 

-9a; 

We see that the sum of the two cross products + 4 a; and ~ 9 x is — 6 a;, 
which is the middle term of 6 a^ — 5 x — 6. Hence 

6a^-6a?-6=(3x + 2)J2x — 3). 

53. Factor: 

1. 6aj* + 5a? — 6. 6. 15 a? + 14 oa? - 8 a*. 

2. 5aj«4-14a?-3. 7. 17 a«aj* - 36 oa; + 4. 

3. 5aj2-14a?-3. 8. 20r» + 19r« + 35*. 

4. 8y*-14y-39. 9. 9 aj» - 36 a^ - 13 y*. 
6. 18m* + 37m + 19. 10. 28m2-48m + 17. 

VII. Type form: a* + o^ft* + &*• 

a* + a»6« + 6* = a* -h 2a*b^ + 6' - a^b^ = (a« + 6')* - a«&» 

= (a« + a6 -f 6*)(a« - aft + b^)- 

54. Factor: 

1. a;*-H«* + l. 6. a?* — 14 aj*y* + y*. 

2. 9m* + 8mV4-4w*. 7. m* — 38mV-Hn*. 

3. 16a* + 20a«6« + 96*. 8. 4 a* - 13 a'6« + 9 6*. 

4. r8 + r*5* + «*. 9. 25c* + 26c«cP + 9d*. 

6. a* - 5 a«6* 4- 4 6*. 10. 49a* + 110a*62 + 81 6*. 

55. In factoring, first take out monomial factors. Then 
inspect the resulting polynomial, ascertaining to which type 
form it belongs, and factor it accordingly. In the final form, 
all factors should be prime. 



36 ELEMENTARY ALGEBRA 

ORAL BXBROI8BS 

56. Factor: 

1. rn^-a}. 9. 20-9j5i-«*. 17. a^ + yhi. 

2. ^a^— y^. 10. m* — n*. 18. aj* — ^a?. 

3. a« + 6a!> + 96*. 11. p^-j:^. 19. a5»-4ajy». 

4. 9aj»- 6ajy + y*- 12. aj» + 2aj-35. 20. 4a»-a6*. 
6. 4aj*-12a?5+9«*. 13. aj«-2a?-35. 21. a^6-9&». 

6. oaj*— 16ay«. 14. m*-2m — 3. 22. 5a' + a*6. 

7. ca^ + cy\ _ 15. p^-Qp-^- 5. 23. a* - 2 a*6» + &*. 

8. y«-5y + 6. 16. 4c*±4cd + cP. 24. 9a*±12a6+4&*. 

WBITTBN BXBBOI8BS 

67. Factor: 

1. 4a* — 20a6. 6. a*hc — ah*c. 9. 16 + 8 c + c«. 

2. 6»-d». 6. 25r*-36«*. 10. l~a«. 

3. m* — 9n*. 1. f -^^--y — z. 11. a^^ — 6*. 

4. 4a*-.8a«6. 8. mV— 2mn*4-l. 12. a* - 5 a6 + 6 6». 

13. cflj -f cy — cte — dy. . 24. m* — 2 mn + n* + m — n. 

14. 6*-|.66« + 5. 26. v^'{-7*'\-r^ + r. 
16. 64-&». 26. aj* + 2aj« + l. 

16. 7 a? — 12 — aj*. 27. c*-cP — c — d. 

17. a*®— a*. 28. 6* — 26c + c» + 5& — 5c. 

18. 3aj» + 3aj«-27aj-27. 29. 60* + 13a6 + 66*. 

19. a?*" + 2af-15." 30. a?" - ajyi*. 

20. aj»-6aj*4-12aj— 8. 31. r* -4r» + 6r» — 4r + 1. 

21. a^^a^\, 32. (a? — y)* - 9(aj + y)*. 

22. r»-«*-2««-<». 33. a* + 12 a6 + 36 6*. 

23. 26-4A:* + 4^*^-^«. 34. (a + 6)* - 7(a + 6) + 12. 

36. (a? - 13 aj -h 42)(aj»+ 3 a? - 10). 
36. (6-c)*+36-12(6-c). 



ELEMENTARY DEFINITIONS AND OPERATIONS 37 

37. (a + («)*+ (6 + c)* - 2(o + d)(6 + c). 

38. 25 a* - 41 a*6* + 16 6*. 

39. a*-2a6 + 6«-2ac + 26c+<^. 

40. c" — (?•. 

41. a?-49a? + 360. 

42. i^-47r»+l. 

43. ab—abc^. 

44. m* + 64n*. 

46. (m — n)* — 1 + a(m — n -f- 1). 

46. 3a*-10a6-86«-h3ad-H2W-9ac-6&c. 

47. a?*-3iB* + 4. 

48. 4a* + 816*. 

49. 4a*~37a26«-H96*. 
60. 64y*+128.v»2!2 + 81»*. 

« 

FRACTIONS 

58. KfrcLCtion is the indicated quotient obtained by divid- 
ing one number by another. 

The fundamental principle of operations with fractions is — 
Both numercUor and denomincUor of a fraction may be multiplied 
or divided by the saws number, vnthout changing the value of the 
fraction. 

Thus, I = li, 2^ = ?, 8(^ + y ) =-2_. 

8 16 2a a 4(x+y)(x-y) x-y 



38 ELEMENTARY ALGEBRA 

BXEROISES 
59. Reduce to lowest terms : 

1. — — -n-n' O. 



6a»&c* 3aj* + 3a;-18 

2 ?i^. 7 (a-m«-i-ft)v 

2a-26 g 4-(m-ny 

* a«-6« ' * n*-(m--2)«' 

- 5 ggg — 10 gy g a* — 2 a — 1 

aj* — 5ajy4-6y«' " a» — 20*4-1 

g 8(a^-6*) j^ aa-^c«-9-2a5-6c+&V 

24a? + 246' ' 62-.a«-9-26c-6o+c*' 

60. From the laws of signs in multiplication and division it is 

evident that 

a — a a . — a a a 



6 -6' 6 b ' b -b 

From these relations we see that 

1. The signs in both numerator and denominator may be 
changed without changing the value of the fraction. . 

2. The sign of the numerator and of the fraction may be 
changed without changing the value of the fraction. 

3. The sign of the denominator and of the fraction may be 
changed without changing the value of the fraction. 

61. The following principles may be used effectively in 
operations with fractions. 

1. The signs of an even number of factors may be changed 
* without changing the sign of the product. Explain. 

Thus, a • 6 • c = ( — a) • (— 6) • c = abc. 

2. The signs of an odd number of factors may be changed, 
provided the sign of the product is changed. Explain. 

Thus, G'b- c= — (— a) 'b ' c = abc ; a'b 'C=^ (—«)(— b)(—c). 



ELEMENTARY DEFINITIONS AND OPERATIONS 39 

MULTIPLICATION AND DIVISION OF FRACTIONS 

62. In multiplication and division of fractions in algebra 
we have the same rules of operation as in arithmetic; 
namely, 

1. To find the product of two fractionsy mtdtiply the numerators 
together for a new numerator , and the denaHrainaJtora together for a 
new denominator. 

2. To find the quotient y invert the divisor and then proceed as 
in the multiplication of a fraction by a fraction. 

It is easy to establish the truth of these rules by means of 
the equation. 

For example, the second rale may be proved as follows : 



Let 


a c 


then 


a = &£, c = dy, 


and 


ad = hdx^ he = hdy. 


Dividing equals by equals, 




ad bdx X 




be hdy y 


But 


x_a . c 
y h' d' 


Hence 


a , c __ad 
h d he 



This proves the rule for the division of one fraction by another. 

Proceeding in a similar manner, prove the rule for the 
multiplication of fractions. 

Multiply m±^ by ^^'-^y\ 

hx -{-hy ax-\- ay 



40 ELEMENTARY ALGEBRA 

BXEROISBS 
63. Perform the indicated operations : 

^ a b d ^ r-'S r* — «* 

1, -■ • — I — • 6. • • 

c c T^ + ra r* — rs 

^ 2a Sbx 2h ^ x^-3x-{-2 x'^-Sx 



X a 3aa a^ — 6x ijfi-\-x-'42 

• 26-2'^6— l' ' a + b'^(a + by 

. 16m*n 8 m' ^ a^ + ah a-\-b 

45 fl^ ^ ^ <^* — y* {'^ — yy 

« 

jj a^~-4a; + 3 gg-12a?4-35 5ar-10 

aj2-7aj + 10' aj2H-2a?-3 '«2__3a.* 

--. a^ + oft a'c + 2 a^c + aft^c 
a2 4- 62 a^c — b^c 



13. 



m 



*— n* j6 (m' — m^n + mn^ — n*) 



14. 



3 m2 4-3 n2 m^ — 2 mn + n« 

18ag-9a , 2a2-f5a-3 



r2-l "^r^-llr + io' 

gg — c2 -t- 2 a& 4- ft^ ?f^lAzi£. 
a2— c2 — 2 6c — 62'a + 5_c* 

2a2 4-a-10 3a2 4-4a-4 



17. 



a2_4 2-3a 



ELEMENTARY DEFINITIONS AND OPERATIONS 41 
05*— J/* x^ — xy a?^ — 2 a?y + y^ 



18. 



19. 



20. 



(x-hyy x-^-y a^-\-2xy + y^ 

9 gg - 16 feg , 6 gg - 8 a5 
3g2-llg6-20&2 • g2-5g6 * 

&2-136 + 42 a2 + 6g-27 
a5-3 6-7.g + 21 * g2-81 



COMPLEX FRACTIONS 

64. A complex fraction is one which contains one or more 
fractions in its numerator, or denominator, or both. 

Complex fractions may be simplified in two ways : 

1. Simplify the numsrator and denominator, then divide the 
numerator by the denominator, 

2. Multiply both numeraJbor and denominator by the I. c. d. of 
their fractional terms. 

The pupil is probably familiar with the first method from 
his study of arithmetic. In simplifying complex fractions 
arising in algebra, the second method is usually the easier. 
It should be thoroughly m^^stered. 

Simplify, i-Zljk. 

i ~i r 

The l.c.m. of 2, 8, 4, 9 is 86. 

Multiplying the nomerator of the complex fraction by 86 gives 18 — 12. 

Multiplying the denominator of the complex fraction by 36 gives 9 — 4. 

Hence, Izil = l^-Jl? = ? . 

J-J 9-4 6 

BIXBBCISES 

65. Perform the indicated operations and simplify : 
1. 



L o + ft *J a»-6» 



42 ELEMENTARY ALGEBRA 

3 ^ 'f ^ M 

■ 1 - a* \1 - a 1 4- ay 

4. (2aj*-6a? + 3)-!-/'i-iy 

5. fl + i + lY?+l?+^y?iy:£. 



7. 



\n sj \m r) 

\» y ^J \y ^ 



10. (m^ 



11. 



i+i 



12. 



a 
b 



13. 



I a 
" + 6 



11 

a & 



m + 1 


V(m + 1)« m + i; 




14. 


2 + 2 
a 




-r 




15. 


1+^ 

s 










a + 2> q~ & 




16. 


a — b a+ 6 




a — 6 , a4- ^ 



a 6 a+b 'a—b 



ELEMENTARY DEFINITIONS AND OPERATIONS 43 

a?-3 — 

17. ^• 



a? + 4 





a« 


+ ab 




19 


a« 


-6* 




X V. 


a 
a + b 


a — 


b 




r 


3- 
r 


r 


20. 


S + r 




r 


+'- 


r 



18. -3. 

x + y-{- — 

y r + 3 r 

HIGHEST COMMON FACTOR AND LEAST COMMON MULTIPLE 

66. Only expressions which contain no fractions or radicals 
will be considered here. 

The highest common factor (h. c. f .) of two or more expres- 
sions is the product of the prime factors that occur in each 
expression, every factor being taken the least number of times 
it occurs in any one expression. 

This definition indicates the process of finding the h. c. f . 

The lowest common multiple (1. c. m.) of two or more expres- 
sions is the product of all the prime factors that occur in the 
expressions, every factor being taken the greatest number of 
times it occurs in any one expression. 

This definition indicates the process of finding the 1. c. m. Observe 
that the 1. c. m. is the least expression which is exactly divisible by each 
of the given expressions, while the h.c.f. is the expression of highest 
degree by which each of the given expressions is exactly divisible. 

EXERCISES 

67. 1. Find the h. c . f . and 1. c. m. of 

4aa:- 4a3^, Sa***- Say, 12a;»- 122^. 

Factoring, 4 ax — 4 ay = ^a(x — y) . 

8 a%e2 - 8 a^j^ = 2^a^(x -\-y){x-y). 
12 ac» - 12 y8 = 28 . ^x - y){V^ + «y + V^)- 
The h. c. f. = 22(a; - y) = 4(x - y). 

The 1. cm. = 2« . 3 aHx - y)(x + y)(x2 + xy + y2) 

= 24a2(x + y)(«»-y»). 



44 ELEMENTARY ALGEBRA 

Find the h. c. f. and 1. c. m. of : 

2. 9 a6c, 18 a«6c», 27 a»6*c. 

3. 16 aj»y V, - 30 a^i^z, 45 «V»' 

4. (a - 6)*, a* - 6«. 

5. 3a«+6a6 + 36«,9a«-96«. 

6. 2(a - 6)»(a 4- 6), 3(a -f 6)»(a - 6). 

7. a^-5»+6,aj* + 2aj-8. 

8. 3a»-3a6«, 3a(a+&)«. 

9. 3aaj»-15aaj4-18a, 6aV + 24a*a;-126a«. 

10. sfi — y\ X — y, ax ^ ay ^ ex -{- cy. 

11. a(a - 1)«, a«(a* - 1), 2 a(a« + 2 a - 3). 

12. 5(aj» - y»), 10(aj + 3^)«, 15(x - y)«(a; + y). 

13. A«-6A-14, ^'-10^ + 21, A«-49. 

14. 16(ab 4- b), 8 a(a« - 6«), 24 ab(a} + 2ab + 6*). 
16. m\m — 1)*, m(m' — 1), m\fn}n — w). 

16. (y + 4)(y2-16),y«-y-20,ajy + 4aj-ay-4a. 

ADDITION AND SUBTRACTION OF FRACTIONS 

68. The process is the same as in arithmetic. If the frac- 
tions do not have the same denominator, they must be reduced 
to fractions which do have the same denominator. The lowest 
common denominator (1. c. d.) is obtained by finding the lowest 
common muLtiplk of the given denominators. 

1. Perform the indicated addition and subtraction : 

1 — X . a-{-x 5a — X 

' "^ — • 

2 ax 3 aa^ 6 a*x 



ELEMENTARY DEFINITIONS AND OPERATIONS 45 

The 1. c. d. is 6 a^^. The redaction of the fractiomi to the 1. c. d. is 
effected as follows : 

1 — « Sax 3 ax — 3 ax^ 



6a^^-i-2ax =3 ox, 



2 ax Sax 6 a^^ 



6a%c«-^3ax»=p2a, a+| . 2a^ 2a^ + 2ax , 

Sax^ 2 a QaH^ 



6 a^x^ -i- 6 a^x = «, 



5a — X X _ 6 ox — x^ 
6a*e * x"~ 6a2x2 



We obtain ^~^ 4. <^^ + ^ _ 5a — x _ 3 ax— 3 ax^H-2 a'^H-2 ax— 5 ax-\-oi^ 
2ax 3ax2 Ca^x 60^x2 

^2a«-3e^^ + x^ ^ns. 

In practice, much of the work can be done mentally and need not be 
written down. 



2. Perform the indicated subtractions : 

^ 1 J^ 

a-b 2a 46* 

The 1. c. d. is 4 ab(a — b) . Reduce the fractions to the 1. c. d. : 

1 4ab 4ab 



4 a6(a — 6) + (a — 6) = 4 ab, 



a — b 4a6 4a6(a— fr) 



4a6(a-6)^2a =26(a-6), -1 . 2^(<»- ^) = 2a6 - 2 6^ 
^ ^ '^ ^' 2a 26((r-6) 4a6(a-6) 

4a6(a-6)-f.46 =a(a--b\ i- • <»(«-^) = <»" " «^ . 

^ ^ "V« >'» 4^ a(a-6) 4a6(a-&) 

Subtracting and simplifying, ^^^t.^^^7,^\ -4n». 

4 ao(a — 0) 

3. Perform the indicated additions : 



a* — 1 aj(aj + 1) 1 — a? 

The lowest common denominator is x(x^ — 1). Before reducing the 
last fraction to one with the denominator x(x2 - 1), it is a convenience to 
be able to write x - 1 in place of 1 — x. As 1 — x differs from x — 1 only 



46 ELEMENTARY ALGEBRA 

in algebraic sign, we can do this, provided we change also the sign of the 
fraction or else change the sign of the numerator. The former change is 
simpler. Thus we have, 

3^2 1 



a;2 - 1 x(x + l) x-1 

The reduction to the lowest common denominator, x(x* — 1), is effected 
as follows : 



x(x2-l)+a;(a;+l)=a;- 1, 



«« - 1 z »(aJ« - 1) 
2 x—l 2a; — 2 



x{z+l) « — 1 a(a:« — 1) 



x(a«-l)-i.(x-l) =x(« + l), — ^ • x(x_±lX__3i^±x_^ 
^ ^ ^ ^ ^ ^' x^l xix+1) «(a;2_i) 

Weobtain-i- + — ^ L- = 3xH- 2x- 2- x«- x 

x2-l^a;(a+l) x-1 x(a^-l) 

4 X - 2 - x« .^. 
= x(x«-l) - ^'^^ 

4. Perform the indicated addition and subtraction : 

1 1 + 1 



(a - 6)(c - a) (a - 6)(c - 6) (c - 6)(a - c) 

Here the factor (a — b) occurs twice, and both times with the same 
order of the letters. The same is true of (c^b). But in the first de- 
nominator occurs the factor (c — a), in the last denominator occurs {a—c). 
It is a convenience to have (c — a) in both denominators. Since (a — c) 
differs from (c — a) only in sign, we write in the last denominator (c — a) 
and at the same time change the sign of the fraction. We obtain 



(a-ft)(c_a) (o-&)(c-6) (c-6)(c-a) 

The 1. c. d. = (a — 6)(c — a)(c — 6). Reducing to a common denom- 
inator and adding, 

e — b i 



= 0, Ana, 



ELEMENTARY DEFINITIONS AND OPERATIONS 47 

BXEBCISBS 

69. Perform the indicated operations : 

, 2x.^x . 6,6 
1- -TT^—r* 4. - + 



3 4 a?-l aj + l 

2. • 5. 



y X X — y X — 2/ 

3. JL 8_. 6. ^^^ + 5i^. 

aj + 1 x + 1 a + b a—^b 

„ a — b , b — c , c — a 

7. 1 -f- • 

ab be ac 

(y -»)(»-») («-»)(» -y) (» - 2/)(2/ - «) 

g __3 7 4- 20a 

' l-2a l + 2a l-4a«* 

10. -^+ 1 1 



11. 



a + 6 (a + 6)» a« — 6« 
11 .V 



2(a) -y) 2(3! +y) as'-y' 



12. -J- + ^ll ?-+ 1 



m+w m + 3n m — n m — 3n 
j„ g 6 , a' + ft' — oft 



a* - 62 a* + 62 ■ ft4 _ ^4 (a _^ ft)(a2 _|. 52) 

14. ?-t^ + 1±1 + ^+^ 



{x-yXx-z) (y-^z){y-x) (z'^x)(z-y) 

io« — — -f- 



(a-bXa-c) (6-.c)(6-.a) (a-cXc-b) 



48 ELEMENTARY ALGEBRA 

,^ a*4-2a-f4 a*-2a4-4,4-.a» 
iv. — — -f- — * 

a + 2 2-a a-2 



17. l^+.,-l^- 1^+ 1 



4a + 4 4-4a 12-12a«3-3a 
18. -^-i—.-^-J— ^+ 1 



a* — a; — 6 aj^+6a5 + 8 12 — as — a^ 

19. a;-.y-g ^ y-^g-,a; ^ g-g-y 
(aj-y)(aj-2) (y - z){y - x) (z - x)(z -- y) 



20. — 

PiP-gXp-r) q(q- p) (q - r) pqr 

21. :^-f ^ -^^ 



2-a a + 3 a» + a-6 

22. ^ +-^ nlf. 

a? + l 1 — a^ 05—1 

23. ^ ^ ,+ ^ 



24. 



2-a:-aj» aj*+3a;+2 1 - a?* 

2 _3 4 

(a-2)(a-3) (3-a)(a-l) (l-a)(2-'a) 



1.1.1 
25. -— -H-: — -H- 



(a — 6)(6 — c) (c — a){b — a) (a— c)(6 — c) 

QUADRATIC EQUATIONS 

70. The equation ao* + 6a5 + c = is a complete gt^odrottc 
equation. It is a quadratic equation because the highest power 
of the unknown x is the second ; it is complete because it con- 
tains a term involving the unknown x to the first power and a 
term c (called the ahsolvie term) which is free from x. 



ELEMENTARY DEFINITIONS AND OPERATIONS 49 
Quadratic equations of tlie forms 

are called incomplete quadratic equationSj because either the term 
involving x to the first power or the absolute term c is absent. 

71. Incomplete quadratic equations are easily solved. 

Take the form »* = c. 

Extracting the square root of both sides of the equation, x = ± Vc. 

The form ic* + 6a; = is solved by factoring, but may be solved 
also by the method of " completing the square," to be explained 
later. 

Factoring, we obtain, x(x + 6) = 0, 

Make the first factor equal to zero, x = 0. 

Make the second factor equal to zero, x + & = 0. 

x=— h, 

72. The solution of quadratic equations leads to two values 
of the unknown quantity. As both of these values are usually 
of interest and importance in the solution of problems, it is 
customary, in the extraction of square roots, to write down 
both results, the principal value and also the second value. This 
is indicated by the use of the symbol ± . Thus, in " «= ± Vc," 
+ Vc is the principal root, — Vc is the second root. 

Since, in finding a? = ± Vc, the square root of both sides of 
the equation has been extracted, it might be claimed that the 
sign ± should be written on both sides, giving 

± aj=:^ Vc. 

But this result is the same as when we write a? = ± Vc. 

For, the equation ± « = ± Vc means here 

(1) +a;= + \/c. (8) ■\-x= — y/c, 

(2) -x = ->/c. (4) — a;=H->/c. 

Of these four sets, the first two are the same, and the last two are the 
same. Hence, %-=i±y/c gives all the values of x. 

E 



50 ELEMENTARY ALGEBRA 

73. A complete quadratic equation may be solved in three ways : 

(1) Bj factoring, 

(2) By completing the square, 

(3) By substitution in a formula obtained by the method of 
completing the square. 

1. With our present knowledge of factoring, the first method is 
applicable only when the roots of the quadratic equation are ra- 
tional. The method depends upon the principle that, if the prod- 
uct of two or more factors equals zero, one factor must equal zero. 

Solve 3iB« = 2 + 5a;. 

Transpose all the terms to the first side, Sar^ — 5a; — 2 = 0. 

Factor, (3 a; + 1) (x - 2) = 0. 

Place the first factor equal to zero, 8 x + 1 = 0. 

x = -J. 

Place the second factor equal to zero, a; ^ 2 = 0. 

a; = 2. 
The required roots are 2, — |. 

II. The second method depends on the type form of a tri- 
nomial which is a perfect square : a^ ±2ah-\- ¥. 

If the b^ is lacking, we may take the square root of the first 
term, double it, divide the middle term by this, and square the 
quotient. 

Take, for illustration, x^ ± 6x, To complete the square, we take the 
principal square root of x^, which is x ; double it, 2 x ; divide 5 x by 2 x, 
{ ; square j, ^. 

Hence x^±6x + ^\aa, perfect square. 

This results in the following rule for the solution of a quad- 
ratic equation : 

> 1. Transpose all terms containing a^ and x to the left side of the 
equation; all others to the right side. 

2. Dimde both sides by the coefficient of a?. 

3. Add to both sides the square of half the coefficient of x. 

4. Extras the square root of both sides and solve the resulting 
equations. 



ELEMENTARY DEFINITIONS AND OPERATIONS 51 

Solve 3aj«-5aj-2 = 0. 

Transpose, Sx^ — bx = 2. 

Divide by 3, a;2 _ j j. - |. 



Add(J.})2, x2-4a; + jj = 

Take the square root, x — f =; d: J. 

Whence, « = 4 ± {. 

a = 2, - }. 
^n<., 2, - J. 

III. The third method depends upon the formula derived 
from the solution of the type form of the complete quadratic. 

Solve aa^ + bx^ c=:0. 

Transpose c, ox^ H- 6a; = — c. ^' . , (^^ 



Divide by a, ^ 



'^^ *aiL^__c 



(2 'a) 



Addfl.^V 



» 



a - a 

o 4a^ 4a2 



/', 



Take the square root, » + ;^ = ± ^^^ - 4 gc 

2a 2a 



- »:fc yy - 4 oc • 

2a 

In using the formula, transpose all terms of the given equa- 
tion to the left side. Why ? 

Solve 3 aj* - 5aj = 2. 

Transpose the 2, 3 a;8 -. 5 » — 2 = 0. 
Here a = 8, 6 = — 5, c = — 2. 

.^^^6±v^6TM^i±7^2or-i/ 
6 6* 

BXBBCISES 

74. Solve ; if a numerical equation has irrational roots, aiy- 
proximate their values, to three decimal places. Use the table 
of square roots in § 197. 

1. x^ + 4a; = 6. 3. «« — 8a: = — 11. 

2. 3iB«4.aj-14 = 0. 4. a?»-2a;-15 = 0. 



52 ELEMENTARY ALGEBRA 

5. aj2+6aj = -9. 10. «* — 2 ooj = 6» - a*. 

6. aj* + aj4-l = 0. n. m»aj« - 2 ma? = n - m^. 

7. x^ — ax=c. „u 

12. aa? -f- -^^ = (6 + c)a;«. 

8. 6a?» — caj=:d. 6 + c 

9. 3aj»-h»4-7 = 0. 13. (3 a: - 5)» = 4 ic 
14. 12«»4-13aj-35 = 0. 

16. -i^+ 1 1 



05 — 1 05 — 2 a?— 3 

16. (a: + 2)(aj-2)=7(aj-f 2)-6. 

17. (ic-f 3)2-2(a;4-3)H-l = 0. 

18. 5aj(aj-3)-2(aj2-6) = (a;H-3)(a; + 4). 

19. l«^^±2 ^?^5 ^, (<^'-f)(y;-^^) = 2y. 

2a; + l a;-3 c* 4- cP ^ 

(a? 4- m)* (a; — 7i)* 2y — a+2c 

23. (aj-l)(a:+2)(aj* + aj-l)=0. 

24. (a?*-7a:+12)(ar^+3a;+2)=0. 

PROBLEMS 

Ascertain in each problem whether both roots of the quadratic equation 
are applicable to the problem. 

75. 1. Find two consecutive numbers whose product is 992. 

2. Find the length and breadth of a rectangle whose area is 
375 sq. in., and whose length exceeds its breadth by 10 in. 

3. Express 71 as the sum of two numbers whose product is 
448. 

4. If the length of a square be increased by 2, and the width 
be increased by 3, the area of the resulting rectangle is 40. 
Determine the length of a side of the square. 



ELEMENTARY DEFINITIONS AND OPERATIONS 53 

5. The height of a triangle exceeds its base by 8 ; if the 
area of the triangle is 1209, what is its base ? 

6. The diagonal of a square is 2 ft. longer than the side. 
Find the side. 

7. A cylinder 12 ft. in height has a capacity 125 cu. ft. 
Determine the diameter of its base. 

8. When the edges of a cube are each increased by 6 in., 
the volume is increased by 936 cu. in. Find the dimensions of 
the original cube. 

9. The sum of the numerator and denominator of a fraction 
is 77. If the numerator is increased by 111 and the denomi- 
nator is increased by 40, the fraction is doubled. Find the 
fraction. 

10. Find two numbers which differ by 2, the cubes of which 
differ by 296. 

11. The radius of one circle is twice the radius of another. 
Find the radii of both, if the difference of their areas is 75. 

12. The difference of the volumes of two spheres is 100 cu. 
in. ; the difference of their radii is 5 in. Find their radii, 
correct to two decimal places. 

4irr8 



The volume of a sphere is 



3 



13. A woman paid $ 64 for silk. If she had bought 4 yards 
less for the same money, she would have paid $ 1 J more per 
yard. How many yards did she buy ? 

14. The longer leg of a right triangle, exceeds the shorter 
leg by 3 ft. The area of the triangle is 135 sq. ft. Find the 
length of each leg. 

15. A bookdealer sells a number of algebras for $ 87. Had 
he reduced the price of each book by 12 ^, he would have sold 
16 more books for the same sum of money. How many books 
would he have sold at the reduced rate ? 




54 ELEMENTARY ALGEBRA 

16. If a man's daily wage had been $ 1 less, it would have 
taken him 15 days longer to earn $ 180. How many days did 
he work to earn $ 180 ? 

17. A picture 10" X 14" is placed in a frame of uniform 
width. If the area of the frame is equal to half the area of 
the picture, how wide is the frame ? 

Draw a figure. 

18. Find two consecutive even numbers whose product is 
528. 

19. Find two consecutive odd numbers whose product is 
8099. 

20. If the difference between the parallel sides of a trape- 
zoid is 5 ft., and, the altitude of the trapezoid is equal to the 
longer of the parallel sides, find the lengths of the parallel sides 
when the area is 2375 sq. ft. 

21. A flower bed is 15' x 20'. How wide a walk, must sur- 
round the bed, to increase the total area by 770 sq. ft. ? 

22. A tinner makes a square box 3 in. deep, with a capacity 
of 1587 cu. in. From each comer of a square sheet of tin a 
3-inch square is cut and the four rectangular parts of the tin 
are turned up. What are the dimensions of the square sheet 
of tin? 

Draw a figure. 

23. If a square has its length reduced by 7 in. and its width 
by 10 in., what are the linear dimensions of the resulting rec- 
tangle, if its area is 8370 sq. in. ? 

24. An oil tank can be filled by one pipe in 2 hours less 
time than by another pipe. If both pipes are open 1^ hours, 
the tank will be filled. In what time can the tank be filled by 
each pipe ? 

25. A number of postage stamps can be arranged in a rec- 
tangle, each side containing 60 stamps. If the same number 



ELEMENTARY DEFINITIONS AND OPERATIONS 55 

of stamps be arranged in two rectangles so that each side of 
one rectangle will contain 12 more stamps than each side of 
the other, how many stamps does a side of each of the latter 
rectangles contain ? 

26. A boat's crew can row at the rate of 9 miles an hour. 
What is the speed of the current in the river if it takes them 
2 hours and 15 minutes to row 9 miles up stream and back ? 

27. Divide $ 1248 among three persons, so that the second 
shall have $ 3 more than the first, and the third shall have as 
many times the share of the second as there are dimes in the 
first person's share. 

28. The population of a city increases from 20,000 to 20,808 
in two years. What is the annual rate of increase per hundred ? 

29. A sum of $ 2000 drawing interest that is compounded 
annually, amounts to $ 2142.45 in two years. Find the rate of 
interest. 



CHAPTER II 

MORS ADVANCED THEORY AND OPERATIONS 

FUNDAMENTAL LAWS OF ALGEBRA 

76. The operations of algebra obey certain fundamental 
laws which we have not formulated thus far. Nevertheless 
we have so accustomed ourselves to follow them, that we find 
it difficult to see how a new algebra might be made, in which 
a different set of laws would prevail. We shall now explain 
the laws which underlie our algebra. 

If several positive and negative numbers are added or sub- 
tracted, it matters not in what order the operations are per- 
formed ; the numbers may be commuted at pleasure. 

For example, 4 + 7 — 6 = 6, 

or 7 + 4-6 = 6, 

or -6 + 4 + 7 = 6. 

This is called the commutative law for addition.* Using let- 
ters, the law may be stated thus, 

a + b =:b + a. 

* In our algebra, addition and subtraction may be represented geometrically 
by the addition and subtraction of distances along a straight line. Let a and 
b represent distances measured off toward 

the right, then a + 6 and 6 + a both repre- — ^ ,.- ^ ^ ^ 

sent the same distance OC in Fig. 3; the Q I i C 

commutative law is obeyed. \y a 

Suppose, now, that a and b are assigned Fia. 3. 

meanings entirely different from those given 

above; suppose a means a rotation about the line OA as an axis, through 
90 degrees, and b means a rotation about OB as an axis, through 90 degrees. 

56 



MORE ADVANCED THEORY AND OPERATIONS 57 



Again, 5 + — 2 = 12, 

or (5 + 9)- 2 = 12, 

or 6-f(9-2)=12. 

That is, the final result is the same, whether the 9 be asso- 
ciated with the 5 or with the — 2. 

This is called the associative law for addition. Using letters, 
it may be expressed thus, 

a + b + c =(a -\- b) + c 
= a+(6 4-c). 

If an expression contains two or more factors, it matters 
not in what order the multiplications are performed. 

For example, 2.7 = 7-2. 

This is called the commutative law for miUtiplication,* Using 
letters, the law may be expressed thus : 

a ' b = b • a. 

Let a rectangle be the figure rotated (Fig. 4). Then a-\-b (i.e. the rotation 
about OAy followed by a rotation about OB) brings the rectangle in a position 
where " Alg/' is horizontal, as in Fig. 5. On the other hand, b-\-a (i.e. a 
rotation about OB, foUowed by a rotation about OA) brings the rectangle in 






Fig. 6. 



a position in which ** Alg." is vertical (Fig. 6) . Since the final positions of the 
rectangle are different, it follows that in this case, a+b^b+a. That is, the 
commutative law is not obeyed. (The symbol ^ means " is not equal to.") 

Whether the commutative law for addition is obeyed or not depends there- 
fore upon the definitions given to a and 6, and to the processes of addition 
and subtraction. 

* There is an advanced algebra, called quaternions, in which (;' =^ji; that 
is, the commutative law for multiplication does not generally hold true. 
Quaternions are used in the study of mathematical physics. 



68 ELEMENTARY ALGEBRA 

Again, it matters not how the factors are associated or grouped, for 

6 > 6 . 3 = 90. 

6. (6. 8)= 90, 

(6 . 6) . 8 = 90. 

This is called the associative law for mtUtiplication. Using 
letters, the law may be expressed thus : 

a 'b • c=r:a» {b • c)={a • 6) • c. 

Again, a factor placed before or after a parenthesis contain- 
ing two or more terms may be distributed among the various 
terms without any change in the final result. 

That is, 6(9 - 4 + 6) = 6 X 9 - 6 X 4 + 5 X 6 = 66. 

This is called the distributive law for multiplication. Using 
letters, the law may be expressed thus : 

a(6 + c)==a6 + ac. 

raSTORICAL NOTE 

77. It is a carious fact that, in the development of algebra, the funda- 
mental laws were the last things to be explained. The beginnings of 
algebra can be traced back to ab6ut 2000 years before Christ, but not 
until the nineteenth century were the fundamental laws of algebra formu- 
lated. In the earlier treatment the laws were tacitly assumed to be 
true. For instance, it was assumed that ab = hat a{b -\- c) = ab + ac, 
(ab)c=:a(bc)f without special attention being called to this matter nor 
special names being given to the relations assumed. The need of an 
explicit statement of the fundamental laws came to be recognized when 
it was perceived that, besides the algebra which we are studying, in which, 
for example, ab is always equal to ba, there could be established other 
algebras in which ab is not always equal to ba. Among those who helped 
to perfect the science of algebra along these lines were the Englishmen, 
George Peacock, D. F. Gregory, Augustus £>e Morgan, and Sir William 
Rowan Hamilton ; the Frenchmen, F. J. Servois, and A. L. Cauchy ; the 
Germans, Martin Ohm, Hermann Grassmann, and Hermann Hankel; 
and the American, Benjamin Peirce. The names "commutative law," 
" distributive law," were first used by Servois in 1814. Among the first 
to use the name ** associative law" was Sir William Rowan Hamilton. 




AnODSTUS DB MOBOUI (1806-lti71) 

Was professor of matheinaticB In London and wrote papers on Che tounda- 
tioDS of algebra and on loglo. To as English Journal, the Aihenieum, he con- 
tributed amaxing articles on circle squarers. These articles were aCterwarde 
ooUected iu a book, entitled A Budget of Paradozet. 



« 41 W 

V 



• » 
• w ». ^ C » . 



V «. 






k >. V. i. 






MORE ADVANCED THEORY AND OPERATIONS 59 

REMAINDER THEOREM AND FACTOR THEOREM 
78. Divide aj* — 5a^ + 7aj — 2bya? — a. 

X — a 






x2 +(a - 6)x + (a-* - 6 a + 7) 



(a - 6)a;2 + 7 x - 2 
(a~6)a;'^-(a2-6a)ag 

(a2 _ 5 a + 7)a; - 2 

(q^ - 6a + 7)a; - q8 + Sgg - 7 q 
a8-6a2 + 7a-2 

Observe that the remainder, a' — 5 a^ + 7 a — 2, is the same 
as the dividend, if in the dividend a is substituted for x. This 
illustrates the 

Remainder Theorem : If a rational integral expression in x is 
divided by x— a, the remainder is the same as the original ex- 
pression with a substituted for x. 

Now aj' — 5aj* + 7a? — 2 would be exactly divisible by a? — a, 
if a» - 5 a2 + 7 a - 2 = 0. 

Hence such a polynomial will be exactly divisible by a 
binomial of the form a: — a, if , when a is substituted for x, 
the polynomial vanishes, i,e. equals zero. This illustrates the 

Factor Theorem : If a rational integral expression in x becomes 
zero when a number a is substituted for x, then x — a is a factor 
of the expression. 

Thus, the expression a:P -\-2sfi —05—2, fora;=l, becomes 1 + 2—1—2=0, 
hence x — 1 is a factor of the expression. 

Again, for a; = — 2, this expression becomes 64 — 64 + 2 — 2 = 0, hence 
X — (— 2)ora; + 2isa factor of it. 

A third factor can be found by dividing x' + 2a^ — x — 2by the prod- 
uct of a; — 1 and a; + 2 ; that is, by x^ + x — 2. The quotient is 
a:* + x' + a;^ + x+l. Hence we have, 

a:6 + 2a*-x-2 = (x- l)(x + 2) (xl^ -\- ofi + x^ + x -^ 1). 

In the example above we took x = 1 and a; = — 2. Notice 
that 1 and — 2 are both integral factors of the absolute term 



60 ELEMENTARY ALGEBRA 

— 2 in the expression aj'4-2a^ — aj — 2. There is a theorem 
bearing on this point which we shall use along with the Factor 
Theorem. It relates to rational integral expressions, 

a?" + oo;"-! + 6aj— 2 H \' kx + k. 

The theorem, which we give without proof, is as follows : 

When a rational integral eacpression has 1 as the coefficient of 
the highest power ofx, and the otJier coefficients a, 6, "•, h, k are 
aU integral numbers (either positive or negative) , then, in search- 
i'f^g for rational va^vss of x that will make the given escpression 
zero, only integral factors of the absolute term k need he tried. 

Factor sc* + 6 x« + 42 a; - 49. 

The coefficient of a:* is 1 ; the other coefficients, 6, 42, — 49 are integers. 
Hence both conditions are satisfied, and we need to try only integral 
factors of 49 ; namely, ± 1, ± 7, ± 49. 

When « = !, 1+6 + 42 — 49 = 0. Hence, by the Factor Theorem, 
a; — 1 is a factor of the expression. 

When* a: =— 1, 1 — 6 — 42 — 49 ^ 0; a; + lis noJ a factor. 

Whenx = 7, 2401 + 2068 + 294 - 49 z^fc 0, a: — 7 is noJ a factor. 

When X = — 7, 2401 — 2068 — 294 — 49 = 0, a; + 7 is a factor. 

In the same way, a; — 49 and a; + 49 are found not to be factors. 

Dividing the expression by (x — l)(x + 7) yields another factor, aj2+7. 

Hence, a^ + 6x» + 42x- 49= (x- l)(x + 7)(x2 + 7). 

BXBBOISES 

79. 1. Divide ^a^ — 1 a^ -{• ba^ — x — l\yj x — 6, then com- 
pare the remainder with the dividend. How does this illus- 
trate the Kemainder Theorem ? 

2. Divide 5a!^ — 4aj8 + 6a? + 2bya? — a and illustrate the 
Remainder Theorem. 

3. Using the Factor Theorem, factor 

aj4 _ 5 aj3 + 9 aj2 - 15 a? + 18. 

4. Factor oj^ — 17 oj^ + 3 a? +- 54. 

* The symbol ^ means * * is not equal to." See § 76, footnote. 



MORE ADVANCED THEORY AND OPERATIONS 61 

6. What is the remainder when 2aj3 — 3a;2 + 5aj— 1 is 
divided byaj — c? By x — b? By a? — 1? 

6. What is the remainder when 5a^-{'2x^-{-Sx + 6 is 
divided by x -f-w ? By a? + 1 ? 

7. Is a? + 2 an exact divisor of aj* — » — 6 ? 

8. If a rational integral expression, with 1 as the coefficient 
of the highest power of a?, and with all other coefficients 
integers, is exactly divisible by a? — a, where a is an integer, 
what relation is a to the last term ? 

9. Find a factor ofa?*^ — aj*-f-aj3 — aj^-f-aj — 1. 

10. Find a factor of aj3 — 9 aj2 + 17 a? — 6. 

11. Find a factor of 2 aj* - a^ - 2 aj2 - 3 a? - 10. 

12. Find a factor of a^ — 13 a; -f 12. 

13. Factor a^ — a**. 

If a is substituted for x, then a* — a^ = 0. Hence, by the Factor 
Theorem, x — a is a factor of ar^ — a*. Division of a^ — a^ by x — o 
yields a second factor. Hence 

05*^ — a* = (x — a) (x* + x'a 4- «^a^ 4- ««* + «*). 

The quotient may be obtained by the following special method of short 
division : 

x^-4-x = x*. x*-^x•a = x*a. x^a -^ x • a = x^a^. 

x^a^ -t- X • a = xa**. xa^ h- x • a = (i*. 

14. Factor a;*^ 4- a^ 

If —a is substituted for x, then (— a)* + a^ = — a* + a^ = 0. Hence 
by the Factor Theorem, x-^(— a) orx + aisa factor of x^ + a*. By 
division, we find a second factor. Hence 

x^ 4- a^ = (x + a) (x* — x^a 4- sc^a^ — xa* 4- a*) • 

The special rule of division given in Ex. 13 applies to this case, except 
that the signs in the quotient are alternately 4- and — . 



62 ELEMENTARY ALGEBRA 

16. Factor a^ + a*. 

If ± a is substituted for x, we obtain ( i a)® + o* = o* + o* t^ 0. 

Hence, neither a; + a nor a; — a is a factor of x* + o^. But afi + <jfi may 
be considered as the sum of two cubes (x^)^ + (a^)' and can therefore be 
factored, as j^ + 0^ is factored. 

16. Explain the following results pertaining to rational and 

real factors : , . , . i. t. u i. j 

a* -f- 52 cannot be factored. 

a* 4- &* cannot be factored. 

a« + 6« = (a2 H- 62) (a* - a^b^ 4- ^O- 

a' 4- 6' cannot be factored. 

Note. The sum of the same two even powers of two numbers may be 
factored, if the exponent is the product of an odd and an even factor. 
One factor is the sum of the numbers with exponents equal to the even 
factor ; the other factor may be found by inspection or by long division. 
Thus, in a^^ + 6^^, the exponent 12 = 8 • 4, where 3 is odd, 4 even. 

One factor of a^^ + h^^ is a* + 6*. The other factor is a* — a*6* + &». 

Hence, a^ + h^^= (a* + 6*)(a8 - a*6* + b^). 

If numerical cubic equations of the form ofi + dx^ + 6x + c = 0, a, 6, c 
being positive or negative integers, have at least one root rcUional, then 
the equation can be solved vnth the aid of the Factor Theorem. 

0^ + 4 x^ — 2x — 5 becomes zero when — 1 is substituted for x. Hence 
X 4- 1 is a factor, the quotient x' 4- 3 x — 6 is another factor. 

Write the cubic equation in the form (x + 1) (x^ + 8 x— 5) = 0.^ 

Make the first factor equal to zero, x 4- 1 = 0, 

and X = — 1. 

Make the second factor equal to zero, os^ + Sx — 6 = 0, 

Q_i_ "v/29 

and solve the quadratic equation, x = — . 

Hence the roots of the cubic are -1, --3 + \/29 -3---\/29 

'22 

17. Solve (a? - l)(a;2 4- 6 a; 4- 8) = 0. 
Solve the following cubic equations : 

18. aj»4- 5aj24-7a;4-3 = 0. 20. aj» -6aj2 - 8a? 4- 7 = 0. 

19. a;'4-5aj2-3aj-22 = 0. 21. a;» 4- 18 02^32 a? 4- 55=0. 



MORE ADVANCED THEORY AND OPERATIONS 63 



\ 



80. Factor: 

1. aj2 — 16y2. 

2. 25a2-6l 

3. 4a^ — 9y2. 

4. 05^ + 3^. 
6. 7? — j^. 

6. aj' - 1. 

7. 14-2/*. 

8. aa;2 — ayK 

9. aj2 _ 100. 

10. iB» — 1000. 

11. m» 4- 8. 

12. m'-8. 



ORAL REVIEW 

13. 27- a'. 

14. 27 + a». 
16. m^ + n*. 

16. m*^ — w^ 

17. a* — 6*. 

18. a* + &*. 

19. 2^ — 1. 

20. 2^-y». 

21. 05*4-0*. 

22. x*—a?. 

23. 10 aj' - 10. 

24. 10 01? + 10. 



25. iB»4-3ar2 4.3aj4. 1. 

26. 2/»-3y24-3y-l. 

27. a* 4- 2 a^* 4- 3^. 

28. 0^4- 82/«. 

29. 7? — 21v?. 

30. a?* — 162^. 

31. 10a^4-20ajy4-102^^ 

32. «« — 1. 

33. a:«4-l. 

34. a?— (2y + a;)'. 
36. aj2 — (2y-2)2. 
36. 2a* — 2a. 



WRITTEN REVIEW 



81. Factor: 
1.. 8 a« 4- hK 

2. ci«-di2, 

3. ci2 + c2i2. 

4. 27 a^- 1252^. 

5. 27a^-2^. 

6. 8m«4-27n». 

7. 27a^-64 2/». 

8. m*^ — n^^ 

9. a* - h\ 

10. c«4-64d«. 

11. aj^— (2a?— y)*. 

12. aj»+(y + «)». 

13. a*+4a^-2a;2-4a;4-l. 



14. aj«-4a:*-7aj2 + 28. 

16. aj*-3aj» + 7a?-13a?4-6. 

16. a?— a^ — 5 a^x 4- 5 a\ 

17. m'- 6m* 4- 12m— 8. 

18. ajio— yo^ 

19. a.** 4- 35*2^* + ^. 

Hint. X* + ojV 4 y* 

= x* + 2icV4y*-a:*y*. 

20. a*4-a' + l. 

21. 4a?*4-3a;y 4-92^. 

22. a* — 7 a262 4- 64. 

23. 49p*H-68p2g*4-369*. 

24. 64aj* + 55a^.y2 4-252^. 



64 ELEMENTARY ALGEBRA 



IRRATIONAL AND IMAGINARY FACTORS 

82. Thus far all factoring has been confined to the discovery 
of rationed factors. But it is sometimes advantageous to re- 
solve expressions into factors which involve irrational or even 
imaginary terin?»>^ Expressions like a' — 2 6*, a* — 3 b*, a? + 6^, 
or a* + 6* can be factored, if the restriction is removed that 
the factors must be real and rational. 



BXBROISBS 

83. 1. Factor a» — 3. 

This may be done by the type form a^— 62=:(a -f- 6)(a — 6), where 
62 = 3 and h = VS. We obtain 

a2-3=(a + V3)(a-V8). 

2. Factor aj2 -f y\ 

This, too, may be factored by the type fonn a* — 6^ =(a + 6)(a — 6). 
Write «2 f y2 in the form x^-{-y^). Since V- y2 := yy/ZTi = fy, 
where % = V— 1, we have, 

x^ + y^^(x + %y){x-'iy). 



Kesolve into irrational or imaginary factors : 

3. m«-5. 7. a* + 62. 11. 9a« + 166*. 

4. a*- 12. 8. 7^-bsK 12. a*-36^ 

5. a2 4.4. 9. 2a2-6». 13. a2— 6. 

6. m2-h9n2. 10. aj^ — 8y*. 14. a* — 46*. 

16. By factoring, find all four roots of the equation 
a^-l = 0. 

16. By factoring, find all four roots of the equation 
a^-16 = 0. 



MORE ADVANCED THEORY AND OPERATIONS 65 



THE BINOMIAL THEOREM 
84. By multiplication, we find that 
(a ± by= a±b. 

(a ± 6)»= a» ± 3a26 + 3a5* ± ¥. 

(a ± 6/= a^ ± 4a»6 4- ea^fe^ ± 4a6» 4- 6*. 

(a ± 6)«= a» ± 5 a*6 + 10 a.»62 ± lo a^b" 4- 5 a5* ± 6». 

(a ± 6)«=:a» ± 6a»6 4- 15 a*^^ ± 20a'&» 4- l^a^b* ± 6ab^ +b^, 

etc., wherein a and 6 represent the first and second terms, 
respectively, of any binomial. 

A careful study of these products will show that they follow 
certain laws, by which they may be written down without 
recourse to laborious multiplications. 

In the products we observe the following laws : 

I. The first term isTyraiaed to the same power as that of the 
binomial. In each succeeding term the easponent of a decreases 
byl. 

II. The factor b does not appear in the first term. The ex- 
ponent of b in the second term is 1 and increases by 1 in each 
succeeding term, 

III. The coefficient of the first term is 1. The coefficient of 
any term after the first is found by multiplying the coefficient of 
the preceding term by the exponent of a in that term^ and divid- 
ing by 1 more than the exponent ofb, ' 

IV. If the binomial is a -i- b, the signs in the product are. 
all plus; if the binomial is a — 6, the signs are alternately 4- 
and — . 

V. The number of terms is one more than the exponent of the 
binomial, 

VI. Each term is of the same degree as the binomial, 

p 



66 ELEMENTARY ALGEBRA 

Using these laws, find without actual multiplication the 
product of (a 4- by. 

The first term is by I, a', 

1x7 
The second term has by III the coefficient or 7, by I the factor j 

cfi^ by II the factor &. Hence the term is +1 <fib. 

The third term haa by III the coefficient l^i^ or 21, by I the factor 

a^, by II the factor h\ Hence the term is + 21 a^h^, ' 

The fourth term has by III the coefficient ^^ ^ ^ or 35, by I the fac- 

o 

tor a^, by II the factor &*. Hence the term is + 36 a*&*, and so on. By 

y the final result has 8 terms. We obtain, 

(a + by- aT + 7 (i«6 + 21 <fib^ + 86a*6» + 86a«6* + 21 a^lfi + 7 a6« + b\ 

All terms are of the seventh degree, as required by VI. 

85. If n is a positive integer, then the products found above 
may all be represented by one general formula, as follows : 

(a + by = a- 4- m-»6 + 5^?^^ a-^fe^ + n{n^l){n^2) ^,.3^, 

This formula is called the Binomial Theorem. By actual 
multiplication it was shown above to be true for all positive 
integral valueg of n, up to 6. We found the expansion of 
(a + by on the tacit assumption that the formula holds for 
n = 7. Really we had no right to take for granted, without 
proof, that the laws given above do hold true for n>6. 
Frequently certain relations hold true up to a certain point, 
but no further. 

For example, the first three integers 1, 2, 3 are all prime numbers. A 
careless reasoner might be tempted to juntp to the conclusion that all ; 

integers are prime numbers, which is, of course, not true. 

That the Binomial Theorem is true for any positive integral 
value of n will, be proved later in § 193. 



MORE ADVANCED THEORY AND OPERATIONS 67 

Expand (2 a; — 3 yy. 

Here a = 2 x, b = Sy. For conyenience in expanding, place 2 x and 
8^ in parentheses, thus (2 x) and (3y). By Law IV of § 84, the terms 
in the expansion are alternately +, — . We obtain, 

(2a;-3y)*=(2a;)*-4(2x)8(8y)+6(2a;)2(3y)2-4(2aj)(3y)»+(3y)*. 

Simplify each term in the expansion : 

(2x - 3y)*= 16 X* - 96x8y + 216 xV - 216xy» + 81 y*. 



EZEBOISES 

86. Expand: 

1. (x-hyY- 4. (a-Sby. 7. (ox — yY^. 

2. (a -6)8. 6. (c2-d2)4^ 8. (3a2 + 2 6»)». 

3. (2x + yf. . 6. (3a- 62)6^ 9. (m — 2ny. 

10. (2r3-3s2y. 

11. Write the first three terms of (a—b)^, 

12. Write the first five terms of (x + y)^. 

13. Compare the coefficients of the first and last terms in 
the given expansions in § 84 ; of the terms next to the first 
and last. 

14. What is the third term of {mx -f- ny)^ ? 
16. Write the 9th term of (2 a^ - 3 by. 

SYSTEMS OF LINEAR EQUATIONS. PETERMINANTS 

87. Systems of n independent linear equations ir. n xm- 
knowns may be solved by eliminating tho same unknown from 
all the equations. The resulting system contains in general 
one less equation and one less unknown. 

From this resulting system eliminate a second unknown, and 
continue this process until one equation containing only one 
unknown is obtained. 



68 ELEMENTARY ALGEBRA 

BXBBOISES 

88. 1. Solve the system of equations 

« H- y + « = 9, (1) 

x-h2y^z = S, (2) 

2aj-33^-f 42 = 7. (3) 

SoluHon. Subtract (2) from (1), - y + 2 « = 1. (4) 

Subtract (8) from 2 . (1), 6 y - 2 = 11. (6) 

Add (4) and (6), * 4 y = 12, 

y = 3. 
Substitute 8 for y, in (4), «; = 2. 

Substitute 2 for z, and 3 for y, in (1), x = 4. 

Check by substituting the answers in all three equations (1), (2), and (3). 

2. aj-y-h« = 20, S. 5x — 6y-{'Tz = 105, 

2aj-3^--« = -30, 3aj-6y + 82; = 103, 

3a. — 43^4.22; = 10. 3y — 4« = -44. 

4. « -I- y -f 2; 4- to = 10, 
« — yH-2« — 3«; = — 3, 
3aj — 2y + 2— w = 5, 
3aj-3y-f-22 — 4m?=— 3. 

89. In previous parts, three methods of elimination have 
been explained : (1) by addition or subtraction, (2) by sub- 
stitution, (3) by comparison. 

A fourth method of solution is by means of a formula, called 
a determinant. In deriving this formula we use the old 
methods of solution. 

Solve : ax + by = c (1) 

dx-{-ey=f. (2) 

Multiply (1) by d, (2) by a, adx + bdy = cd (3) 

adx + aey = c^f .. (4) 

Subtract (3) from (4), (ae — bd)y = c^f— cd 

ae — bd 

Similarly, a;=£lzi_St. 

ae — bd 



MORE ADVANCED THEORY AND OPERATIONS 69 



The numerator af—cd may be written 



a c 
d f 



The term af is the product of a and / which lie on the diagonal from 
the upper left comer to the lower right corner. 

The term cd is the product of c and d which lie on the diagonal from 
the upper right comer to the lower left comer. 

The second product is subtracted from the first. 

3 4 



Similarly, 



5 6 



= 3x6-4x6. 



The numerator ce — bf may be written 

The denominator ae — bd may be written 

Expressions written in this form are called determinants. 



c 


b 




f e 




kTI 


a b 


jXI. 


d 


e 



Hence, 



a? = 



c b 

f e 



a b 
d e 



(5) 



y = 



a 


c 


d 


f 


a 


b 


d 


e 



(6) 



The determinants in (5) and (6) can be used conveniently 
as formulas for solving two linear equations in two unknowns. 

The determinants can be written down at once, according to 
the following rules : 

I. In the two denominators the determinant is the same. It is 
fortned by writing the coefficients of x and y as they stand in the 
equations (1) and (2). 

II. The determinant in the numerator of (p) is formed from 
the denominator by replacing , the coefficients of x, by the abso- 
lute terms . 

/ 



70 



ELEMENTARY ALGEBRA 



III. The determinant in the numerator of (6) ia formed by 

b c 

replacing , the coefficients ofy, by the absolute terms, 



Example. Solve 



3a?-f 4y = 34, 
5 a? — 3 y = — 11. 



The denominator is 



3 4 
5 -8 

The numerator for the value of x is 



34 4 
-11 -3 



3 34 
6-11 



The numerator for the value of ^ is 

Hence, 

34 4 

■ ^ 84.(-8)-4.(-ll) _ -102-f-44 ^ 
3(«3)-4.6 -9-20 



flC = 



-11 -3 



8 4 
6 -8 



-68 

-29 



= 2, 



y = 



8 


34 


6 


-11 


3 


4 


6 


-8 



3(^ 11) - 84 . 6 
-29 



- 33 - 170 - 203 



-29 



- 29 



= 7. 



EXERCISES 

90. Find the values of the following determinants : 
1. 



2. 



3. 



9 6 

8 7 


.9 .2 
.5 .1 





4. b 


. 3 • 
4 


-2 
10 




6. 


9 
a b 




6. 


la 86 


46 


-3 


a 



7. 


a» 


62 
6-1 




8. a 


be 
ae 


-6c 


9. 


a-2 


6-' 





Solve, by determinants : 

10. 2aj-h32/ =4, 
5 oj -♦- 6 y = 7. 



11. aia; + 6iy=Ci, 
a^ 4- 62^ = C2. 



MORE ADVANCED THEORY AND OPERATIONS 71 

12. 2iB — 5y= — 13, 14. x + my = ^m*, 
3 oj — 4 y = — 9. oj 4- ny = — w*. 

13. aa?-f 6y = a — 6, 16. aa; + 62/ = 2a6, 
oa? — 6y = a -f 6. 6a? -|- ay = a* + 6*. 

Solve, by any method : 

16. 3a;-f 4y = 7, 20. 3a?-f-4y4-5 = 0, 
4aj4-y=s3. 6a? + 7y4-8 = 0. 

17. aj-|-2y = 8, 21. aaj + 6y=l, 
3 a; + y = 9. ca? -|- dy = 1. 

18. 8 a: — y = 34, 22. cx + dy=sa, 
x + SyssoS. mx -i-ny^b. 

19. 3a? = 4y, 23. 2(a?-2) = 3(y + 3), 
5aj=6y-4. 3(a?- 2)-|-2 = 6(y +3). 

24. (r-l)(a + 2)=(r-3)(«-l)4-8, 
4(2r-l)-16(«-2)=20. 

25. (6 — a)aj4-(a + % = a'4-ftS 
6a?-2ay = 6«-2a*. 

26. m — w 33 6 — a, 
am — ac = 6/1 — 6c. 

27. (m 4- n)x -f- (m -h />).y = m -|- n, 
(?/i + p)a: +(m + n)y = m 4-i>. 

91. The method of solution by determinants may be ex- 
tended to systems containing three or more equations. 

For example : aa? + 6y + caj = d, (1) 

ex -hfy + gz = hy (2) 

kx + ly + rnz = n, (3) 



72 



ELEMENTARY ALGEBRA 



Using the addition and subtraction method, we obtain after some 
prolonged effort, 

__ €(fm 4- hlc + ngb — cfn — gld — mhb 
afm + 6/c + kgh —cfk — fjria — meb 



(4) 



__ g/im -^^ enc -\- kgd — cftA; — gna — med 
cj/w* + elc 4- A;(76 — c/A; ^gla — W€6 

_ fl^n + el d + ifc^fe — djk — /iZg ~ ne6 
afm 4- eZc 4- A;(76 — cfk — gla — meb 



(5) 



(6) 




These numerators and 
denominators may be 
written down more con- 
veniently in the form of 



1^ determinants. The six 



terms of the denomi- 
nator may be written 



a 


b c 


e 


f 9 


k 


I m 



provided that the f ollow- 
ing rule of expansion is 
observed (Fig. 7) : 

(1) Take the positive of the products of the terms marked 
by the 4- arrows. 

The three products are 4- <0», 4- e^c, 4- kgb, 

(2) Take the negative of the products of the terms marked 
by the — arrows. 

The three products, with their signs changed, are — c/fc, — gla^ — meb* 
It follows then that 



a h c 

e f 9 
k I m 



= afm -\- elc 4- kgb — cfk — gla — meb. 



MORE ADVANCED THEORY AND OPERATIONS 73 



This process of expansion must be fixed thoroughly in the 
mind. The numerators of (4), (5), and (6) can be expressed by 
the determinants in the same way. 

Three independent linear equations in three unknown 
quantities can be solved by determinants according to the 
following rules : 

I. In the three denominators of the vahies of x, y, and z, the 
determinant is the same. It is formed by writing the coefficients 
ofx, y, z, as they stand in equations (1), (2), (3). 

II. The determinant representing the numerator of (4) is 
formed from the denominator by replacing the column containing 

a, e, k, the coefficients ofx, by the absolute terms d, h, n. 

III. The determinant representing the numerator of (5) is 
formed from the denominator by replacing the column containing 

b, f I, the coefficients of y, by the absolute terms d, h, n. 

« 

IV. TVie determinant representing the numerator of (6) is 
formed from the denominator by replacing the column containing 

c, g, m, the coefficients ofz, by the absolute terms d, h, n. 



Example, Solve : 



The denominator is 



1 
2 
3 



2x-y-\-z = b, 
3x-4y4-22! = 3. 

1 1 



-1 
-4 



1 
2 



= lx(-l)x2+2(-4)xH-3xlxl-lx(-l)x3-lx(-4)xl 

-2x2xl=-4 



The numerator for x is 



6 1 
5-1 
3 -4 



1 
1 
2 



=6x(-l)x2+6x(-4)xl+3xlxl-l>^(-l)x3-lx(-4)x6 

-2x6xl=-12. 



16 1 

2 5 1 

3 3 2 



The numerator for y is 

=lx6x2 + 2x3xl4-3xlx6-lx 5x3-1x3x1-2x2x6= -8. 



74 



ELEMENTARY ALGEBRA 



The numerator otz\& 



1 1 6 
2-16 
8-4 8 

= lx(-l)x3+2x(-4)x6+3x6xl-6x(-l)x8-6x(-4)xl 

-8x2xl = -4. 
Hence x =(- 12) + (- 4)= 8. 

y=(-8)+(-4)=2. 

*=(-4) + (-4)=l. 



BXBBOI8B8 

92. Solve: 

1- a + y + » = 9, 
« - y + 2 = 3, 

05 4- y — 2 = 1. 

2. 2aj-3y + 2;= — 1, 
3a; + 2y — 2 = 4, 
^x — y-\- 5 2 = 17. 

3. a + 26-c = l, 
8a4-464-3c = 29, 
3a-56-4c = -ll. 

4. wi + n = 3, 
j!> 4- m = 12, 
n 4"i> = 6. 

6. r + 2 « = 10, 
3r-|-4« = 12, 

6 « + 6 « = 22. 



6. aj + y + « = 2, 
3aj — 4y-f-62=41, 
9aj-13y=-57. 

7. a.-y-» = -^, 
2aj — 3y4-42 = l, 
4aj4-6y + 2 = 4^. 

B. aj + y4-2 = a, 
aj - y + 25 = 6, 
ic + y — « = c. 

9. ax + by = m, 
c^ -I- cf2 = n, 
eaj 4-/2 =1>. 

10. r4-» + «4-w=sO, 

2r + 3a4-4iJ4-5w= — 6, 
3r-4a+5^-6u = 2, 
47.4.55 — 2« — tt = 2. 



PROBLEMS 

93. 1. A man has 20 coins, all dollars and quarters, amount- 
ing to % 13.25. How many coins of each denomination has he ? 

2. Weights of 13 and 17 lb. are fastened to the ends of a 
9-foot pole. Where must the pole be supported in order that 
the weights will balance ? 



MORE ADVANCED THEORY AND OPERATIONS 75 

3. Two men carry a weight of 300 lb. by means of a pole 
from which the weight is suspended. One man holds the pole 
at a distance of 3.7 feet from the weight, the other at a dis- 
tance of 4.3 feet. How much of the weight does each man 
carry ? 

4. Two unknown weights balance when suspended 11 and 
12 inches respectively from the fulcrum. If the weights are 
interchanged, 23 ounces must be added to the lesser weight 
to restore the balance. Find the two weights. 

6. A farm laborer received $ 2.75 and board for every day 
he worked, and paid $.75 for board every day he was idle. 
After GO^days he received $154.50. How many days was 
he idle? 

6. A •man has $12,000 at interest in two investments. 
From one he receives 5 % interest, from the other 6 % . His 
income from these investments is $ 645 per year. How is the 
money divided ? 

7. Find the capital and the receipts of a railway company, 
if its receipts are distributed as follows : 

(1) Payment of a guaranteed dividend of 4 % on ^ of the 
capital stock. 

(2) Payment of a dividend of 3 % on the remaining capital 
stock, this dividend amounting to $ 162,000. 

(3) To working expenses and a reserve fund, which absorb 
55 % of the entire receipts. 

8. The area of a trapezoid is the product of half the sum 
of the two bases and the altitude. Find the two bases, if the 
altitude is 12 in., the difference between the bases is 3 in., and 
the area of the trapezoid is 150 sq. in. 

9. How many gallons of each of two liquids, one 25% 
alcohol, and the other 75 % alcohol, must be taken for a 17- 
gallon mixture that is 45 % alcohol ? 



76 ELEMENTARY ALGEBRA 

10. How many ounces of silver, 75 % pure and 85 % pure, 
must be mixed to give 20 ounces of silver, 83 % pure ? 

11. Tin and lead occur in the following ratios : In plumber's 
solder 1:2; in soft solder 2 : 1 ; in common pewter 4 : 1. How 
many pounds of common pewter and plumber's solder must be 
taken to make 10 lb. of soft solder ? 

DIVISION BY ZERO IMPOSSIBLE 

94. To avoid perplexing errors in algebraic transformations 
it should be made very clear that it is not permissible to divide 
by zero. 

Let a and h be two fixed numbers. Both a and b^ie finite 
numbers, because in elementary algebra there are no fixed num- 
bers that are infinite. , 

a CL * * 

Letting - = a;, we define the quotient - as the value of x in 

b b 

the equation bx = a, 

I. As long as a and b are fixed numbers different from zero, 

it is always possible to find a value for x. 

II. If a is not zero, but b is zero, we are asked to find a 
number x which, multiplied by zero, gives a as a product. 

But every fixed number, when multiplied by zero, gives the 
product zero. 

Hence no number x exists which satisfies the equation •x=a. 

If, therefore, we perform operations which amount to divi- 
sion by zero, we must not be surprised if we obtain absurd 
results. 

Point out the error in the following : 

Let x = c. 

Multiply both sides oix = chy x^ x^ = ex. 

Subtract c^ from both sides, x^ — c^ = cx— c^. 
Factor, (x + c) (ic — c) = c(x — c). . 

Divide by (x — c), x-\- c = c. 

Since x = c, 2 c= c. 

Divide both sides by c, 2 = 1. 



MORE ADVANCED THEORY AND OPERATIONS 77 

EQUATIONS WITH FRACTIONS 

95. When an equation involves fractions we usually begin 
its solution by clearing it of fractions. Tbis operation rests 
on the principle that when equals are multiplied by equals, the 
results are equal. 

1. Solve . -^ n = -^?^. 

m + n m — n 

Observe that this equation is meaningless and absurd when m ^n , 
for in that case it involves a division by zero. For the same reason we 
cannot have m:=^ n^ for then m + n = 0. Let us assume that neither 
denominator is zero. 

Then the 1. c. d. = (m + n) {m — n). Multiply both sides by 
(m + n)(riii — n). 

We obtain, (m — n)x — n(m^ — n^) z= m (m + n). 

Transpose, (m — n)x = n(m^ ^ n^)-\- m(m -\'n). 

Divide both sides by (m - n) , x = n(m2~ n2)+ m (m -f n) ^ 

m — n 

2. Solve _^ + .^ZLi=:^. (1) 

Observe that if jc = 1, or x^ = 1, this equation represents an absurdity, 
since it involves division by zero. 

On the supposition that x^ =^ 1, the 1. c. d. = 4{x^ — 1). 

Multiply both sides by 4(x2-l), 4x(x + l) + 4(x-3) = 6(x2-l). (2) 

Simplify, 4x« + 4x + 4x— 12=5x2-5. 

Transpose, x* — 8 x + 7 = 0. 

Factor, (x — 7) (x - 1) = 0. 

Hence, x = 7, x = 1. 

Substitute in the original equation (1) ; we find x = 7 satisfies it, but 
X = 1 does not satisfy it, since it gives rise to division by zero. We 
must reject x = 1 ; x = 7 is the only correct solution of (1). 

It is of interest to notice that, while x=l does not satisfy (1), 
it does satisfy (2). In other words, equation (2) is satisfied by 
two values of a, while equation (1) is satisfied by only one 
value of a?. 

It is clear that the new root was introduced in the act of 
multiplying both sides of (1) by 4:(x^ — 1). Such new roota 



78 ELEMENTARY ALGEBRA 

which do not satisfy the original equation, but do satisfy an 
equation that is derived from it, are called extraiieoua roots. 

When we multiply both sides of an equation by an expres- 
sion involving x, we obtain a new equation some of whose roots 
may not satisfy the original equation and are not correct solu- 
tions of it ; they are extraneous roots. This may happen even 
though all operations are performed free of error. 

Hence, to make sure that our values of x are correct, we must 
mihstitute the values found for x in the original equaJtioUy to see 
whether it is satisfied or not. 

BXBBOI8B8 

96. Solve and check : 

2 1 .43 



1. 



oj— 1 aj-3 7« + 3 6«4-2 

2. 5+4 = ^ + 6. 6. ^ = ^. 

y- y x-\-2 x + b 

8 3 ^ 1 ^ 3m-l _ 3m-5 

2r + l r-1 * 3m + 4 3m-|-2 

1 1 



7. 



{x - l){x - 4) (oj - 2){x - 6) 



8. 4-A__=7-.?i^. 
y4-l y-3 



10. (m* — 4)y = m ~ 2. 



m + n 



x 1 16. ^4.5^=3. 



11. - = - 

a 



a c 



-„ ex 1 16. — - — = — 

^2- J="^' Ax-^S 6aj-l 

18. ?-l=a. 17. -^+ ^ '"^ 



0? — 2 a: — 3 a — 4 



MORE ADVANCED THEORY AND OPERATIONS 79 
5 4 1 



18. 



19. 



05— 5 a; — 4 05 — 3 
1111 



20. 



a?— 1 x—3 x — 5 x — 7 

_3 1 ^ 5r-H5 

r4-3 3-r r*-9 



Solve and tell, by substitution in the original equation, 
which of the roots, if any, are extraneous. ' 

21 ^_+^_ = S. 
x-1 a!-2 2 



05 + 6 « — 7 (a! + 6)(a! — 7) 

23. .i'^r^ =a;+3. 
6a!« — 9a;4-6 



a!«-9 



26. 3ir^^^y_ 

y 2y-l 



26. 2« 8 -3 



x + 1 2a!«-a!-3 2a!-3 



27. 



3 _ -a!» + 34a!-31 



2a;-3 3a;4-4 6aj2-.aj-12 



oo 1 7. 2 6 
28. r — 6 = 



y — 5 ^2 _ 52 



80 ELEMENTARY ALGEBRA 

m 

MISCELLANEOUS PRACTICAL PROBLEMS 

97. Applications of algebra, such as occar in the problems which 
follow, are frequently made by builders and manufacturers. Certain 
formulas which have been established by engineers and mathematicians 
are used in computing the desired results. We shall assume the formulas 
as true without giving the mode of deriving them. Care must be taken 
to use in every problem the proper units of measure. 

1. Find correctly, to two decimal places, the side of a 
square whose area is 35.06 sq. in. 

2. Compute the altitude of an equilateral triangle whose 
sides are a inches. 

3. If one side of a rectangle is a inches long, and one of 
the adjacent sides is three times as long, find the length of the 
diagonal. 

4. A ladder of the length 9 a is placed against a wall, with 
its foot at a distance 4 a from the wall. How high above the 
ground is the top of the ladder ? 

6. In placing blackboards in schoolrooms, architects deter- 
mine the height of the chalk rail above the floor by the 

formula A = 26-hf (^-4), ' 

where h = height of the chalk rail in inches, g = the number 
of the grade to which the pupils in the room belong. Find 
the height of the chalk rail for pupils in the 5th grade ; also 
for pupils in the 8th grade and in the 3d grade. 

6. The Baldwin Locomotive Works in Philadelphia have 
derived a formula, i2 = 3-|-^, for finding the approximate 

relation between the resistance, B, offered by a railway train, 
and the velocity, the train traveling on a straight and level 
track. In this formula B is the resistance in pounds per ton 
weight of the train, v is the velocity of the train in miles per 
hour. What is the resistance per ton of a train running at the 



MORE ADVANCED THEORY AND OPERATIONS 81 

rate of 40 mi. an hour ? What is the resistance of this train, 
if it weighs 100,000 T. ? 

7. The load that may be safely applied to an iron chain is 
given by the formula, I = 7.11 d^, where I is the load in tons 
(2000 lb.), and d is the diameter of the iron chain in inches. 
Find the largest safe load that may be applied to a chain for 
which (f is f in. Find d when Z = 13 T. 

8. The distance which one can see from an elevation at the 
sea has been found to be cZ = 1.23 V^, where d is the number 
of miles one can see, and h is the elevation of the observer in 
feet. How far is the horizon from a man standing at the sea- 
shore, whose eye is 6 ft. from the ground ? 

• 9. In Ex. 8, how far can one see from a cliff 27 ft. high ? 
67 ft. high? How high must a cliff be to afford a view of 
20 mi. ? 

10. From a cliff I can just see the lights of a seaport 12 mi. 
across the sea. If these lights are 20 ft. above the sea, what 
is the height of my eye ? 

11. According to a rule sometimes used by architects, the 
"rise" (r inches) in the steps of a stair- 
case (Fig. 8) is connected with the " tread " [| T 

(t inches), by the formula r = ^ (24 — *), I 

where r and t are also subject to the limita- ' l \ u '* 

tions, 7 > r > 5|, 12 > « > 10. Explain the fw 8 

formula in words. What is the " rise " when 
the tread is 10^' ? Hf"? 

12. Engineers have determined that the velocity (v feet per 
minute) of a stream at the bottom of a river is connected with 
the velocity ( V feet per minute) at the surface, by the formula 
V = F+ 1 — 2VF. Find the velocity at the bottom when the 
velocity at the surface is 6 ft. per minute, 8 ft. per minute, 
95 ft. per minute. 

G 



82 ELEMENTARY ALGEBRA 

13. A rectangular beam (Fig. 9) is supported at the ends 
and loaded in the middle. The weight it will just bear, with- 

out breaking, is w; = , where w 




is the weight in pounds, b the 
breadth of the beam in inches, d 
-p Q is the depth of the beam in inches, 

I the length of the beam in feet, 
and A: a coefficient equal to 3470 for wrought iron, 2540 for cast 
iron, 450 for red pine. Find the greatest weight which can be 
hung in the middle of a wrought-iron bar 5 ft. long, 1 in, wide, 
and 2 in. deep. 

14. In £x. 13, find w when the beam is cast iron, also when 
it is red pine. 

16. In Ex. 13, how deep must a beam of red pine be, to sup- 
port 2 tons, if it is 6 ft. long and 3 in. wide ? 

16. At a curve in a railway track the outer rail is raised 
above the inner rail by an amount indicated by the formula 

h = , where h is the number of inches that the outer rail 

1.25 r' 

is elevated above the inner, w is the width between the rails in 

feet, V is the greatest speed of a train in miles per hour, r is 

the radius of the curve in feet. If r = 2000 ft., w = 56^ in., 

V = 60 mi. per hour, what is ^ ? 

17. The safe load which can be applied to a rope is i = ^iC*, 
its breaking load is 6 = k^c^ where I and b are measured in 
tons, and c (the circumference of the rope) in inches. For 
common hemp ropes ki = .032, Ajj = .18 ; for best hemp rope, 
ki = .100, k2 = .60 ; for steel-wire rope, ki = .450, Aij = 2.8. Cal- 
culate the safe load (a) for a common hemp rope 3.5 in. in 
circumference, (6) for a steel-wire rope 2.5 in. in circumference, 
(c) for a best hemp rope 4 in. in circumference. Calculate the 
breaking load for each of these. 



MORE ADVANCED THEORY AND OPERATIONS 83 

18. Draw a graph showing the safe load of steel-wire ropes 
for different circumferences, up to 4 inches. 

19. The breaking load of the best hemp rope is given by 
the formula, 6 = .60 c*, where b is the breaking load in tons, 
c is the circumference of the rope in inches. Draw a graph 
showing the breaking load from c = .3 in. to c = 4 in. From 
the graph determine the breaking load for three values of c 
not used in constructing the graph. 

20. In the flow of water through pipes, a certain head 
is necessary merely to overcome the friction of the water 
against the pipes. This head is given by the expression, 

1/ 0043 \ s* 

hss-l .0036 4- )— - , where a is the speed of the water in 

A -y/s /32 

feet per second, I is the length of the pipe in feet, d is the 

diameter of the pipe in inches, h is the head of the water in 

feet. Find the head required to overcome the friction in a 

3-inch pipe, 3000 feet long, in which water is running at the rate 

of 3 feet per second. 

21. A house is to rest on piles as a foundation. In the erection 
of it one must know how much each pile can support. This is 
computed from the performance of the " pile driver," a machine 
that lifts a heavy " ram " and drops it on the pile. The max- 
imum load Z, in tons, that a pile will bear is obtained from the 

formula, Z= ^ ^ — -, where w is the weight of the ram in cwt., 

h is the height in feet from which the ram falls, d is the dis- 
tance in inches that the pile was driven in by the last blow, 
p is the weight of the pile in cwt. If lo = 6, A = 4, p = 15, 
d = 1^, what is the maximum load the pile will safely bear ? 

22. A steam engine is operated by steam which enters a 
cylinder C (Fig. 10) and presses against one side of the piston 
P and moves it ; the motion is transmitted by the piston rod B 
to the crank shaft i^i, which turns the wheel around. The rate 



84 



ELEMENTARY ALGEBRA 



at which the engine does mechanical work depends upon the 
size of the cylinder, the pressure of the steam, and the number 
of revolutions per minute. The rate of doing work is measured 
in horse powers, H. If d is the diameter of the cylinder in 



/ 






V 




Fig. 10. 



inches, I the length of the piston stroke in feet, n the number 
of revolutions of the wheel per minute, and p the mean steam 

pressure per square inch, then ^^ ^JL^. ' If w= 22, d = 55", 

I = 61", p = 20 lb., find H. If n = 25, p = 15 lb., I = ^ ft., 
what must d be, that the horse power may be 30 ? 

23. The horse power required to move a ship is given approx- 
imately by the expression H = .0088 s^ (.05 A -h .005 S), where 
s is the speed in knots, A is the immersed cross section of the 
ship in square feet, S is the wetted surface of the ship in 
square feet. A and S remaining the same, state the nature 
of the variation of ^ as a function of s. What horse power 
is needed to move a steamer at 12 knots, when A = 230 and 
fi' = 3100? 



CHAPTER III 

PROPORTION, VARIATION, FUNCTION 
PROPORTION 

98. A proportion expresses the equality of two fractions. 
The fractions are sometimes called ratios. 

Thus, — = £ is a proportion. It is written also a\h — cid. 
b d 

In a/b = c/dy the terms a and d are called the extremes, the terms b 
and c the means. In the ratio a/b, a is sometimes called the antecedent, 
b the conseqiient, but ordinarily there is no need of these terms. 

A mean proportional between two numbers a and b is the number m 

which satisfies the equation — = ^ . 

m b 

Since a proportion is really an equation, it is treated like an 
equation. 

Thus, in the proportion, ? = ^. 

b d 

1. Multiply both sides by bd, ad = be. 

This relation is expressed by the theorem : 

In a proportion, the product of the means is equal to the product of the 
extremes, 

2. When only three terms of a proportion are known numbers, the 
fourth tenn may be found. 

Thus, if a, b, and c are known, we obtain from ^ = -, 

b d 

ad= be 

and d = -. 

a 

3. Add 1 to both sides of the equation, - + 1 = - -h 1, 

b d 



Whence, the new proportion, 

85 



a-\-b c + d 
b d 



86 ELEMENTARY ALGEBRA 

BXBBOI8BS 

99. 1. Prove that if the product of two numbers, e/, is equal 
to the product of two other numbers, gh, one pair may be taken 
as the means, the other pair as the extremes, of a proportion. 

Prove that, if - = -, then 

b d 



2. 


a — 6 c — d 
b d 


7. 


c d 


3. 


a — b c — d 
a + b c4-d 


8. 


6* d* 


A 


a c 


9. 


a» c» 


'M» 


a -\-b c4-d 


6» (f 


6. 


a c 


10. 


a* c" 


a — b c — d 


6" d" 


6. 


a 4- b _c + d 
a^b c—d 


11. 





Simplify the following ratios by writing each in the form of 
a fraction and reducing the fraction to its lowest terms : 

12. 126:675. 16. (a - 6)* : a* - 6». 

13. 69 a»c» : 46 a*6c. 16. (8m« + n») :16(2m4-w). 

14. a*-6*:(a + 6)c. 17. 4(p^ - 8 g*) : (p - 2 g). 

18. (a* -h a*6* + 6*) : (a« - a5 + &«). 

19. (aj» — ay — 2i^=):3(a; — 2y). 



22. (aJ«-2ary4-y*}:(«' + »y-l-3/*). 

23. (a?* + 1) : (ic* - a? V2 -f 1). 



1 



PROPORTION, VARIATION, FUNCTION 87 

24. Find the mean proportional between 31.2 and 0.96. 
26. Find the mean proportional between 3 a^c^ and 7 c^d^. 

26. Determine which is the greater of the ratios 19 : 25, or 
66 : 74. 

27. Two numbers are in the ratio of 3 : 4, and their difference 
is to their product as 1 is to 18. Find the numbers. 

28. Find the last term of a proportion whose first three 
terms are 27, 6, and 18. 

29. Divide $ 644 among A, B, and C, so that A's share is to 
B's in the ratio of 2 to 3, and B's share is to C's in the ratio of 
6 to 7. 

30. The income of an estate should be divided between A 
and B in the ratio of 6 ; 4 ; $ 1100, however, is paid to A, and 
$ 700 to B. Which has been unjierpaid, and by how much ? 

FUNCTIONS 
100. The formula 8 = 16.1 1^ 

is used in finding the distance (space) through which a body- 
falls from rest, in time t, where the time is measured in seconds 
and the distance in feet. 

During the fall, the time t and the distance 8 both increase ; 
t and 8 are variable8y while the number 16.1, of course, is con- 
8tant. 

For every value of t it is possible to compute », 8 being de- 
pendent for its value upon the value of ^ ; 8 is said to be a 
function of t 

A variable i8 a function of another variable when its value i8 
determined for every value of the other variable. 

We have used variables x and y in the drawing of graphs. 
In the equation of the straight line, y = 3 a? -|- 1, a; and y are 
variables ; 3 and 1, of course, are constants ; y is a function of 
oj, since for every value of x there exists a definite value of y. 



88 ELEMENTARY ALGEBRA 

ORAL EXERCISES 

101. 1. The velocity of a body falling from rest is given by 
the formula, ^ ^ ^^.2 t, 

where v is the velocity in feet per second and t is the time of 
fall in seconds. 

Which are the variables ? Which is the constant ? Why is 
V a function of ^ ? 

2. The area of a circle is given by the formula 

Why is u4 a function of r ? Is tt a constant or a variable ? 

3. If a man's monthly salary is % 125, his earnings may be 
expressed by the equation, 

E = 125 w, 

where n is the number of months during which he draws his 
salary, and E the earnings expressed in dollars. 

In this formula explain " constant," " variable," and " func- 
tion." 

4. The area of a rectangle is given by the formula 

A = hh, 

where h and h are the " base " and " height " in linear units, 
and A is the " area " in the corresponding square units. Here 
A varies when h varies and also when h varies. That is, A is 
a function of the two variablesy b and h. Find A, when 6=2, 
^ = 3 ; also when 6 = 4, ^ = 2. 

DIFFERENT MODES OF VARIATION 

102. When one variable is a function of another variable, 
the variation takes on different forms in different cases. In' 
working problems it is necessary to know the exact mode of 
variation, else absurd results will be obtained. 



PROPORTION, VARIATION, FUNCTION 89 

In geometry it is proved that the length of a circle varies as 
its radius, that the area of a circle varies as the sgware of its 
radius, that the volume of a sphere varies as the cvbe of its 
radius. 

In symbols, C : Ci = r : rj, 

V: Fi = r^ : Vi^, 

Hence, if the second radius is twice the first (ri = 2>), then 
the length of the second circle is twice that of the first, the 
area of the second circle is four times that of the first, the 
volume of the second sphere is eight times that of the first. 

Of special interest and importance are the problems in which 
one variable varies directly as another, or inversely as another. 
The words " directly " and " inversely " have the same signifi- 
cance in algebra as in arithmetic. 

For example, at any moment, the length (L) of the shadow of a ver- 
tical post cast upon level ground depends upon the height {H) of the post ; 
the taller the post, the longer the shadow. Here L varies directly as H ; 

we wriiie. _ _ _ . _ . _ _ __ _ _ . 

L:Li = n'.nuOT L = kJS, or Lccff, 

where kis9, constant and where L x ^signifies ^' L varies asH,^^ These 
three notations denote the same relation between L and H; all three in- 
dicate that L is a function of H, 

The time {t)it takes a train to travel from New York to Chicago de- 
pends upon its speed (v) ; the greater the speed, the less the time. Here 
t varies inversely as « ; we write 

t-.ti = Vi'.Vy OTt=k-<,OTtcC-, 

V V 

From t = k*-we obtain vt = k; that is, if one variable varies inversely 

V 

as another, the product of the two variables is a constant. 

Sometimes there are three variables. When one variable de- 
pends upon two other variables in such a way as to vary as the 
product of the two variables, we say that the first variable varies 
jointly as the two other variables. 



90 ELEMENTARY ALGEBRA 

Ex. 4 in § 101 was of this sort. If we doable the length of the base (6) 
of a triangle and at the same time doable its height (A), the area (^) is 
four times greater. In symbols, the variation is expressed thus, 

AiA\-=hhi h\hu or ji = ib • &A, or ji K hh. 

There are many other kinds of variation. A variable may 
vary directly as a second variable and inversely as a third, as in- 

dicated by the formula y = A;-. Or, a variable may depend 

z 

upon three or more other variables. Thus, the price of lace 
depends not only upon the number of yards, but also upon the 
quality of the article, the cost of labor, the rate of profit ex- 
acted vby the seller. Practical problems involving variables as 
functions of other variables and exhibiting many different 
forms of variation were given in § 97. 



VARIATION SHOWN IN GRAPHS 

103. Consider a proportion in which two of the terms are 
variables. 

An automobile travels 29 miles in 2 hours ; at this rate, how 
far does it travel in h hours ? 

As the distances are proportional to the times of travel, we have the 
proportion, d:29 = A:2, 

where d indicates the number of miles and h the number of hours. Here 
d and h are variables ; when h increases, d increases also, the value of d 
being dependent upon that of h. 

The relation between h and d can be expressed equally well 
by changing the equation to the form 

^ 29, 
2 

This shows that h and d vary in such manner that the number 
d is always — as large as the number h\ d is a function of h. 

*4 



PROPORTION, VARIATION. FUNCTION 91 

Thia variation is exhibited by tlie tollowing graph (Fig. 11). 
Hours are measured along the horizontal line OX; miles aie 
measured along the ver- 
tieal line Y. For cou- 
venieuce the divisions 
representing bours are 
taken ten times longer 
than the divisions rep- 
resenting miles. The 
line OA is constructed 
ae follows : 

When h = 0, then 
ds=0. This determines 
the point 0. 

When ft = 4, then 
d = 5S. This deter- 
mines the point A. 

Through and ^ 
draw the line OA, 

Using this graph we 
can tell by inspection 
the distance traveled 
during a time not ex- 
ceeding 5 hours. For 
instance, to tell the 
distance traveled in 
3.2 hr., we find on OX 
the point C, then pro- 
ceed parallel to OT to 

the point B; and from B to the left, parallel to OX. The 
answer is 46*^ miles. Actual multiplication gives 46.4 miles. 

How accurately can the distance be measured by this graph ? 

How accurately can the time be measured by this graph ? 

How can a graph be drawn that will give distances correctly 
to ^ of a mile ? 



92 ELEMENTARY ALGEBRA 

EXBBOISB 

104. 1. From the graph, Fig. 11, tell the distance traveled 
in 2.5 hr., 3.4 hr., 4.1 hr., 4.9 hr. 

2. From the same graph tell the time needed for traveling 
24 mi., 50 mi., 35 mi., 65 mi., 48 mi. 

3. Draw a graph that will show the railroad fare for trav- 
eling short distances at the rate of 2| cents a mile. 

4. Draw a graph showing corresponding values of x and y 
in the proportion x:y = 1.9 : 2.3. 

6. A clerk in a New York store converts prices in " marks- 
per-meter" into ^* dollar s-per-yard.'' He draws a graph and 
roughly " checks " his computations by it. If 1 m. = 1.1 yd., 
and 1 M. = $ .24, draw the graph. 

Hint. 1 M. per 1 m. = |l .24 per 1.1 yd. = $ ? per 1 yd. 

6. One knot (nautical mile) is equal to 1.15 miles. Draw 
a graph for changing from one unit to the other and then tell 
from it the speeds, in miles per hour, of cruisers making 21, 
22, 24, 27, 29 knots per hour. 

7. In lifting weights with the aid of a small, well-lubricated 
screw jack, part of the power varies with the weights that are 
raised, and the rest of the power is a fixed amount to overcome 
friction. It was found by trial that 5 lb. was necessary to lift 
a weight of 100 lb., and that 2.8 lb. was necessary to lift 50 lb. 
Derive an equation and draw a graph showing the relation 
between the power P and the weight W. 

By the conditions of the problem, the relation between P and TTcan 
be expressed by a linear equation of the form P= aW-\- b, where P and 
W are the variables, and a and b are constants to be found from our 
data. Observe that aW is the part of the power which varies with the 
weight, and that b is the constant part that is expended to overcome 
friction. Notice that P is a function of W. 

When P = 6 lb., W = 100 lb., hence 6 = 100 a + 6. 

When P= 2.8 lb., Tr= 60 lb., hence 2.8 = 50 a + 6. 



PROPORTION, VARIATION, FUNCTION 



93 



Solving for a and b we obtain a = .044, 5 = .6. 

Substituting these values we obtain F = .044 Tf 4- .6, 

which expresses ** the law of the machine." The graph of this equation 
is shown in Fig. 12. 

8. In Fig. 12, tell 
by inspection the 
power, P, necessary 
to raise a weight, TT, 
of 30 lb., of 40 lb., of 
65 lb., of 75 lb. To 
what fraction of a 
pound can you esti- 
mate the power by 
this graph? 

9. In Fig. 12, tell 
by inspection the 
weight Wy which can 
be raised when P is 
2^ lb., 3 lb., 4| lb. 
By this graph, can 
W be determined as 
closely as 1 lb. ? How 
would you draw a graph yielding more accurate results ? 

10. In a system of well-lubricated pulleys a weight of 
50 lb. is raised by a power of 10.7 lb., and a weight of 80 lb. 
is raised by a power of 16.7 lb. If part of the power varies 
as the weight, and part of the power is fixed, find the equation 
or " law of the machine " which shows the relation between 
P and W, Draw a graph exhibiting simultaneous values of 
PandTT. 















































Y 












































100 








































































/ 














90 




























J 


f 








































/ 
















80 


























/ 








































J 


f 


















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Fig. 12. 



94 ELEMENTARY ALGEBRA 

PROBLEMS 

105. 1. If XQcy, and x^^ when y = 6, find x when y = 7. 
First Solution. Since xocy, the yariation is direct. 



Hence, 


y:yi^x:xi, 


Taking y,=6, a?i=5, y=7, 


7:6=:aj:5. 


Hence, 


a5 = 5f. 


Second SoltUion. Since 


««y, 


we have 


o^sA;^. 



To determine the constant k, use the given simultaneous 
values of the variables x and y, viz., x = 5 when y = 6. We 

^^*a^ 5 = A;. 6. 

Hence, ^ = f . 

Substituting the value of /c, xss^y. 

When y = 7, we have a? = ^ x 7 = &J. 

The student should master both solutions. 

2. If x varies inversdy as y, and a; = 5 when y s= 6, find a; 
when y = 7. 

i^ir«< Solution. Since a? oc -, we obtain the proportion 

7 : 6 : 5 : a;. 
x==^. 

Second Solution. Since a; x - , 

a; = — • 

Taking a? = 5, y = 6, ^ "^ fi' 

A; = 30. 

30 

Substituting the value of A;, a; = — • 

y 

When y = 7, a; = -8^ = 4^. 



PROPORTION, VARIATION, FUNCTION 95 

3. If z varies jointly as x and y, and 2 = 5, when xssS and 
y = 4, find z when a? =a 7 and y = 6. 

First Solution, By proportion, z:zi = xy: x^yi. 

Take ajj = 3, yi = 4, «i = 5, a; = 7, y = 6. 

Then, « : 5 = 42 : 12, 

2; = 17f 
Second Solution, Since zccxy^ 

z = kxy. 

To determine A;, substitute the simultaneous values, 

2 = 5, a? = 3, y = 4, 
5 = A: X 3 X 4, 

I. — . 6 
a;— ly. 

Hence, « = -ji^ ajy. 

When aj= 7, y = 6, 2 = ^ x 42 = 17f 

4. If z varies directly as x and inversely as y, and 2 = 5, 
when a? = 3, y = 4, find 2; when a; = 7 and y = 6. 

Solution. Here 2 oc -, hence 

z = k>-. 

y 

Substitute the simultaneous values 2 = 5, a; = 3, y = 4, 

5 = A;.f. 
Hence, ^ = ^, 

A 20 a? 

ana 2 = — • - • 

3 y 
Whena? = 7,y = 6, 2; = ^.-J = 7f 

Let the student give the solution by proportion. 

6. The distance described by a body falling from rest varies 
as the square of the time. If in 2 seconds it falls 64.4 feet, 
find the distance it falls in 6^ seconds. 

Solution* The distance (d) varies directly as the square of the time {t). 
Hence, d = kfi. 



96 ELEMENTARY ALGEBRA 

When « = 2, d = 64.4, hence, 64.4 = ifc . 2^, 

A: = 16.1, 
and d = 16.1 <2. 

When t = 6i seconds, d = 680.2 + ft. 

6. The brightness or intensity of lig:ht varies inversely as 
the square of the distance from the source of the light. How 
many times brighter is the page of a book illuminated at a 
distance of 3 ft. from the source than at a distance of 5 ft. ? 

Solution. The intensity (i) varies inversely as the square of the dis- 
tance (d). 

That is, t = — , or i.ii = di^.d^, 

d^ 

Since we do not have sufficient data to determine the numerical value 
of k, it is easier to work the example by proportion. 

We obtain i :ii = ^: 3«. 

Or 1 = 25 = 24. 

ii 9 

At a distance of 3 ft. the page is 2} times brighter than at 5 ft. 

7. If a: varies as y, and if x is 144 when y is 3, find the value 
of y, when x is 360. 

8. If X varies as the square of y, and if x is 144 when y is 
3, find the value of y when x is 360. 

9. If x varies jointly as y and z, and op = 10 when y = 15, 
and z = 18, find x when y =5 and z = 24. 

10. If oj varies inversely as the cube of y, and a; = 54 when 
y = 3, find x when y = .1. 

11. The areas of similar triangles vary as the squares of 
homologous sides. If homologous sides of two such triangles 
are 7 and 10, and the area of the larger triangle is 13.5, what 
is the area of the smaller triangle ? 

12. The areas of two similar triangles are 36 and 43 ; one 
side of the smaller triangle is 5. Find the homologous side of 
the larger triangle, correct to two decimal places. 



PROPORTION, VARIATION, FUNCTION 97 

13. Two pieces of round timber (right cylinders) have the 
same altitude ; their girths are 3.4 ft. and 3.1 ft. Find the 
ratio of their volumes. 

14. If a carriage wheel 4 ft. 2 in. in diameter makes 240 
revolutions in going a certain distance, how many revolutions 
will a wheel 4 ft. 8 in. in diameter make in going the same 
distance ? 

15. The weight of a sphere of given material varies as the 
cube of the radius. If a sphere having a diameter of 3 in. 
weighs 4 lb., find the weight of a sphere of the same material, 
but with a radius of 2 in. 

16. The rents of an estate should be divided between A and 
B in the ratio of 4:5. However, A is paid $ 250, and B is 
paid $ 275. Which has been overpaid, and how much ? 

17. If a train, whose speed is 55 miles per hour, makes a 
certain trip in 3^ hr., how long will it take a slower train to 
travel | of that distance, its speed being to that of the fast 
train, as 4 is to 9 ? 

18. If the pressure p exerted upon the air in a bicycle pump 
and the volume of the air v obey Boyle's law, accordiug to 
which pv = constant, and if the pressure is 25 lb. per square 
LQch when the volume is 27 cubic inches, what will the pres- 
sure be when the volume is reduced to 20 cubic inches ? 

19. Part of the expenses of a certain school was fixed ; part 
of them varied with the number of pupils. If the yearly ex- 
penses were $ 75,000 when the number of pupils was 710, and 
$ 85,000 when the number of pupils had increased to 900, 
find the yearly expense when the number of pupils was 800. 

20. If the ratio ofSx + ytoAx — Sy equals the ratio of 21 
to 2, what is the ratio oix to y? 

21. At the end of 12 days, 15 men finish one fourth of a 
piece pf work. How many additional men must be engaged 
in order to complete the remaining work lq 18 days ? 

H 



98 



ELEMENTARY ALGEBRA 



22. If a body falls from rest through a distance of 16.1 feet 
during one second, how far will it fall in 8.3 seconds ? 

23. Compare the intensity of illumination of the page of a 
book at a distance of 6.3 feet from a source of light with the 
intensity at a distance of 4.9 feet. 

24. If the surface area of a certain sphere is 35 sq. in., what 
is the surface area of a sphere having a radius 4.7 times longer ? 

26. If the volume of a certain sphere is 500 cu. in., what is 
the volume of a sphere whose radius is 2.9 times longer ? 

26. If the base of a triangle is increased 10-fold and its alti- 
tude is reduced to -j^ its original length, find the ratio between 
the initial and final areas. 

27. Two casks of similar shape have their homologous linear 
dimensions in the ratio of 2 : 3.4. Find the ratio of their 
capacities. 

28. The radii of the bases of two cylindrical tanks are 1 ft. 

5 in., and 1 ft. 11 in., respectively ; the heights of the tanks are 

6 ft. and 5 ft., respectively. Compare the capacities of the 
tanks. 

29. The area of a circle is found by the formula A = wR*, 
where v = 3.1416. Draw a graph by which the areas can be 
determined by inspection. 



B 


A 


Point 


i 


.79 


A 


1 


3.14 


B 


li 


4.91 


C 


H 


7.07 


D 


U 


9.62 


E 


2 


12.56 


F 


3 


28.27 


Q 


4 


60.27 


H 



Assuming values of i?, as shown in the 
table, compute the values of the area A, 
Explain the construction of Fig. 13. 



J 



PROPORTION, VARIATION, FUNCTION 



99 



30. In Fig. 13, 
find by inspection 
the areas of circles 
having radii of 2J, 

2^, 2f , 3j, ^, ^. 

31. If a garrison 
of 2000 men is pro- 
visioned for 40 days, 
how long will their 
provisions last if 
they are joined by 
375 other men ? 

32. The surface 
of a sphere varies 
directly as the 
square of the diam- 
eter. If the diam- 
eter is 2 inches, the 
surface is 12.666 
square inches. Find 
the surface when the diameter is 7 inches. 

33. The value of a piece of land has risen in ten years from 
$ 1200 to $ 3200. How would you calculate the rise in value of 
another similarly situated piece which was worth $1750 ten 
years ago ? 

34. The rent of a house in a town has gone from $ 85 to $ 70 
per month during the last 15 years. If the value of other houses 
in that neighborhood has decreased in the same ratio, what was 
the decrease in the rent of a house now rented at $ 42 ? 

36. The diagonal of a cube varies directly as the length of 
an edge. When the edge measures 3 inches, the diagonal 
measures 5.196 inches. What does the diagonal measure when 
the edge is 3.8 inches ? 





































































































































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lA 






















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sU 




















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Fj 


/ 


























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E 


/ 




























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T 


)/ 


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c 


/ 










































B>f 






































A 




/^ 




































^ 


2C 


























_, 





2 S 

Radius of Circle 

Fia. 13. 



100 ELEMENTARY ALGEBRA 

36. The weight of a copper coin varies directly as its thick- 
ness and also as the square of its diameter. A coin weighing 
11 grams has a diameter of 3 cm. and a thickness of .2 cm. 
What is the weight when the diameter is 2.5 cm. and the 
thickness .3 cm. ? 

37. A's rate of working is to B's as 3 : 4 ; B's rate of work- 
ing is to C's as 6 ; 5. How long will it take C to do work 
which A can do in 8 hours ? 

38. The distance through which a body falls from rest varies 
as the square of the time of falling. If a body falls 257.6 ft. 
in 4 seconds, how far will it fall in 5^ seconds ? 

39. If a stone weighing 5 pounds falls from rest through a 
distance of 144.9 ft. in 3 seconds, how far will a stone fall from 
rest in 3 seconds, if its weight is 500 pounds ? See Ex. 38. 

40. The tractive force or pull necessary to move a vehicle at 
a uniform rate (say 3 miles an hour) varies directly as the 
pressure (weight of vehicle and load) and inversely as the 
square root of the average radius of the wheels. If on level, 
paved, or macadam roads a pull of 57 pounds per ton of pres- 
sure is necessary when the front and rear wheels average 50 
inches in diameter, what pull is necessary to move 3500 pounds 
when the wheels of the vehicle average 38 in. in diameter ? 

41. If a tractive force of 75 lb. is necessary to keep a load 
of 1 ton in motion in a vehicle whose wheels average 38 in. in 
diameter, on a dry and hard earth road, what is the tractive 
force for 5 tons when the wheels average only 26 in. in 
diameter ? See Ex. 40. 

42. If on an earth road, in sticky mud \ in. deep, the trac- 
tive force per ton is 119 lb. when wheels 38 in. in diameter are 
used, what tractive force is required to move 4 tons when the 
diameters are 50 in. ? 

43. The tractive force per ton over dry, cloddy plowed ground 
is 252 lb. for wheels 50 in. in diameter ; what is the tractive 
force for J of a ton when the wheels are 26 in. in diameter ? 



PROPORTION. VARIATIpkf rtkc'TtON^ '|' \ IQl 

GRAPHS EXHIBITINO EkplRICAL 'iJATA 
106. 1. By a certain plan of life inaurance a single premium 
is paid in order that the insured may receive $100 when he ia 
60 years old or that Ms beneficiaries may receive $100 if he 
dies before that age. The premium depends upon the age at 
which the i: 



Age next birthday : 

15 20 25 30 36 40 45 60 

Premium: $41.20 $46.25$49.30 953.60 $60.10 $65.10 $72.05 $80.20 

Draw a graph and 
use it to find the pre- 
mium one would pay 
at the age of 18, 23, 
32, 38, 43. 2 

It U convenient to g 

mark oB along the z-axie J 

the ages above 16 ; along a 

the {(-axis the premiums -g 

aftoiie $40. By this <le- J 

vice a much smaller sheet ** 
of Bquare paper can be 
used. See Fig. 14. 

2. In a city in the 
northern part of the 
United States the 
times of sunrise on 
certain days in May, June, and July 



Miv 

;20 
.04 



Jf: 



i^es above 16 years 
Fig. 14. 

as follows ; 



46 



Draw a graph showing the hour of sunrise from May 10 to 
July 30. 

Mark oS on one axis the number of days after May 10, od the other 
axis the number of minuteB after 3 : 44 a.h. 



• • • * • . 



• • - 

« 






CHAPTER IV 

LOGARITHMS 

107. In an equation 10^ = N 

the exponent L is called the logarithm of the number N, to the 
base 10. 

For example, 

102 = 100, hence 2 is the logarithm of 100, to the base 10. 

We write, 2 = log 100. 

10> = 1000, hence 3 is the logarithm of 1000, to the base 10. 

We write, 8 = log 1000. 

10^ = 3.16227+ (verify this), hence .6 is the logarithm of 8.16227+, to 
the base 10. ^^ ^^ite, .6 = log 3.16227+. 

10* = 10i(10)i = 81.6227+, hence 1.5 is the logarithm of 81.6227+, to 
the base 10. ^^ ^^te, 1.6 = log 31.6227+ 

As in this chapter all logarithms are taken to the same base 10, no con- 
fusion will arise from the omission, hereafter, of the phrase *' to the base 
10." 

Verify the following statements : 

10» =1000 10* = 10(10)* = 31.6227+ 

10* =100 IQi =3.16227+ 

10^ =10 1 

lft-i = — =.316227+ 
10^ =1 ^^ lOi 

10-1 = .1 .1 

10-. = .01 io-i = ^-i|=. 0316227^ 

10-» = .001 1 

lO-f = A = .00316227+ 
10* = 100(10)* = 316.227+ 10* 

102 



LOGARITHMS 103 

Hence, by the definition of logarithms, 

3 = log 1000 ^.5 = log 316.227+ 

2 = log 100 1.5 = log 31.6227+ 

1 = log 10 .5= log 3.16227+ 

O = logl -.6= log ,316227+ 

- 1 =. log .1 - 1.5 = log ^0316227+ 

- 2 = log .01 - 2^5 = log .00316227+ 

- 3 = log .001 

LOGARITHMIC CUBVB 
108. On a piece of square paper as small as that in Fig. 15, it is 
not convenient to draw a curve that will exhihit the logarithms 
of numbers ranging from ,001, the smallest, to 1000, the largest 
number considered above. The curve shows the logarithms of 
positive numbers between .1 and 50. The numbers are laid off 
on the as-&xiB, their respective logarithms on the y-axis. 



Fig. 15. 

— 1 = log .1 looatea the point A 

= log 1 locates the point B 

.6 = log 3.16+ IncatPB the point C 

1 = log 10 locates the point D 
1.6 = log 31.6-^ locates the point E 

curve, some additional values were used. 



104 ELEMENTARY ALGEBRA 

ORAL BXBBCISBS ON THB LOGARITHMIC CXJBVB 

109. 1. Where does the curve cut the ataxia ? How much 
is log 1 ? 

2. What is the algebraic sign of the logarithms of all num- 
bers larger than 1 ? 

3. For what range of numbers are the logarithms negative ? 

4. Does the logarithmic curve extend to the left of the y-axis ? 
6. Do negative numbers have real logarithms ? 

6. Does the logarithm increase as a variable number in- 
creases ? 

By inspection of Fig. 15, find approximately the logarithms 
of the following numbers : 

7. 5. 10. 35. 13. 2. 

8. 15. 11. 45. 14. 4. 

9. 25. 12. 50. 16. 6. 

By inspection of Fig. 15, find approximately the numbers 
corresponding to the following logarithms : 

16. 1.6. 20. 1.2. 24. .8. 

17. 1.5. 21. 1.1. 26. .3. 

18. 1.4. '22. 1. 26. 0. 

19. 1.3. 23. .7. 27. —.5. 

Find by inspection, how many times greater 

28. log 4 is than log 2. 33. log 27 is than log 3. 

29. log 8 is than log 2. 34. log 16 is than log 4. 

30. log 16 is than log 2. 36. log 25 is than log 5. 

31. log 32 is than log 2. 36. log 36 is than log 6. 

32. log 9 is than log 3. 



1 



LOGARITHMS 105 

Show by Fig. 15 that 

37. log 50 = log 5 4- log 10. 40. log 48 — log 8 = log 6. 

38. log 40 = log 5 + log 8. 41. log 24 - log 4 = log 6. 

39. log 42 = log 6 4- log 7. 42. log 32 — log 4 = log 8. 

ASSUMPTIONS 

• 

110. In drawing the logarithmic curve in Fig. 15 we located 
a few points and then drew a smooth curve through those points. 
This process is based on the following assumptions which 
admit of proof : (1) All positive numbers, whether rational or 
irrational, have logarithms ; (2) In all cases, the logarithm of a 
number increases when the number itself increases. 

FUNDAMENTAL THEOREM 

111. In studying the logarithmic curve we noticed that 
log 50 = log 5 4- log 10. 

This illustrates an important theorem. 

Let N and Ni be any two positive numbers. 

Also, let JVr=10% 

and Ni = 10^. 

Then their product is, N- Ni = lO^+^i. 

By definition of logarithms, L = logarithm of N, 

and Zi = logarithm of Ni, 

X + Zi = logarithm of N • ^i. 
Hence, the theorem. 

The logarithm of the product of two positive numbers is equal 
to the sum of the logarithms of the numbers. 

112. The integral part of a logarithm is called its charac- 
teristiCy and the decimal part is called its mantissa. 

For example, log 31.6227+ = 1.5. 
The characteristic of log 31.6227+ is 1. 
The mantissa of log 31.6227+ is .6. 




106 ELEMENTARY ALGEBRA 

The tables of logarithms give only the mantissa ; the charao- 
teristic can be supplied by two easy rules. 

It has been found convenient to take the mantissas of all 
logarithms positive. The characteristic is positive in the 
logarithms of numbers larger than 1, and negative in the 
logarithms of numbers smaller than 1. That is, the char- 
acteristic of log, 100 is positive; the characteristic of log .01 
is negative. If the characteristic is negative, the negative 
sign is placed over the figure, as a reminder that the — does 
not apply to the mantissa. Thus the characteristic of log .06 
is 2. 

For the purpose of deriving the rule for determining the 
characteristic of the logarithm of a number, we restate some 
of the relations in § 107 : 

1000 = 10* 1 =10° 

100 = 10^ .1 =10-^ 

10 = 10^ .01 =10-2 

1 = 10° .001 = io-» 

Since 569.5 lies between 100 and 1000, its logarithm lies between 2 and 8. 
That is, log 569.6 = 2 + a mantissa. 

Since 86.6 lies between 10 and 100, log 86.6 = 1 + a mantissa. 
Since 7.03 lies between 1 and 10, log 7.03 = + a mantissa. 

Since .673 lies between .1 and 1, log .673 = ~ 1 + a mantissa. 

Since .046 lies between .01 and .1, log .045 = ^ 2 + a mantissa. 
Since .0078 lies between .001 and .01, log .0078 = — 8 + a mantissa. 

By inspection of these relations we obtain the rule : 

1. If the first Bigni&CAnt figure of a number is n places to the 

' left 1 

/ ' of units' place, the characteristic of the logarithm is 

I vOflv 

( ■ ^ 

I 4- w 1 
— n 



LOGARITHMS 107 



The following illustrations will make this rule plainer : 



J 

S 

1^ . Characteristic 

6 6 9.6 2 . 

8 6.6 1 

7.03. 

0.673 -1 

0.046 -2 

0.0078 -3 

It will be seen that when a number is 10 or larger than 10, the first 
significant figure is to the left of units ^ place ; if the number is less than 
1, the first significant figure is to the rig?U of units' place. This is shown 
by the dots placed above the digits ; the number of dots placed over a 
number indicates the number of units in the characteilstic. 



BXBBCISBS 

113. By inspection, tell the characteristics of the logarithms 
of the following numbers : 

1. 123. 4. 17.754. 7. .091. 10. 2000. 

2. 97. 6. .7894. 8. .00034. 11. .00005. 

3. 2.345. 6. .005. 9. .0034. 12. 13.764. 

114. The following is an important property possessed by 
logarithms constructed to the base 10 : 

If two numbers contain the same figures in the same order , hut 
differ in the position of the decimxxl point, their logarithms to the 
base 10 have the same mantissa. 

Suppose the two numbers are 7896.1 and 7.8961. 

We have 7896.1 = 7.8961 x 1000. 

By §111, log 7896.1 = log 7.8961 + log 1000. 

But log 1000 = 3. 

Hence, log 7896.1 = log 7.8961 + 3. 



108 ELEMENTARY ALGEBRA 

Since the two logarithms differ by the integer 3, the decimal parts (the 
mantissas) of the logarithms must be the same. 
For example, log 7896.1 = 3.8974. 

log 7.8961 = 0.8974. 
How the mantissa is found from the tables will be explained next. 

FINDING LOGARITHMS 

115. 1. Find the logarithm of 789. 

The characteristic is 2. 

The mantissa is taken from the table, p. 110. The column on thcr ex- 
treme left contains the first two significant figures of the number whose loga- 
rithm is sought. In this case we look for the figures 78. In the same row 
with 78, and in the column headed * * 9 ' ^ ( 9 being the third digit in 789) , is the 
number 8971 ; this is the mantissa sought, expressed to four decimal places. 

Hence we have log 789 = 2.8971. 

2. Find the logarithm of .02738. 

The characteristic is 2. The significant figures are 2738. On p. 109 
look in the column on the extreme left for 27 ; in the column headed 
** 3 '* and in the same row with 27 is the mantissa .4362. 

Hence log .0273 = 2.4362. But we want log .02738. This is not found 
in this table and must be determined by a process called *^ interpolation.'^ 

The table is so constructed that interpolation is made easy. In the 
column on the extreme right, headed **789,*' we find, under the 8, and 
in the same row with 4362, the number 13. 

Add 13 to 4362. We obtain 4376. This is the mantissa, obtained by 
interpolation. 

Hence, log .02738 = 2.4376. 

116. It may be added that logarithmic computation in gen- 
eral is only approximate. With a four-place table like the 
one in this book, logarithms of numbers are found to four 
places only. Absolute accuracy is, therefore, out of the ques- 
tion. This fact does not materially diminish the value of 
logarithms, for the reason that in practical problems, results 
correct to three, four, or five decimal places are satisfactory. 
For instance, in the computation of the interest on a given 
sum, a result correct to the nearest cent is all we want. 



LOGARITHMS 



109 



117. 



Logarithms 



No. 

10 
11 
11 
IS 
14 

16 
16 
17 
18 
19 





1 


S 


S 

0128 
0681 
0S99 
1239 
1668 


4 

0170 
0569 
0984 
1271 
1684 

1875 
2148 
2405 
2648 

2878 


6 

0212 
0607 
0969 
1308 
1614 


6 


T 

0294 
0682 
1088 
1867 
1678 


8 


9 


1 S S 


4 6 6 


T 8 9 


0000 
0414 
0792 
1189 
1461 

1761 
2041 
2804 
2653 

2788 


0048 
0458 
0828 
1178 
1492 

1790 
2068 
2880 
2577 
2810 

8082 
3248 
8444 
3686 
8820 


0086 
0492 
0864 
1206 
1528 

1818 
2096 
2856 
2601 
2888 

8064 
8268 
8464 
8666 
8888 

4014 
4188 
4846 
4502 
4664 


0268 
0646 
1004 
1886 
1644 


0884 
0719 
1072 
1899 
1708 

1987 
2268 
2604 
2742 
2967 


0874 
0765 
1106 
1480 
1782 


4 812 
4 811 
8 710 
8 610 
8 6 9 


17 2126 
1619 28 
14 17 21 

18 16 19 

12 15 18 


29 88 87 
26 80 84 
24 28 81 
28 26 29 
2124 27 


1847 
2122 
2880 
2625 
2866 


1908 
2176 
2480 
2672 
2900 


1981 
2201 
2455 
2695 
2928 


1950 
2227 
2480 
2718 
2945 


2014 
2279 
2629 
2766 
2989 

8201 
8404 
8698 
8784 
8962 


8 6 8 
8 5 8 
2 6 7 
2 6 7 
2 4 7 


11 14 17 

11 18 16 

10 12 15 

9 12 14 

9 1118 


20 22 26 
18 2124 
17 20 22 
1619 21 
16 18 20 


80 
SI 
SS 
SS 
S4 


8010 
8222 
8424 
8617 
8802 


8075 
8284 
8488 
8674 
8856 


8096 
8804 
8502 
8692 
8874 


8118 

8711 
8892 

4065 
4282 
4898 
4548 
4698 


8189 
8846 
8641 
8729 
8909 


8160 
8865 
8660 
3747 
8927 

4099 
4265 
4426 
4579 

4728 

4871 
5011 
6146 
6276 
6408 


8181 
8386 
8679 
8766 
8946 


2 4 6 
2 4 6 
2 4 6 
2 4 6 
2 4 5 


81118 
81012 
81012 
7 911 
7 9 11 


15 17 19 
14 16 18 
14 15 17 
18 15 17 
12 14 16 


86 
S6 
ST 
SS 
S9 

SO 

S4 

S6 
S6 
ST 
S8 

S9 

40 
41 
4S 
4S 
M 

46 
46 
4T 
48 
49 

60 
61 
6S 
6S 
64 


8979 
4160 
4814 
4472 
4624 


8997 
4166 
4880 

4487 
4689 


4081 
4200 
4862 
4518 
4669 


4048 
4216 
4878 
4688 
4688 

4829 
4969 
6105 
5287 
6866 


4082 
4249 
4409 
4664 
4718 

4867 
4997 
5182 
5268 
6891 


4116 
4281 
4440 
4594 
4742 


4188 
4298 
4456 
4609 
4767 


2 8 6 
2 8 6 
2 8 6 
2 8 6 
18 4 


7 910 
7 810 
6 8 9 
6 8 9 
6 7 9 


12 14 15 
1118 16 
11 18 14 
11 12 14 
10 12 18 


4771 
4914 
5061 
5186 
6816 

5441 
5668 
6682 
6798 
6911 

6021 
6128 
6282 
6386 
6185 

6582 
6628 
6721 
6812 
6902 


4786 
4928 
5065 
6198 
6828 

5468 
6576 
6694 
6809 
6922 

6081 
6188 
6248 
6845 
6444 

6642 
6687 

67aio 

6821 
6911 


4800 
4942 
6079 
5211 
5840 


4814 
4955 
6092 
6224 
5858 


4848 
4988 
5119 
6260 
5878 


4886 
6024 
6169 
6289 
6416 


4900 
5088 
5172 
6802 
6428 


18 4 
1 8 4 
18 4 
18 4 
18 4 


6 7 9 
6 7 8 

6 7 8 
6 6 8 
6 6 8 


10 11 18 

10 11 12 

9 1112 

9 10 12 

9 1011 


6465 
5587 
6706 
6821 
6988 

6042 
6149 
6268 
6855 
6464 


6478 
6599 
5717 
6882 
6944 

6068 
6160 
6268 
6865 
6464 

6561 
6666 
6749 
6889 
6928 

7016 
7101 

7186 
7267 

7848 


5490 
5611 
5729 
6848 
5956 


6602 
6628 
5740 
6866 
6966 

6076 
6180 
6284 
68S5 
6484 

6580 
6676 
6767 
6857 
6946 

7088 
7118 
7202 
7284 
7864 


6514 
6685 
6762 
6866 
6977 

6086 
6191 
6294 
6895 
6493 


6527 
5647 
5768 

5877 
5988 


5589 
5668 
5776 

6888 
6999 

6107 
6212 
6814 
6416 
6618 

6609 
6702 
6794 
6884 
6972 


5661 
6670 
6786 
5899 
6010 


1 2 4 
1 2 4 
1 2 3 
1 2 8 
12 8 


6 6 7 
6 6 7 
6 6 7 
6 6 7 
4 5 7 


91011 
81011 
8 9 10 
8 910 
8 9 10 


6064 
6170 
6274 
6876 
6474 

6671 
6665 
6768 
6848 
6987 

7024 
7110 
7198 
7276 
7866 


6096 
6201 
6804 
6405 
6503 

6599 
6698 
6T85 
6875 
6964 

7050 
7185 
7218 
7800 
7880 


6117 
6222 
6826 
6425 
6522 


1 2 8 
12 8 
12 8 
12 3 
1 2 8 


4 6 6 
4 6 6 
4 6 6 
4 6 6 
4 5 6 


8 9 10 

7 8 9 
7 8 9 
7 8 9 
7 8 9 


6661 
6646 
6789 
6880 
6820 

7007 
7098 
7177 
7269 
7840 


6690 
6684 
6776 
6866 
6955 

7042 
7126 
7210 
7292 
7872 


6618 
6712 
6808 
6898 
6981 

7067 
7162 
7285 
7816 
7396 


12 8 
12 8 
12 8 
12 3 
12 8 


4 6 6 
4 5 6 
4 6 5 
4 4 5 
4 4 5 


7 8 9 
7 7 8 
6 7 8 
6 7 8 
6 7 8 


6990 
7076 
7160 
7248 
7824 


6998 
7084 
7168 
7251 
7882 


7060 
7148 
7226 
7808 

7388 


12 8 
1 2 8 
12 2 
12 2 
12 2 


8 4 6 
3 4 6 
8 4 6 
8 4 5 
8 4 5 


7 8 
6 7 8 
6 7 7 
6 6 7 
6 6 7 



110 



ELEMENTARY ALGEBRA 



Logarithms 



No. 




7404 


1 


6 
7419 


8 


4 


6 
7448 


6 
7451 


7 
7469 


8 
7466 


9 


1 6 


8 


4 


6 


6 


7 8 9 


7412 


7427 


7485 


7474 


1 2 


2 


8 


4 


ft 


5 6 7 


M 


7482 


7490 


7497 


7505 


7618 


7520 


7528 


7686 


7548 


7661 


1 2 


2 


8 


4 


5 


5 6 7 


67 


7069 


7566 


7674 


7682 


7589 


7697 


7604 


7612 


7619 


7667 


1 2 


2 


8 


4 


5 


5 6 7 


M 


7634 


7642 


7649 


7657 


7664 


7672 


7679 


7686 


7694 


7701 




2 


8 


4 




5 6 7 


M 


7709 


7716 


7728 


7781 
7808 


7788 
7810 


7746 


7752 


7760 


7767 
7889 


7774 




2 


8 


4 




5 6 7 


60 


7782 


7769 


7796 


7818 


7825 


7882 


7846 




2 


8 


4 




5 6 6 


61 


78m 


7860 


7868 


7876 


7882 


7889 


7896 


7908 


7910 


7917 




2 


8 


4 




5 6 6 


6S 


7924 


7981 


7988 


7945 


7962 


7969 


7966 


7978 


7980 


7987 




2 


8 


8 




5 6 6 


66 


7998 


8000 


8007 


8014 


8021 


8028 


8065 


8041 


8048 


8066 




2 


8 


8 




5 5 6 


64 

66 


8062 


8069 


8075 


8082 


8089 


8096 


8102 


8109 


8116 


8122 




2 


8 


8 




6 5 6 


8129 


8186 


8142 


8149 


8166 


8162 


8169 


8176 


8182 


8189 




2 


8 


8 




5 5 6 


66 


8195 


8202 


8209 


8215 


8222 


8228 


8285 


8241 


8248 


8264 




2 


8 


8 




5 6 6 


67 


8261 


8267 


8274 


8280 


8287 


8293 


8299 


8806 


8812 


8819 




2 


8 


8 




6 5 6 


66 


8825 


8881 


8888 


8844 


8861 


8867 


8868 


8870 


8876 


8882 




2 


8 


8 




4 5 6 


66 


8888 


8895 


8401 
8468 


8407 
8470 


8414 
8476 


8420 

8482 


8426 


8482 
8494 


8489 


8446 
8606 




2 


2 


8 




4 6 6 


70 


8451 


8457 


OvOO 


8600 




2 


2 


8 




4 5 6 


71 


8618 


8619 


fmt> 


8681 


86B7 


8648 


8549 


8666 


8661 


8667 




2 


2 


8 




4 5 5 


76 


8678 


8679 


8686 


8691 


8697 


8606 


8609 


8616 


8621 


8627 




2 


2 


8 




4 5 5 


76 


8688 


8689 


8646 


8651 


8657 


8668 


8669 


8676 


8681 


8686 




2 


2 


8 




4 5 5 


74 


8692 


8696 
8766 


8704 
8762 


8710 

8768 


8716 

8774 


8722 
8779 


8727 


8788 


8789 


8746 
8802 




2 


2 


8 




4 5 6 


76 


8751 


8785 


8791 


8797 




2 


2 


8 


8 


4 5 5 


76 


8806 


8814 


882(r 


1B825 


8831 


8887 


8842 


8848 


8864 


8869 




2 


2 


8 


8 


4 5 5 


77 


8866 


8871 


8876 


8882 


8887 


8898 


8899 


8904 


8910 


8916 




2 


2 


8 


8 


4 4 5 


78 


8921 


8927 


8982 


8988 


8948 


8949 


8954 


8960 


8965 


8971 




2 


2 


8 


8 


4 4 6 


76 

80 


8976 
9081 


8982 
9086 


8987 
9042 


8998 


8998 


9004 


9009 


9015 


9020 
9074 


9026 
9079 




2 


2 


8 


8 


4 4 5 


9047 


9058 


9058 


9068 


9069 




2 


2 


8' 


8 


4 4 5 


81 


9085 


9090 


9096 


9101 


9106 


9112 


9117 


9122 


9128 


9188 




2 


2 


8 


8 


4 4 5 


86 


9188 


9148 


9149 


9164 


9160 


9166 


9170 


9175 


9180 


9186 




2 


2 


8 


3 


4 4 5 


66 


9191 


9196 


9201 


92()6 


9212 


9217 


9222 


9227 


9232 


9288 




2 


2 


8 


8 


4 4 5 


64 


9248 


9248 
9299 


9258 


9258 


9268 


9269 
9820 


9274 


9279 
9880 


9284 
9885 


9289 
9840 




2 


2 


8 


8 


4 4 5 


86 


9294 


9804 


9809 


9815 


9825 




2 


2 


8 


8 


4 4 5 


66 


9845 


9850 


9856 


9860 


9865 


9870 


9876 


9380 


9886 


9890 




2 


2 


8 


3 


4 4 5 


87 


9895 


9400 


9405 


9410 


9415 


9420 


9425 


9480 


9486 


9440 


1 




2 


2 


3 


3 4 4 


86 


9445 


9450 


9465 


9460 


9465 


9469 


9474 


9479 


9484 


9489 


1 




2 


2 


8 


8 4 4 


66 


9494 


9499 


9604 
9652 


9609 


9518 


9618 


9623 
9671 


9528 
9576 


9688 


9638 


1 




2 


2 


8 


8 4 4 


90 


9542 


9547 


9567 


9562 


9566 


9581 


9586 


1 




2 


2 


8 


8 4 4 


91 


9590 


9596 


9600 


9605 


9609 


9614 


9619 


9624 


9628 


9683 


1 




2 


2 


3 


8 4 4 


96 


9688 


9648 


9647 


9662 


9657 


9661 


9666 


9671 


9676 


9680 


1 




2 


2 


3 


8 4 4 


98 


9685 


9689 


9694 


9699 


9708 


9708 


9718 


9717 


9722 


9727 


1 




2 


2 


8 


8 4 4 


94 


9731 


9786 


9741 


9745 
9791 


9760 


9764 


9769 
9806 


9768 


9768 
9814 


9773 

9818 


1 




2 


2 


8 


3 4 4 


96 


9777 


9782 


9786 


9796 


9800 


9809 


1 




2 


2 


8 


3 4 4 


96 


9828 


9827 


9882 


9886 


9841 


9846 


9860 


9854 


9869 


9868 


1 




2 


2 


3 


3 4 4 


97 


9868 


9872 


9877 


9881 


9886 


9890 


9894 


9899 


9903 


9908 


1 




2 


2 


3 


8 4 4 


96 


9912 


9917 


9921 


9926 


9980 


9984 


9989 


9948 


9948 


9962 


1 




2 


2 


3 


8 4 4 


99 


9956 


9961 


9966 


9969 


9974 


9978 


9983 


9987 


9991 


9996 


1 




2 


2 


3 


3 8 4 



LOGARITHMS 111 

BXBBCISBS 

118. Find the logarithms of : 

1. 870. 4. .01235. 7. 13.8964. 

2. 63. 6. .003986. 8. 1089. 

3. 708. 6. .7943. 9. 1008. 

FINDING ANTILOGARITHMS 

119. 1. Given log n = 1.2355, find the antilogarithm n. 

The number n, frequently called the arUilogarithm, may be found by 
steps which are the reverse of those taken in finding the logarithm. 

At first consider only the mantissa 2366. On page 109 find the mantissa 
2366. It occurs in the same row as the number 17 on the extreme left, 
and in the column headed ** 2.^* ^ The order of the figures in the answer 
is 172. ' 

Where should the decimal point be ? That is determined with the aid 
of the characteristic. All we need to do is to apply the rule relating to 
characteristics. Place the decimal point so that the characteristic of the 
resulting number is 1. 

That number is 17.2. Thus, n = 17.2. 

2. Given log n = 2.8764, find the antilogarithm n. 

Look in the table for the mantissa 8764. It cannot be found. 

Look then for the next smaller mantissa. It is 8762 ; the order of 
figures in the antilogarithm of 2.8762 is 762. But we want the antiloga- 
rithm of 2.8764. This is found approximately by the following ** inter- 
polation.*' 

Find the difference between 8762 and the given mantissa. This differ- 
ence is 2. 

Look in the same row with 8762, and in one of the right-hand columns 
for the number 2. You see it in the column headed ** 3 '* and also in the 
column headed "4." 

Take either 3 or 4 as the figure found by interpolation. Which of these 
is the more accurate, we cannot tell from these tables. Take 3. 

Write 3 after 752, and we have the required order of figures ; namely, 
7623. 

Place the decimal point so that, by the rule, the characteristic of the 
resulting number is 2. We obtain n = .07623, nearly. 



112 ELEMENTARY ALGEBRA 

EXEBCISBS 

120. Find the antilogarithms of the following : 

1. 1.7931. 4. 1.7605. 7. 3.4567. 10. 3.6709. 

2. 2.3181. ' 6. 0.9064. 8. 0.7064. 11. 3.5374. 

3. 3.9876. 6. 2.1907. 9. 2.3706. 12. 5.9860. 

121. 1. Using logarithms, find the product n = 7893 x 879G. 

By § 111, the logarithm of a product is the sum of the logarithms c 

the factors. We find , „^^„ ., „^«„ 

log 7893 = 3.8973 

log 8796 = 3.9448 

log n = 7.8416 

We find the antilogarithm n = 69430000. 

Observe that this answer is only approximate. The true answer is 
69426828. Closer approximations may be obtained by logarithms, by 
using a table containing figures to 6, 6, 7 or more places. 

2. Find the product x = .9062 x .007362. 

The characteristics of the factors are I and 3. When logarithms are 
added or subtracted, it is easier to write a negative characteristic as the 
difference of two positive integers, thus : 1 = 9 — 10, 3 = 7 — 10. 

Accordingly, we get log .9062 = 9.9572 — 10 

log .007362 = 7.8670-10 
Add log X = 17.8242 - 20 

Or loga:= 3.8242 

X = .006671, nearly. 

BXEBCISBS 

122. Using logarithms, compute the following products : 

1. 123 X 978. 4. 37.61 x 3.945. 7. 9999 x 7777. 

2. 12.34 X 75.34. 6. 1479 x 6984. 8. 19283 X 87045. 

3. .0009638 X 7894. 6. 2673 x 7654. 9 93069 x 80493. 



LOGARITHMS 113 

FUNDAMENTAL THEOREMS 

123. In § 111, we proved the important theorem that 

Tlie logarithm of a product is equal to the sum of the logarithms 
of the factors. 

We proceed to establish two other theorems that are no less 
fundamental. 

The logarithm of a quotient is equal to the logarithm of the 
dividend minus the logarithm of the divisor. 

The proof is similar to that of the first theorem. 
Let N and Ny^ be any two positive numbers. 

Let also JV^=10^ 

and ^1 = 10a. 

Divide N by iVi, — = lO^-^i. 

By definition of logarithms, 

. L = logarithm of Ny 
Li = logarithm of Ni, 

and L — Li= logarithm of — 

Hence the theorem is proved. 

The logarithm of a positive number with the eocponerU n is n 
times the logarithm of the number. 

In this theorem n may be a positive integer or a positive 
fraction. 

Let -^be the number, and let 

JV^=1(P. 

Raise both sides to the nth power, 

N^ = (1(F)». 
Simplify N^ = lO"^, 



114 ELEMENTARY ALGEBRA 

By the definition of logarithms, 

L = the logarithm of N 
and nL = the logarithm of N\ 

Hence the theorem is proved. 

If w =— , then this theorem gives the logarithm of y/W. 
m 

124. 1. Using logarithms, find the quotient a?= ||^. 

We find log 9876 = 3.9946 

log 6987 = 3.8443 
log a; = 0.1603 
The autilogarithm x = 1.414, nearly. 

2. Find a? = (8.786)*. 

We find log 8. 786 = 0. 9438 

Multiply by 6, 6 

loga; = 4.7190 

X = 52360, approximately. 

3. Compute aj =(.01237)*. 

The characteristic of .01237 is 2 ; we write it 8 — 10. 
Hence, log .01237= 8.0923-10 

Multiply by 7, 7 

66.6461 - 70 

Divide by 3. Since 70 -»- 3 introduces a fraction, it is more convenient 
to write the characteristic, 66 — 70, in the form 16 — 30. 

The logarithm is then 16.6461 -- 30. 

Dividing by 3, log a; = 6. 5487 - 10. 

X = .00003537, nearly. 

BXBBCISES 

jl25. Evaluate the following by means of the four-place table 
of logarithms : 

1. 6.823 X 2.315. 4. 2293 x 4489 -^ 7895. 

2. 46420 X 27.49. 6. .943 x 9855 -^ .0896. 

3. 32.25-^6.923. 6. (2.519)*. 



LOGARITHMS 115 

7. (.007668)'. ^^ 98.6 x 7.639 

8. a/4978000000. -^^ 



9. V752 X V3450. .. (273)«(743)« 

10. (1.63)*. ' (897)V 

11. (3.197)1 ^^ V3:598 ' -^479 , 

12. V;03276. * V933 

PROBLEMS 
126. In the following problems let w = 3.1416, B = radios. 

1. Find the length of a circle (i.e. the circumference) whose 
radius is 3.17 ft. 

2. Find the area of a circle whose radius is 9.795 in. (Area 

3. Find the area of a circle whose radius is 7.891 cm. 

4. Compute the curved surface of a right circular cylinder 
whose height is 7.9 in. and whose base has a diameter of 
79.86 in. 

6. Calculate the volume of a sphere whose radius is 2.97 yd. 

Volume of a sphere = . 

3 

6. What is the area of the surface of a sphere when R 
= 0.7 in. . Area of spherical surface = 4 wlfi, 

7. Compute the leg of a right triangle when 
the hypotenuse is 25.97 in. and the other leg is 
17.8 in. 




In Fig. 16, X = Vh^ - 63 = V(h + 6) (A - 6). 

8. Compute the area of a triangle whose sides are, respec- 
tively, 17.89, 20.96, 19.78. 



Area of triangle =>/«(« — a) (« — 6) (« — c) , where a, &, e are the sides 
of the triangle and s is half the sum of the sides. 



116 ELEMENTARY ALGEBRA 

9. In a right triangle the hypotenuse is 9.675, one leg is 
7.98 ; find the length of the other leg. 

10. Find the area of the surface of a hemispherical dome, 
the diameter of which is 40 feet. 

11. A triangular piece of ground measures along the edges 
315 yd., 541 yd., 479 yd. Find the area. 

12. The volume of a pyramid is one third the product of 
its base and altitude. What is the volume of a pyramid whose 
height is 7.82 ft. and the sides of whose triangular base are 
6.92 ft., 5.83 ft., and 4.91 ft. ? 

EXPONENTIAL EQUATIONS 

127. An exponential equation is one in which the unknown 
occurs in an exponent. 

Thus, 12* =15, a**+'=6 are exponential equations which can 
be solved with the aid of logarithms. 

Since log 12* = x log 12, 

we obtain from 12* = 15, ' x log 12 = log 16. 

Divide both sides by log 12, x = l^i? . 

log 12 

Finding the logarithms, we obtain x = — . 

1 i7fii 

is computed like any other quotient. 
1.0792 y J ^ 

log 1.1761 = 0.0705 

log 1.0792 = 0.0.382 

Subtracting, log x = 0.0373 

X = 1.090, nearly. 

Note that in computations that are only approximate the answer 1.090 
receives a different interpretation than the answer 1.09. The difference 
is this : the answer 1.090 shows that an effort was made to ascertain the 
third decimal figure and that it was found to be 0. On the other hand, 
1.09 means that no effort was made to ascertain the value of the third 
figure. Consequently 1.090 signifies an answer carried to 3 decimals, 
while 1.09 signifies an answer carried to but 2 decimals. 



LOGARITHMS 117 

EXERCISES 

128. Solve for a?; 

1. 7* = 17. 3. 9»+^ = 87. 6. 25*"+» = 795. 

2. 126* = 98. 4. IS*' = 25. 6. 9* = 13. 

PROBLEMS 

129. 1. In what time will $ 1 double itself at 4 %, interest 
compounded ajinually ? 

At the end of the first year the interest is $ .04, the amount is $ 1.04. 

At the end of the second year the interest is $ (1.04)(.04), the amount 
is $1.04 + (1.04) (.04) = I (1.04)2. 

At the end of the third year the interest is $ (1.04)2 (.04), the amount 
is $(1.04)8. 

At the end of x years the amount will be $ (1.04)'. 

We have the equation, (1.04)* = 2. 

x= 17.7 nearly. 

2. In what time will $ 1 treble itself at 4 %, interest com- 
pounded annually ? 

3. A sum of $ 1000 bears 5 % interest compounded annually. 
What is the amount after 20 years ? 

4. Compute the compound interest on $ 879 at 4 % for 13 
years. 

HISTORICAL NOTE 

130. Logarithms constitute one of the most useful inventions in 
mathematics. A great French astroaomer once said that logarithms ^* by 
shortening the labors, doubled the life of the astronomer.*' Logarithms 
were invented by John Napier, Baron of Merchiston, in Scotland, who in 
1(314 published a table of logarithms and described their use. An inde- 
pendent inventor of logarithms was the Swiss watchmaker and astron- 
omer, Joost Burgi, who published a table in 1620, at a time when Napier's 
logarithms were already known and admired throughout Europe. Napier's 
and BtLrgi's logarithms differed somewhat from each other and from the 
logarithms described in this book. Our system of logarithms to the base 



118 ELEMENTARY ALGEBRA 

10 was designed by John Napier and his friend Henry Briggs, conjointly, 
before the year 1617. At one time Napier lived in a beautiful castle on 
the banks of the Endrick. Tet even in such surroundings he was not 
free from annoyances which hindered intellectual work. On the opposite 
side of the river was a lint mill, and its clack greatly disturbed Napier. 
He sometimes desired the miller to stop the mill so that the train of his 
ideas might not be Interrupted. Much of Napier^s time was taken up in 
the management of his estate. Joost BUrgi was not rich, like Napier ; 
the necessity of earning a livelihood greatly hampered his scientific work. 
But both Bttrgi and Napier were men who loved mathematics and were 
able to achieve great results in spite of many hindrances. 






y 



7 



iti 






CHAPTER V 

QUADRATIC EQUATIONS AND THBIK PKOPBKTIXS 

REVIEW 

131. In solving a quadratic equation of the form (j^ ^hx 
-h c = by " completing the square," the first step in our ex- 
planation was to divide both sides of the equation by the coeffi- 
cient of aj*. There are other methods of procedure which possess 
certain advantages ; fractions having large denominators may 
be avoided, or, fractions may be avoided altogether. These 
methods will be explained in § 191. 

In the following exercises solve by any method. To find 
the square roots of numbers, use the tables in § 197. 

1. 6aj2-f-20a; = 17. 6. 35 a?,- 3a^= 11. 

2. aaj» -I- afta; -f- 4 aft = 0. 6. 3aa^-f- 6 a; — 5a* = 0. 

3. 5aj«-h21a;=30. 7. 40a?-3a^ = 12. 

4. oo* -f- o&c -t- a« = 6». 8. (a+6)«*-|-2(a— 6)aJ-f-a=0. 

EQUATIONS QUADRATIC IN FORM 

132. Equations like o^ -f- osff" -f- 6 = are said to be quad- 
ratic in form, because the unknown x occurs only with the 
exponents n and 2n, where one exponent is twice the other. 
Equations of this form can be solved by the methods used in 
solving ordinary quadratic equations. 

1. Solve a;*-7aj*-f-10 = 0. 

Factoring, (a;« - 6) {x^ - 2) = 0. 

a;3 - 6 = 0, «a - 2 = 0. 

From the tables in § 107 we obtain x = ± 2.236, x =± 1.414, approxi- 
mately. 

119 



120 ELEMENTARY ALGEBRA 



2. Solve aJ»-f-3aj»4-l=0. 

Complete the square, a^ + S a^ + { ^^ i 4. | = |. 



x=-y_,±:^. 



As there are in general three cube roots to a number, and 
there are in this ease two different numbers under the radical 
sign, it is seen that there are six roots of the g^ven equation. 
Of these, four are imaginary. As we wrote the answer, only 
the two principal roots are indicated. 

BXBBOISBS 

133. Solve: 

1. ic* 4- 6 a^ - 16 = 0. 6. 6 m« - 23 m» 4- 20 = 0. 

2. a«-2aj*4-l = 0. 7. y»-2y»-143 = 0. 

3. aJ»-h3aj»-hl = 0. 8. 2;* = 266. 

4. a?«-8aj*-20 = 0. 9. y«"-52r + 6 = 0. 
6. 2a?«-7a^-30 = 0. 

RELATIONS BETWEEN ROOTS AND COEFFICIENTS 

134. From the solution of aac* -|- to -|- c = 0, we obtain 



— 6 ± V6» — 4 ac 

X =! • 

2a 
If we designate the two roots by x^ and ajj, we have 

2a 



V6*- 


-4ac 


2 


a 


V62- 


-4ac 



and Am = — 

^ 2a 2a 



QUADRATIC EQUATIONS 121 



The sum of the roots is 






_6 
a 



The product of the roots is 



^^ /- 6 + V6» - 4 acY^ b - V62-4 ac\ 

^^ =(,^^ — 2^ — ^;(, — Ui — ) 

6* — (6* — 4ac) 4ac c 
4 a* 4 a' a 

If we divide both sides of ax^ -f- 6aj + c = by a, we obtain 

a a 

in which the coefficient of a^ is unity. We see that, in the last 
equation, 

(1) The sum of the roots is the coefficient of x with the sign 
changed; and 

(2) The product of the roots is the absolute term. 

To illustrate: Solve ac^ + 6« + 6 = 0. 

The roots are — 2 and — 8. Their sum is — 6, the coefficient of x with 
its sign changed ; their product is + 0) the absolute term. 

From this relation between the coefficients and the roots, we 
may form an equation, if the roots are known. 
Form the equation whose roots are — 1 and 6. 

We assume that, in the required equation, the coefficient of x^ is 1. 
Then, xi + X2 = 6f the coefficient of x with its sign changed ; xix% = — 6, 
the absolute term. 

Hence, the equation isx^ — 6x— 6=0. 
Or, we might write the equation thus : 

xa-(-l + 6)x+(-l-6)=0. 
Whence, ac* — 6 a; — 6 = 0. 



122 ELEMENTARY ALGEBRA 

135. Form the equation whose roots are : 

1. + 5, 4- 2. 6. 2^, 3|. 9. V2, - V3. 

2. -3,-8. 6. -i, -f 10. I4-V2, 1-V2. 

3. 4,-6. 7. —a, —6. 11. — V5, — VS. 

4. 7, 0. 8. m2, - 3 m\ 12. |V2, - 1 V2. 

If you know one root of a given quadratic equation, how 
can you calculate the other: (1) from the coefficient of x, 
(2) from the absolute term, (3) by the Factor Theorem ? Use 
each of these methods in turn, to find the second root in each 
of the following equations : 

13. aj2 — 8.7 05 -I- 17.6 = 0, one root 5.5. 

14. aj2 — 2.1 a; + .9 = 0, one root 1.5. 
16. a;2 — 1.7 a; — 4.8 = 0, one root 3.2. 

16. a;2 4- 11.8 X + 34.17 = 0, one root — 6.1. 

17. aj2 -f 3.6 a: - 25,92 = 0, one root - 7.2. 

18. a;2 — .085 x + .00175 = 0, one root .05. 

NATURE OF THE ROOTS 

136. We shall discuss the nature of the roots of the quad- 
ratic equation asc* + 6» -f c = 0, in which the letters a, 6, c are 
assumed to be real and rational numbers* When some of 
these letters are imaginary or irrational, the conclusions which 
we shall draw do not necessarily hold. The solution of 

ax2 -j- 6aj 4- c = is . 

— b± Vb^ — 4 ac 
x = • 

2a 

The nature of the roots depends upon b^ — 4:acy which is 
called the discriminant. 

I. When b^—4:ac=0, the roots are real and equal. Explain. 

II. When 62 _ 4 etc > 0, the roots are real and uneqtuil. If 

62 — 4ac is a perfect square, both roots are rational; if 



QUADRATIC EQUATIONS 123 

6* — 4ac is not a perfect square, both roots are irrational. 
Why? 

III. When 6* — 4 ac < 0, both roots are imaginary. Explain. 

To iUustrate: 1. In as^- lOx + 26 = 0, 6^ _ 4 ^j^ = 100 - 100 = 0. 
Hence the roots are real, rational, and equal. The roots are 6, 6. 

2. In flr2 - Sjc - 28 = 0, 62 «. 4 ^ - 9 _|. 112 = 121. Hence, the roots 

are real and unequal; since 121 is a perfect square, both roots are 
rational. 

3. Inx«-2a: + 6 = 0, 63 — 4ac = 4 — 24 =- 20. Hence both roots 
are imaginary, 

EXEBOISBS 

137. Determine, without solving, the nature of the roots of : 

1. aj2-aj-12 = 0. 6. 7a;2- 12ajH-4 = 0. 

2. iB24.8a;-16 = 0. 7. 4 «2 - 12 «'= - 9. 

3. a.2__8ic-25=:0. 8. 4 aj2- 12 a? -55 = 0. 

4. 5aj2-f.7a; + l = 0. 9. a?2_2a;H-3 = 0. 

6. 2aj2 4-3a; = 8. 10. 3aj2-f.l3a:-f-l = 0. 

For what values of a are the roots of the following equa- 
tions equal ? Real and unequal ? Imaginary ? 

11. aaj24.4a; = — 1. 13. 3a«-h3aj + a = 0. 

12. aj^^ cue 4- 5 = 0. 14. ax'^ -{- ax -\- 2 = 0. 

GRAPH OF THE QUADRATIC EQUATION, ax^-^bx-^c = y 

138. The value of the expression, ooj* -^ bx-\-c, for given 
values of a, 6, and c, depends upon the value of the variable x. 
For that reason the expression is called a function of x. If we 
write aoi^ -f- fta? + c = y, then for every value of x there is a 
corresponding value of y. "^ 

The values of x and the corresponding values of y may be 
taken as the coordinates of points on the graph of the equation 
aa? -h 6a5 4- c = y. 



124 ELEMENTAHY ALGEBRA 

1, Draw the graph of a!* — 4 a! — 5 = y. 



" 


V 


Point 




16 


A 




7 


B 







C 




-6 


D 




-8 


B 




-9 


F 




-8 







-6 


S 


-1 





I 


-2 


7 


J 


-8 


16 


. K 



ThuB it will be seen (Fig. 17) that the graph 
crones the x-azis at the points (5, 0) and 
(-1,0). Upon solving ib«-4i-6^0, we 
find tbat X = + 6 and — 1, and these an th« 
values of x which make ^ = 0. 



QUADRATIC EQUATIONS 



2. Draw the graph of a? - 



- 


f 


Poinl 


6 




A 


6 
i 




B 

C 


3 




D 


2 




E 


1 




F 











In rig. 18 It is seen that tbe grapb touches 
the z-axis, but does not ctobs it. The graph 
tumB and proceede upward. 

Instead of two points of intersection with 
the x-asis, as in Fig. IT, there is la Fig. 18 
one point o( contact; the two points have 
UDlted in a single point. Evidently there are 
two roots at that point. Upon solving the 
equation we find that z = 3, S ; 3 is called a 
doable root. 



BLBMBIfTARY ALOBBBA 



3. Diaw the graph of a^ — 2x+ 3 = y. 



=« 


* 


Point 


6 


18 


A 


4 


11 


B 


3 


6 


C 


3 


8 


D 


1 


2 


E 





8 


F 


_ 1 


6 





-2 


11 


H 


-3 


18 


I 



Fig. 19 UluHtratea the fact that not eyeiy 
gnph cati the z-axis. 

K we solve z> - a x 4- 8= 0, we find that 
K = 1 ± v'^. Now V^^ l8 an imaginary 
Dumber ; therefore, there U no real valae of 
z which will make x> - 2 x + 8 equal to 0. 

The grapb makes this fact evident at a 
glance. 

We see that the two niota of the eqoaUon 
ax* + bz -1- c = may be real and unequal, as 
in Fig. IT, leal and equal, as in Tig. 18, oi 
imaginary, as In Fig. 19, 



CHAPTER VI 



SYSTEMS OF EQUATIONS SOLVABLE BY QUADRATICS 

I. ONE EQUATION IN THE SYSTEM IS LINEAR 



139. Solve 



aj« + 3/2 = 26. 
« - 3/ = 1, 



(1) 
(2) 



Let us solve this system, first graphically, then alge- 
braically. 

The equation a; — y = 1 is linear and is represented by a 
straight line. The graph of aj* -f- y* = 25 is a circle. 



X 


-y = 


1 


X 


y 


Point 




1 


-1 




A 
D 



x^ 


+ y2 = 


= 26 


X 


y 


Point 


6 





C 


4 


±3 


D,E 


3 


±4 


F, G 





±6 


H,I 


-3 


±4 


K,L 


-4 


i3 


M, N 


-6 









Fig. 20 shows the graph of x^ + y'^ = 26, (1), and « - y = 1, (2). The 
line (2) intersects the circle in two points (—3, — 4) and (4, 3). 

The coordinates of these points satisfy both equations and are there- 
fore the values sought ; namely, 

05 = 4 and y = 3 ; x = — 8 and y = — 4. 

127 



ELEMENTARY ALGEBRA 



140. Observe that x and y, when subject to the one condi- 
tion a? + t^ = 25, are variables, because they are capable of 
taking on successively a never ending aeries of different values. 
For the same reason x and y are also variables when subject to 
the one condition x~y = l. But x and g can no longer aa- 
same successively a never ending series of different values, 
■when they are subject at the same time to both conditions^ 

('' ~ ( . Hence x and y are now no longer variables, 
x^y=\ f 

but constants. It is the purpose of the solution to find the 

values of these constants. 

In drawing graphs we are dealing with variables; in 

finding the points of intersection of these graphs, we are 

finding the values of constants which, at the outset, are 

unknown. 



SYSTEMS OF EQUATIONS 129 

141. The algebraic solution of 

aj« 4. y« = 26 (1) 

x-y = l (2) 

can be effected easiest by the method of suhstitutixm. 

From (2), x = 1 + y. 

Substitute in (1), 1 + 2 y + y2 _|_ ^a = 26. 
Whence, y^ ^ y _ 12 = 0. 

(y + 4)(y-3)=0. 

y = — 4, or + 3. 
Substitute in (2), x = - 3, or + 4. 

The two sets of roots are, ac = — 3, y = — 4, and as = 4, y = + 3. 

In checking, substitute the values of x and y in both of the given 
equations ; the answers might be wrong, yet might satisfy one of the two 
equations. For instance, x = 3, y = 2 will satisfy (2), but not (1). 

If in Fig. 20 the line should move, parallel to its present position, until 
it were tangent to the cirele, we would have just one set of values satis- 
fying the pair of equations. Show that this is the case, if the equations 
are 052 ^ y2 ::= 26 and ac — y = 6 V2. 

If the line should not intersect the cirele, nor be tangent to it, we 
would have a set of imaginary values satisfying both equations. Show 
that this is the case, if the equations are x^ + y^ = 25, x — y = 10. 

142. Instead of elimination by substitution, some special 
device is frequently used. 

1. Solve ic«-y» = 5, (1) 

x-y = -b. (2) 

Divide (1) by (2), x + y = - 1 (3) 

x-y=-6 (2) 

Add (2) and (3), 2 x = - 6 

X =-3 
Subtract (2) from (3), 2 y = 4 

y = 2. Ans. x=— 3, y = 2. 



ELEMENTARY ALGEBRA 



'- 


-v=- 


6 




j?_^ = 


S 


« 


V 


Po[nt 




- 


V 


Points 



-5 


6 



A 
B 


3 
-8 

« 
-4 




±a 

±2 

±3.3* 

±8.3* 


A,Ai 
S,Bi 
C, C, 

E,E, 











Ezplaiu bow Fig. 

21 exLibita the values 
of X and y which 
nitisEy eqnations (1) 
and (2). 



Square (1), 
(2)- 4, 

Add (8) and (4), 
Extract square root, 

Add (6) and (1), 

Snbtroct (1) from (6), 

The MlB of roots are 



iEys= — 5, 
-ixv + y^ = 36. 



x' + 2aK + v^ = 16. 
x + y = 



a) 

(2) 

(S) 

(<) 

(5) 
(«) 
(1) 



SYSTEMS OF EQUATIONS 131 

8. Solve a' + y» = 20, (1) 

xy = 8. (2) 

MulUpl? (2) by 2, 2 a^, = 16. (3) 

Add (1) and (8), i" + 2 37 + k^ = 36. (4) 

Subtract (1) and (S), a? - 2 xy + v* = 4. (6) 
Extract square root of (4) itnd (6), 

Whence, 2a:=±8, ±4, 

a: =±4, ±2. 
2y=±i, ±8, 

If = ± 2, ±4. 
The sets of rootfl are, i =± 4, v = ± 2; a: =± 2, v = ±*- 
The graphic soIuUon is exhibited in Fig. 22. 



132 



ELEMENTARY ALGEBRA 



X^ 


' + y« = 20 


X 


y 


Point 


±4.47 





A,Ai 





±4.47 


B,Bi 


2 


±4 


C, Ci 


-2 


±4 


A A 


3 


±3.3+ 


^, ^1 


-3 


±3.3+ 


F,Fi 


4 


±2 


a, Gi 


-4 ±2 


H,Hi 





jcy = 8 




X 


y 


Point 


±2 


±4 


a, A 


±3 


±2} 


/, /i 


±4 


±2 


^,^1 


±5 


±^ 


A A 



4. Solve 


aj«-y» = 91, 






a? — 3/ = 7. 


t 


Divide (1) by (2), 


x^^-xy-\-y^ = 13. 




Square (2), 


052 - 2 ay ± y2 -- 49. 




Subtract (4) from (3), 


3 xy = - 36, 
3cy=-12. 




Add (5) and (3), 


a;2±2xy±ya = l. 




Extract square root 


a5 + y = ± 1- 
05 — y = 7. 




Whence 


2 X = 8, 6, 
X = 4, 3. 




And 


2y=-6,. 


-8, 




y=-3, . 


-4. 


The sets of roots are 


aj = 4,y=-3; 


x = 3,y 


6. Solve 


aj*-h 2^ = 97, 
a; 4- y = 5. 





=-4. 



Let x = t« + r; y = w — t>. 

Substitute in (1) and (2), 
«* + 4 m8» + 6 mV 4.4Mt>8 + v*±t«*-4M8v±6 m%2_ 4 u«« + «* 

Whence, 2 u* + 12 u2»2 + 2 1;* = 97, 

and u±t> + M — 1> = 6. 

Whence, 2 u = 6, 



(1) 

(2) 

(8) 
(4) 

(6) 

(«) 
(7) 

(2) 



(1) 
(2) 



97. (3) 
(4) 
(6) 



SYSTEMS OF EQUATIONS 133 

Substitute u = J in (4), a|A + 75 »2 + 2 r* = 97. (6) 

Whence, 16 r* + 600 »2 = 151. 

Completing the square, 16 «* + 600 1?2 4 5625 = 5776, 

4v« + 75=±76, 

4 1?2 = 1, - 151, 

t^ = i -H^» 

Hence, « = 3, 2, J ± J V- 161, 



The sets of roots are 



y = 2, 3, J q: i V- 151. 



05 = 3, y = 2; x = 2, y = 3; a; = J±J V- 151, y = J ^ } V - 151. 
Check each set of real roots. 

The check for the imaginary roots is as follows : 



(1) 3ff.^ i|i V- 161 - u^fiA T ^i^ V- 151 + ^¥<P + W 



:p ijA V-IST _ xi|2A ± ij4 V- 161 + »3^ = 97, 

i|Ji = 97, 
97 = 97. 
(2) jijVirirH-f TiV^=T5l = 5, 

5 = 5. 

143. The equations aj* -|- ^ = 97 and x-\-y = 5 are called 
symmetrical equations, because they remain unaltered when x 
is written for y and y is written for x. 

An expression is symmetrical with respect to two or more 
letters, if it remains unaltered when the letters are inter- 
cttanged. 

Every system of two equations that are symmetrical, or sym- 
metrical concept for the signs, can be solved by assuming x^u-^-v 
and y = u — V, Instances of equations symmetrical except 
for the signs are a* — y* = a, a — y = 6. 



134 ELEMENTARY ALGEBRA 

II. A SYSTEM OF TWO EQUATIONS IN X AND F, BOTH 

QUADRATIC 

144. Any tioo equations of the form aa^ ■i'by^=c,or aa? + hxy = c, 
or oo* + bxy -\- cy^ == d, can be solved by assuming y = vx, and 
determining the constant v. 

It is seen that in these equations all the terms which contain x and y 
are of the second degree in z and y. Some authors call these equations 
homogeneous. We avoid this term for the reason that the word homoge- 
neous is more commonly used to designate equations like ax^'^hxy-\'Cy^=0, 
in which every term of the equation contains x and y^ and is of the same 
degree in x and y. See § 8. 

Solve a^-\-xy = — l, (1) 

y'-2xy = S. (2) 

Let y = vx. 

Then x^ + rx* = - 1, (3) 

and fAc^^2vx^ = 8. (4) 

From (3), (1 + v)x^=-h and «» =-^J-. (6) 

1 + 1? 

From (4), («2 - 2 v)x^ = 8, and x^ = — ^ (6) 

•• i + t,-t^-2i? . ^-^ 

Whence, - «« + 2i? = 8 + 8», (8) 

and t^ + 6i? + 8 = 0, (9) 

and (v + 4)(i? + 2) = 0. 

Then v = - 4, - 2. 

Substitute in (6), x^=^ = l, and a;2 = — = 1, 

K=±jV3, ±1, 
y=zvx = Tiy/S, T2. 
The two sets of roots are a; = ± J VS, y = =F J\/3 ; x =±1, y = T2, 

To find the value of y care must be taken to use the value of v with 
that value of x which was obtained by substituting the value of v. 

When — 4 was substituted for v in (5), yielding x = ±}V3, the — 4 
must be multiplied by ± J\/3 (since y = vx), to obtain the correspond- 
ing value of y. 

Hence the values of x and y must be carefully paired as shown above. 

When ± and =F are used in the answers, the upper signs go together 
and the lower signs go together. 



SYSTEMS OF EQUATIONS 135 

145. The method just explained possesses the great advan- 
tage of always yielding results. Very often, however, much 
shorter solutions can be given by special devices. Frequently 
a third equation can be derived from the two given equations 
which is simpler than one or both of those given. The solu- 
tion is then obtained by the use of this simpler equation along 
with one of the original ones. 



For example, solve a* — 3 ajy = 


-143, 


(1) 


2^ + ajy = 


168. 


(2) 


By adding (1) and (2) the shnpler equation is obtained, 






x^ -2icy-hy2=26. 




Extract the square root of both sides, 


« — y = ± 6. 


(8) 


(3) is a linear equation. It gives 


X = ± 6 + y. 




Substitute in (2), 


y2±5y4-y2=168, 
2y^±6y = l6S, 




Taking the upper sign in ±6y, 


_ 6 ± V1369 




Taking the \ower sign in ± 6y, 


y = ^'r'' 




Hence, 


y = 8, -8,V. -¥• 




Substitute the values of y in (2), 


a: = 18, -13, Y, -^. 





The sets of roots are a; = ±13, y = ±8; x = ±^,y=±^. 

POSSIBILITY OF SOLUTION BY QUADRATICS 

146. It is readily seen that when, in a system of two equa- 
tions, one equation is of the second degree and the other is of 
the first degree or linear, a solution may always be obtained 
by quadratics. 

If, however, both equations are quadratics, this is usually 
not the case. Only special types of quadratic equations, such 
as have been studied in this chapter, and others of similar 



136 ELEMENTARY ALGEBRA 

character, admit of being solved by quadratics. The general 
case is far more complicated. Given 

'^ aal^ + bxy -\- q/^ -\- dx -\- ey -\-f= 

and Oiic* + b^xy + c^ + d^x + e^y +/i = 0, 

to find X and y. If y is eliminated by substitution, the result 
is a quartic equation, which is of the fourth degree and cannot 
be solved by quadratics, except in special cases. It is shown in 
more advanced algebras that the algebraic solution of a quartic 
equation depends in general upon the solution of a certain 
cubic equation or equation of the third degree. The solution 
of cubic and quartic equations is not explained in this book. 

The given equations of the second degree in x and y can be solved by 
quadratics whenever the auxiliary cubic equation here mentioned possesses 
a rational root. This rational root can be found by the factor theorem 
(see § 78) ; the other two roots of the cubic can then be found by quad- 
ratics, as can also the four sets of roots of the given equations. (See 
F. Cajori, Theory of Equations, New York, 1914, pp. 71-73.) 



BXBBCISBS 



147. Solve 



1. 2 ic* — 3 a?y = 5, 6. ic* + 3 a?^ = — 5, 
oj — y = 2. 2xy + y^ = -'l^, 

2. a?-|-y = l, 7. a + 2/ = a, 

iK* + y2 = 85. 4:xy — a^=z — b^, 

X y 15' x^ — y^ = 4: mn. 

1_^1^_34 9. aj*-y* = 2401, 

a^ f 225' 352^2/2^49. 

4. aj» H- y» = 28, lo. ar' + 7 a?y = - 104, 

a + y = -2. 5xy-'y^=:-129. 

6. xy = 45, 11. a5 + y* = 33, 

aj + y = 14. x + y = 3. 



SYSTEMS OF EQUATIONS 137 

12. x-y = ly 15. aj3 -I- y3 «. 109^ 

a , y _. 2 J 05*^ + ojy* = — 36. 

13. L-3y = 0, "• f-y'=10. 

7a^ + 3a,-4.V = 43. 2a- + 6;.^ + 33^' = 14. 

14. a:^-'y^ = m^y 17. 3aj2-32/2 = 9, 

18. a^—7x + y^—7yy 
3(a + 2/) = -2icy. 



PROBLEMS 

148. 1. Prices of two kinds of bicycles are such that 7 
of one kind and 12 of the other kind can be obtained for 
$640. Three more of the latter can be purchased for $180 
than can be purchased of the former for $120. Find the 
price of each. 

2. A person lends $ 2500 in two separate sums at the same 
rate of interest. At the end of one year the first sum with 
interest amounts to $997.50; at the end of two years the 
second sum with (simple) interest amounts to $1705. Find 
the two separate sums and the rate of interest. 

3. Prove that if the sum of two real numbers is multiplied 
by the sum of their reciprocals, the product cannot be less 
than 4. 

Let X and y be the two real numbers, p the product considered ; 
solve for the unknown ratio -. 

y 

4. A traveler starts from A toward B, another traveler 
starts at the same time from B toward A. In two hours they 
meet 20 miles from A. When the second traveler arrives at 
A, the first is still 13^ miles from B. Find the distance be- 
tween A and B. 



138 ELEMENTARY ALGEBRA 

6. If in aoc^ + to -|- c = 0, the coefficients are related to each 
other in such a way that a -h 6 = ^ and a = 2 c, what must be 
the value of a, so that 8 will be a root of the given quadratic 
equation? 

6. The difference between two numbers is 12, and the 
difference between their cubes is 7488. What are the two 
numbers ? 

7. A sum of money at simple interest for four years 
amounts to $2240. Had the rate of interest been 1 % higher, 
the sum would have amounted to $80 less than this in two 
years. Find the capital and the rate. 

8. The sum of the areas of two circles is 694.2936 sq. in. ; 
the sum of their radii is 21. Find their radii. 

9. A circular track is constructed so that the width of the 
track is ^ of the inside diameter. The area of the track is 
2600 sq. yd. What are the inside and outside lengths of the 
track? 

10. Find two numbers such that their sum is equal to their 
product and also to the difference of their squares. 

11. Find two numbers such that the sum of the numbers is 
equal to the sum of their squares and also to twice their 
product. 

12. Find two numbers whose sum is 8 and whose product 
is 26. 

13. Find three numbers such that the sum of the squares of 
the first two is equal to three times the square of the first minus 
the square of the second, and is also equal to the first minus the 
second plus twice the third, and also to the sum of the three 
numbers. 



CHAPTER VII 

EXPONENTS, RADICALS, IMA6INASIES 

MEANINGS OF DIFFERENT KINDS OF EXPONENTS 

149. The different kinds of exponents which have been stud- 
ied thus far have been interpreted in the following manner, m 
and n being taken to be positive iategers : 

a" = a • a • a ••• (to n factors), 

1 -- 1 

a~* = -^i or more generally, a " = — ;;;-, where a ^ 0, 

a •» 
a° = 1, where a =^ 0, 



While a number a has in general n different nth roots, it is agreed that 

1 

only one of them shall be represented by a** or Va, namely, the so-called 
principal root This restriction was made in order to avoid unnecessary 
complication and confusion in the interpretation of expressions and equa- 
tions involving radicals. Accordingly, Vi = + 2, — Vi = — 2, \^ = + 2, 
-\/8=-2, \/^^ = -2, -v^^=4-2, v^=+2, etc. 

The exponents considered above are all rational numbers. But other 
exponents have been brought to our attention. In the study of logarithms, 
mention was made of the fact that logarithms are often irrational numbers. 
Since logarithms are realjy exponents, it follows that exponents may be 
irrational. In more advanced books still another kind of exponent is 
considered, namely, the exponent that is an imaginary number.* 

* For a fuller treatment of the theory of exponents, consult H. B. Fine, 
College Algebra^ 1904, p. 376. 

139 



140 



ELEMENTARY ALGEBRA 



OPERATIONS WITH EXPONENTS 

150. Operations involving exponents are subject to the fol- 
lowing laws : 

1. Law of multiplication : a"^ -a'* = a"*"*^. 

2. Law of division : a"^ -^ a"" = a"*"". 

3. Law of involution : (a*)* = a"»*. 

1 m 

4. Law of evolution : (a") ** = a" . 



DIFFERENT KINDS OF NUMBERS 

151. In the study of arithmetic and algebra several different 
kinds of numbers have come to our notice. These may be 
classified as follows : 



Numbers 



Real 



(Classified according to signs) 



(Classified according to express- 
ibility in Hindu-Arabic nu- 
merals) 



Positive 
Negative 

Rational 
Irrational, as 

V6, ir = 
3.14169... 



Imaginary 



Pure Imaginary, as hV— 1, where &^0 
Complex Numbers, as a-\-hy/— 1, where 6^0 



Arithmetic deals with real numbers, all positive, but some of 
them irrational. 

The irrational numbers of arithmetic arise in the process of 
finding roots. 

In algebra use is made of the numbers encountered in arith- 
metic, but convenience forces upon us the need of considering 
also negative numbers for the purpose of indicating relations of 
opposition, as temperatures above or below a certain fixed point, 
debts or assets, distances to the right or left, etc. In the solution 
of quadratic equations we meet a still different type. If we try 
to solve ic2 -|. 1 = 0, we are confronted with the symbol V— 1. 



EXPONENTS, RADICALS, IMAGINARIES 141 

This result greatly embarrassed mathematicians of the eight- 
eenth and of earlier centuries ; no satisfactory explanation of 
it could be made at first. But now the symbols are recognized 
as constituting a new type of number, the so-called imaginary 
number, which deserves a legitimate place in algebra and is of 
great service in certain advanced developments of algebra, that 
are useful in the study of polyphasy electric currents and of 
other advanced topics in mathematical physics. Just as nega- 
tive numbers have been found truly useful in elementary 
algebra, so imaginary numbers have been found useful in ad- 
vanced algebra. The symbol V— 1, or this multiplied by 
any real number 6, such as 6V— 1, is called a pure imagi- 
nary number. 

Expressions of the type a ± 6V— 1, which are the sum 
or difference of a number a and of a pure imaginary 

b^/— 1 (a and b being any real numbers, but b ^ 0), are 
called GfmSplex numbers. The term " complex " was intro- 
duced because the parts of the expression are partly real 
and partly imaginary. 

SIMPLIFYING RADICALS 

152. A radical is said to be in its simplest form : 

(a) When the index of the root is as small as possible, 

(b) When the expression under the radical sign, called the 
radicandy is integral, 

(c) When the radicand contains no factor with a negative ex- 
ponent, or raised to a power equal to or greater than the index 
of the root. 

-\/25 is not in its simplest form, because the index 4 of the root is not 
as small as possible ; we have y/2b = 6* = 5* = V5. 

^\ is not ill its simplest form, because the radicand, \, is fractional. 

y/c^ is not in its simplest form, because the exponent 4 is greater than 
8, the index of the root ; we have ■\/a*6 = ay/ab. 

\/3 ab^ is in its simplest form, since it fulfills all three conditions. 



142 ELEMENTARY ALGEBRA 



1. Simplify J^^. 

3 gi gj 8 3 * 

Notice that the denominator was rationalized by multiplying numer- 
ator and denominator by 8^. 

2. Simplify </3 a*b-^c. 

6* ft ft 

3. Other illustrations of simplifying are : 

'^7a« ^ 3^ g* _ 3^ ■ q ■ aM _ a j/Wab^ 
V72 - 36 VS = 36^(2 - v^)* = 6 V2 - V3. 



BXBBOISBS 

153. Remove all negative exponents, compute the values, and 
simplify the expressions : 

1. 27*. 4. 16"*. 7. 64"*. ^^' (A)"'- 

2. 16* 5. 64-i. 8. (!)-». 11. (.16)*. 

3. 27-*. 6. 64^. 9. 8"*. 12. (.125)*. 

13. (_| + 6)0. 16. (100«+10)-^ ^g gm-n ^ g-mft 

5-«-25-\ 

14. 70x(.26)-*. 17- — 1 20. ^""^ 

16. (.49/-'. 18. g'-^-g'". 21. «:l±«^+l. 



EXPONENTS, RADICALS, IMAGINARIES 143 



22. V60. 25. V2000. 28. VST. 31. V- 800. 

23. Vm. 26. 7V20. 29. \/48. 32. v^729. 

24. V98. 27. 12 V8. 30. ^"08. 33. \/2450. 

34. 2V6x5>/3. 38. 3Vlbx24V6-f-Vl5. 

36. 5V2x7V6. 39. 3 ViO X 2 Vl5 -4- 3 V2. 

36. 2Vl6 X 5V2 X 2V3. 40. 5Vl6 X 3V6 -4- 2V5. 

37. 2Vl5-j-5V3x2V2. 41. (6.25)-*. 

42. (2.25)1 46. (.216)1 60. * (^)*. 

43. (.25)"*. 47. (.0625)-*. 61. ~ (6.25)i 

44. (.125)"*. 48. (-.125)*. 62. (a*6*)«. 
46. (-.008)-*. ^49. (fj)*. 63. (a26»)-i 

64. (a-«6-»)"'^. S6. c^-^c^xcK 

66. a^-^a^y. a*. 67. 2* x 64* -4- 729*. 

Eationalize the denominators of the following fractions and 
then find their values to three figures : 

58. -i-. Hint. J-=-i..^ = ^. 69. ^- 



V2 V2 V2 V2 2 2V3 

60. -4=- 61. -4=r- 62. V|. 63. ^^^. 

Vl4 V2.3 V:22 

Calculate the values of the following expressions when 
m = V2, n = V3, p = V5, using the tables in § 197 : 

64. 7 m* — n*. 67. phi^ — nha\ 70. (n+J>)w. 

66. 1©* — wiV. 68. — -^ 71. ^ 

^^ 2p pn 

« • . « s «A wi — w »»« ^* — ^** 
66. mH^-\-rM. 69. ^ ^ - 72. 

5jp* — mnp 



144 ELEMENTARY ALGEBRA 

78. Which of the expressions is the larger, V5 or v/lT ? 

V6 = 5* = 5* = 126^ 
\^ = 11*=11* = 121*. 
But 126^ > 12li Hence V5 is the larger. 

Arrange in the order of magnitude in each example : 

74. \/ii, </2l. 77. 2, Vily v^. 

76. V2, -v^^. 78. •^, ^/Vf, VO. 

76. VO, \/9. 79. \/3, -5^, vT7. 

80. Using logarithms, compute the value of y/7943. 

Log 7943 = 3.9000. 
Log \/7943 = 1.8000. 

v'mS = 19.96, nearly. 

81. Using logarithms, compute the value of x =(59.45)"^. 

(59.46)"* = — ^ 



(69.46) 



I 



Log 1 = 0.0000 



Log 69.46 = 1.7742. 

Log (59.46)' = j(1.7742) = 1.1828 

Log X = 2.8172 

X = .00664, nearly. 

Compute the values of the following radicals : 

82. ^/96S, 84. -^790. 86. (154)"*. 

83. VTOSS. 85. (12)A 87. (376)J. 

ADDITION AND SUBTRACTION OF RADICALS 

164. The indicated additions and subtractions of radicals 
can be reduced to one term, only wh^n the terms in the ex- 
pression are similar after they are simplified. 

Thus, \/l8 + V60 = SV^ + 6 V2 = 8 V2; 



EXPONENTS, RADICALS, IMAGINARIES 145 

EXBBCISBS 

155. Perform the indicated operations : 

1. V63+\/784. 3. A/24 + </8i+\/375. 

2. V294+-v^576. 4. 50\/^^+<^7000-^56. 



6. Vr^ - V9a^ - V16 a^dP. 

6. 6V3+8Vf-|-V^. 

7. 4V^-|-15v^-60V|. 

10. w3\/--«S^+wV\/-. 
MULTIPLICATION OF RADICALS 



\ 



156. E.eal radicals of the same index are multiplied to- 
gether according to the formula 

a -y/b X c -y/d = ocVftd. 

When the real radicals have different indices^ it is easiest to 
change to the exponential notation, then perform the multipli- 
cation and, after reducing fractional exponents to a common 
denominator, return to the radical notation. This may be 
shown as follows : 



m n 



Special care must be exercised when the expressions to be 
multiplied involve imaginary numbers. It is customary, for 
brevity, to represent the pure imaginary V— 1 by the single 



146 ELEMENTARY ALGEBRA 



letter. t. Hence the complex number a+V— 16 may be 

written a -f ib. 

The following has been adopted as a definition of V— 1 or f : 
7%e number V— 1 is one whose square is — 1. 



Accordingly, (V— 1)* = i* = — 1. 

All multiplications involving imaginary numbers must be 
made to conform with this fundamental definition. 

Particular care must be exercised to avoid the following procedure: 



V^n. times \/^n[=V(-l)(-l) = >/TT= 1. 



This procedure is in violation of the definition of V— 1 given above, 
and must therefore be avoided. Always take V— 1 x V— 1 = — 1, or 

The multiplication of expressions involving imaginaries can 
be performed easiest^ if we take the precaution to vrrite a radical 

V— a in the form V— 1 Va or i Va. 

For example, 

V— a X V— 6 = i Va X I* V^ = i^ Voft = — VoS. 



1. Multiply 2 Va - 3 V26 + 4 V3 a6 by 5 Va6. 

2 V5-3\/26 + 4 V3a6 



10aV6-166 V2a + 20a6V3 



2. Multiply Va + 2 V- 6 - 3 c by V-a6. 

Va + 2f>/6-3c 
tVa6 



ai 



iVb^2hy/a-Sciy/ab 



EXPONENTS, RADICALS, IMAGINARIES 147 



BXBBOISBS 

157. Perform the indicated operations and simplify the 
result : 



1. V5.V125. 18. V2^^-V2T«. 

2. \/3-\/9. 14. ^^r^.vsTft. 

8. SVa.SVoft. 15. (V2a+V36)». 

4. -^cC^-i-y/a^xVaK le. (a+V6)». 

6. -^ -!- Va X \/a». 17. (a + V^^)*. 

6. aV^x26V^=n:. 18. Vab-^-Va. 

7. 3i.4iV5. 19. </^^Va6. 

8. 4iVc"j-2t. 20. SVx^^'^V23r^. 

9. 5V^V3x6V^^. 21. V^^^a6-4-V^=^ 

10. - M2xV|. 22. iaXi6V2xic. 

11. (^a-bVcy. 23. V^Ta . v"=^ . V^=^. 

12. — V2a' X V2a&. 24. t V« • t V^ • * Vy2. 
26. ^J^^-g X \/^^ X ^v^^=^. 

26. (2Vi — 3Vy+V2«)-5Vc^. 

27. 6^^(2V«+Vy-«). 

28. (2V2-3V^^H-^v/"=^-2</2)\/2. 

29. (2V5-</6-2V^2. 

80. (Vs-sV^^)*. 

81. Change to an equivalent fraction having a rational 

V3+V2. 



denominator, 



V3-V2 



If >/3 ->/2 is multiplied by VS + V2, the product is (>/3)2-(V2)2 
or 3 — 2. Hence, multiply both numerator and denominator of the given 
fraction by VS + V2. We obtain 

(V3+V2)2 ^ 8 + 2V6-h2 ^ 6 + 2v^ ^g ^ g^/g 
(V3)2_(v^)2 3-2 1 



148 ELEMENTARY ALGEBRA 

Change to equivalent expressions having a rational denomi- 
nator: 

82. ' ^ . 34. ^-^-A 86. t±^. 

V3-VO Va-Vft 2-hV3 

33. — 1 36. ^■^^. 37. ^ 



V64.V5 V2-V6 2V5-3V3 

38. Is — 1 -h V5 a root of the equation aj2 4-2aj — 4 = 0? 

Write - 1 + V6 for X, 

(_ 1 4. V6)a + 2(- H-V6)- 4 = 1 - 2V6 + 6 - 2 + 2V6 - 4 = 0. 

Hence, — 1 + V6 is a root of the equation. 

Is the binomial written after each equation a root of that 
equation ? 

39. aj»-4aj-l = 0, 2-hV3. 

40. aj2-a;-3 = 0, |(l-Vi3). 

41. a;2-h3a;-hl = 0, 3-hV5. 

42. 2aj2-aj-2 = 0, i(l-Vl7). 

THE SQUARE ROOT OF a±2y/b 

158. Since ( V3 ± V2)«= 3 ± 2 V6 + 2 = 5 ± 2 V6, it follows 

that V6±2V6 = V3± V2. 

We notice that 5 is the sum of 2 and 3 ; 6 their product. 
This suggests the following rule for finding the square root of 

a-h2V6: 

Find two numbers whose sum is a and whose product is b. 
Write down the square root of the larger number^ plus (or minus) 
the square root of the smaller. 

This rule is of practical use only when the two numbers 
sought are rational. In other cases the resulting expression is 
more complicated than the given expression; hence to be 
rejected. 



EXPONENTS, RADICALS, IMAGINARIES 149 

1. Find the square root of 8 + V^O. 

Writing the expression in the form a ± 2Vbj 8 + 2\/i6. 
The two numbers whose sum is 8 and product 16 are 6 and 8. 

Hence, V8+V60 =Vb + VS. 

2. Find Vg - 4 V2. 

Changing to the form a ± 2 V6, 6 — 2v^. 

The two numbers whose sum is 6 and product 8 are 4 and 2. 

Hence, Vo- 4V2 = Vi - v^ = 2 - V2. 



BXBBCISBS 

159. Find the square root of : 

1. 3-h2V2. 7. 9-h6v^. 

2. 7-4V3. 8. I-V2. 

3. 9-h2Vi4. 9. t^-VJ. 

4. 9 4-2V20. 10. X — Vaj» - 1. 

6. 16 4-6V6. 11. (a2-h4 6)-4aV6. 

6. 8-h2V7. 12. x-^2y + 2V2xy. 

IRRATIONAL EQUATIONS 

160. An irrational or radical equation is an equation in which 
the unknown number x and expressions containing it occur 
under radical signs or with fractional exponents. 

For example, 2 — V« = Va? — 1, aj^ -|- 5 «» = 6. 

It is agreed among mathematicians that in equations of this 
sort, only the principal roots ot^/x, Va; — 1, x^ are to be taken. 

The solution of irrational equations may be explained by 
examples. 



150 ELEMENTARY ALGEBRA 



1. Solve 2 - Vic - Va? - 1 = 0. 

Solution. Transpose one of the radicals, 2 ^ Vx = y/x— 1. 
Square both sides, 4 — 4 V« + x = x — 1. 
Simplify and transpose to the right all 

terms except 4 Vie, 4>/x = 5. 

Sqoare both sides again and solve for x, ^ = il' 

Substitute in the given equation, 2 — Vf | = V{J — 1, 



Hence x = f | is the value sought. 



»=t 



2. Solve 1 -h Va; -h 3 = V«. 

Square both sides, 1 + 2Vx + 3 + x + 3 = x. 

Simplify and isolate the radical, Vx + 8 = ^ 2. 

Square both sides again, x + 3 = 4« 

x=l._ 
Substitute x in the given equation, 1 + \/4 = VI, 

1 + 2 = 1. 

This is absurd ; x = 1 is not a root of the given equation. This is an 
example of a radical equation which cannot be satisfied by any value of x. 
The equation is impossible. It is therefore not true of radical equations 
that values of x satisfying them always exist. 

It will be instructive to examine just how the false value x = 1 was 
obtained. When both sides of an equation involving x are squared, new 
values of x may be introduced, giving rise to extraneous roots. By sub- 
stitution we can ascertain in which place the extraneous root came in. 
It is found that x = 1 does not satisfy equations (1) and (2), but it does 
satisfy 3. Hence, the extraneous root x = 1 was introduced when both 
sides of the equation Vx + 3 = — 2 were squared. 

Extraneous roots arise here in the same way as in the solution of frac- 
tional equations. One simple way of determining whether a value of x 
obtained by squaring both sides is an extraneous root or not, is by sub- 
stitution in the original equation ; if it is not satisfied, then the root is 
extraneous. 

From the explanation given above it is evident that, though all opera- 
tions in finding x may be performed without error, the answer may never- 
theless be false. It is therefore necessary to substitute the value of x in 
the original irratiohdl equation, to ascertain whether that value is true or 
false. 



EXPONENTS, RADICALS, IMAGINARIES 151 

While x = l IB not a root of 1 + Vx + 3 = Vx, it will be noticed that 
it 18 a root of 1 — Vx + 3 = — Vx. The solution of the latter equation 
gives rise to equation (3), and to x = 1. 



3. Solve -Vx'^ -h V3a;-h3 « 3. (1) 

Transpose so that the most complicated radical stands alone on one 

side of the equation, , , 

' Va;-2-3=-V3a; + 3. (2) 

Square both sides, a; — 2 — 6y/x — 2+9 = 3a; + 3. (3) 

Simplify, keep only the radical on left side, — 6 Vx — 2 = 2 x — 4. (4) 

Divide both sides by 2 — 3 Vx-2 = x - 2. (6) 

Square both sides, 9x— 18=x3— 4x+4. (6) 

Solve the quadratic x^ - 13 x + 22 = 0. 

(x-2)(x-ll) 

X = 2, X = 11. 

Substitute x = 2 in (1), y/2^^ + V9 = 3. 

3 = 3. 

Hence X = 2 is a root of (1). 

Substitute x = 11 in (1), Vll-2 + V33 + 3=3. 

3 + 6 = 3. 

Hence x = 11 is not a root of (1). Nor is it a root of (2), (3), (4), and 
(6) . But it is a root of (6). It follows that x = 11 is an extraneous root, 
obtained in squaring (6). 



4. Solve V6 — a; -h V« + 7 = 6. 

Place one radical on one side by itself, V6-x = 6— Vx+7. 



Square both sides, 6 — x = 26 — 10 Vx + 7 + x + 7. 

Simplify and keep only the radical on left side. 



10Vx+7 = 2x + 26. 



Divide both sides by 2, 6 Vx+7 = x + 13. 

Square both sides, 26 x + 176 = x^ + 26 x + 169. 

Solve the quadratic x* + x — 6 = 0. 

(x + 3)(x-2) =0. 

X = - 3, X = 2. 



152 ELEMENTARY ALGEBRA 

Substitute x = ^ 3 in the given equation, 

>/9 + \/4 = 6, 
3 + 2 = 6. 
Hence z = — 3 is a root of the given equation. 
Substitute x = 2 in the given equation, 

V4+\/9 = 6, 
2 + 3 = 6. 
Hence x = 2 is also a root of the given equation. 

BXBROISBS 

161. Solve the following irrational equations, testing care- 
fully each value of x that may arise : 



1. Va? + 4 = 4— Vof — 4. 



2. Va?-2+Vaj-|-3 = 6. 



3. Va;— l4-Va-h6 = V4aj-|-9. 

4. Vic+Vx+-l = V2a;-|- 1, 

6. Vy^^ = Vy-hV2. 



6. vV-hl-Vy+l = 0. 



7. 4V5aj*- 5a: -1-9 + 3 = 31. 



8. Vaj* + 21+a=21. 

IRRATIONAL EQUATIONS QUADRATIC IN FORM 

162. The equation a;« + 5 «» + 6 = is said to be quadratic 
in fomiy because it contains only two different powers of x, one 
exponent of x being double the other exponent. 

If we put x^ = y, then %' = y^ and we obtain the quadratic 

y^ + 5y + Q = 0. 
Solving this we obtain y = — 2 and 2( = — 3. 

Hence, x ■ = — 2 and a;» = — 3. 

Cubing both members, x=—S and a; = — 27. 



EXPONENTS, RADICALS, IMAGINARIES 153 

Substituting x = — 8 in the given equation, 

(_8)*+5(-8)* + 6 = 0. 
Simplify, (-2)2 + 6(-2)+6=0, 

4 - 10 + 6 = 0. 
Hence, x = — 8 is a root of the given equation. 
Substituting x = — 27 in the given equation^ 

(-27)*H-6(-27)t + 6 = 0, 
(-3)2+6(-3)+6 = 0, 
9-16 + 6 = 0. 
Hence, x = — 27 is a root of the given equation. 

Solve a:* - 7 «* = 8. 

The exponent | is double the exponent } ; hence the equation is quad- 
ratic in form. 

Let X* = y, then x* = ^, 

Solving, y2 _ 7 y _ 8^ 

y = 8, y = -l. 

We obtain x' = 8 and x^ = — 1. (1) 

The exponent f in x* signifies the third power of the fourth root of x ; 
we wish to find x itself. 

To find the value of x^, we must extract the cube root of both sides of (1). 

We obtain x* = + 2 and x^ = — 1. (2) 

To find the value of x, we must raise both sides of (2) to the fourth 
power. 

We obtain x = 16 and x = 1. 

Substitute x = 16 in the given equation, (16)^ — 7(16)^ = 8, 

(4)3_7(2)« = 8, 

64 - 66= 8. 
Hence, x = 16 is a root of the given equation. 

Substitute x = 1 in the given equation, (1)^ — 7(1)* = 8, 

1-7 = 8. 

Hence, x = 1 is not a root of a given equation. Where did this false 

value enter ? We see that x = 1 satisfies neither x* = — 1 nor x* = — 1 ; 

hence this extraneous root appeared when the sides of x^ = — 1 were 
raised to the fourth power. 



154 ELEMENTARY ALGEBRA 

In solving (1), we might have reversed the two operations, by first rais- 
ing both sides to the fourth power and then extracting the cube root. But 

the operations would have involved larger numbers iu the case of %* = 8. 

Be careful to avoid mistakes in solving equations like x* = 8. 

A frequent error is to conclude from x* = S 
that X = 8*. 

As a matter of fact, x = 8>. 

The safer way is to proceed by two steps as is done above : 

The first step (extraction of the cube root) changes the exponent of x 

from i to }. 

The second step (raising to the fourth power) changes the exponent 

of x from ^ to 1. 

* 

BXEBOI8B8 

163. Solve: 

1. aj*-7xl4-10 = 0. 

2. 8ajl-66ajl4-8 = 0. 

3. 2* -32* + 2 = 0. 

4. (a? -h 1)* - ll(a; + l)i 4- 30 = 0. 
6. a;*-a;i + 132 = 0. 

6. y* -h 19 yA- 216 = 0. 

7. (a; -h 1)» - 60(a; + l)i = 256. 

8. a;*-h6a?*— 55 = 0. 

GRAPHIC REPRESENTATION OF COMPLEX NUMBERS 

164* The graphic representation of ordinary positive and negative 
numbers is familiar to all students of algebra. On a straight line, 
usually drawn horizontally, a point is chosen as the starting point or 
origin ; positive numbers are shown by distances to the right, negative 
numbers by distances to the left. 

The graphic representation of complex numbers like a + 15, where 

i = \/— 1, is not so obvious. In the seventeenth century John Wallis of 



EXPONENTS, RADICALS, IMAGINARIES 155 

the University of Oxford made attempts to find a geometrical interpre- 
tation of such numbers, but was not able to devise a general and con- 
sistent scheme. Not till the close of the eighteenth century did a 
satisfactory plan suggest itself to mathematicians. In 1797 a Norwegian 
surveyor, by the name of Caspar Wessel, found a graphic representation 
which agrees with that now adopted. But his writings failed to attract 
the notice of mathematicians. A little later a Frenchman by the name 
of Argand had a similar experience. More successful in reaching the ear 
of the mathematical public on this matter was the German mathematician, 
Carl Friedrich Gauss. He placed the theory of complex numbers on a 
firm basis and, by his reputation, induced others to enter upon a more 
careful study of this subject and to adopt the geometrical interpretation 
that is given now in works on algebra. 






4, -3,-2, -i, 
■I i 1 H 



+1, +2, +3, +4, +6, 

■H 1 1 1 H 



-•-2/1 



Fio. 23. 



165. This interpretation of imaginary or complex numbers 
is really quite simple. Two axes are drawn perpendicular to 
each other, one axis horizontal, the other vertical, as in Fig. 23. 
Along the horizontal axis real numbers are marked off, positive 



156 



ELEMENTARY ALGEBRA 



numbers to the right and negative numbers to the left of the 
origin. The pure imaginary numbers -|-V— 1, -|-2V— 1, 
-f 3V— 1, ••• are marked off along the perpendicular axis, from 
the origin up. The pure imaginary numbers — V— 1, 
— 2V~ 1, — 3 V— 1, ••• are marked off along this same 
perpendicular, but from the origin douni. 

Complex numbers are pictured in this diagram by points in 
the plane. In case of 3 -f 1 4, measure off three unit distances 

on the horizontal axis to 
the right, and 4 unit dis- 
tances up. This locates the 
point A in the plane (Fig. 
24). The point ^ in the 
plane is accordingly the 
graphic representation of 

3 + *4. 
To find the geometric 

representation of 3 — 1 4, 
pass from the origin 3 units 
to the right, as before, and 

4 units down; the point B 
in Fig. 24 represents 3— i 4. 

Similarly, C represents 
— 3 — i 4, and D represents 
-3 4-i4. 

The representation is similar for other complex numbers. 
It is seen that every complex number can be represented in this 
way, and that every point in the plane stands for some number. 
All real numbers are confined to the horizontal axis ; all pure 
imaginary numbers are confined to the vertical axis ; all other 
numbers a ± ih lie in one or another of the four quadrants. 

The axis for the pure imaginaries is used in much the same 
way as the y-axis when the graph of an equation containing 
the variables x and y is drawn. Only now the vertical axis 
carries imaginary numbers, while before the y-axis carried real 
values of the variable y. 































" 








"" 




















































































































































































































































































1 


■\ , 














A 






















J 


J ^ 












^ 


A 





























































































































































































































































f^ 1 














, T 
























J< 












' 


>u 













































































































































































































Fig. 24. 



EXPONENTS, RADICALS, IMAGINARIES 157 

BXBBCISBS 

166. Locate the' points whicli represent the following com- 
plex numbers : 

1. 5 4-* 4. 4. 5 — 16. 7. 3_4V31. 

2. _5_i6. 6. -1-h*. 8. 44-5y^^. 

3. -5 4-* 6. 6. 6 4-t6. 9. V3 + V^. 

10. (2-ht3)-h(l4-i2). 

IHint. (2 + i 3) + (1 + i 2) = 3 + <6 ; locate 3 + » 5.] 

11. (4 + t)-(2 + i2). 

12. (5-i2)4-(3-t4). 

13. (6-t5)-'(6-i7), 



CHAPTER VIII 

SBRISS AND LIMITS 
ARITHMETICAL SERIES 

167. The numbers 6, 9, 13, 17, 21 appear upon examination 
to have been selected according to some law and arranged in a 
definite order. Each number after the first is greater than the 
one immediately preceding by 4. Such a regulated succession 
of numbers is called series, , When, as here, the increase is the 
same throughout, the series receives the special name of 
arithmetical series or arithmetical progression. 

An arithmetical series is a succession of numbers in which 
each number after the first minus the preceding one always 
gives the same difference. 

This difference is called the common difference. Instead of 
increasing, the numbers in the series may decrease, so that the 
first number is the largest and the common difference is 
negative. 

Arithmetical series are frequently encountered in the study 
of mathematics, hence it is desirable to develop certain formulas 
relating to such series. 

BXBBCISBS 

168. State which of the following series are arithmetical 
series : 

1. 10,8,6,4,2. 4. a, a-hd, a-|-2d, a4-3d. 

^ 9 7 5 3 1 

2. i, 1, H, 2, 2^, 3. 5. _,_,_,__. 

3. 10, 9, 7, 6, 4. 6,x,x — y,x—2y,x^S y, 

158 



SERIES AND LIMITS 159 

THE LAST TERM 

169. If a stands for the first number or term of an arith- 
metical series, and d for its common difference^ then the series 
may be written in general terms thus, 

a, a -h d, a -h 2 d, a + 3 d, a -h 4 d, etc. 

Let I stand for the last term of the arithmetical series. The 
value of the last term evidently depends upon three things ; 
namely, the value of a, the value of d, and also the number of 
terms in the series. Denote the number of terms by n. 

It is to be observed that the coefficient of d in the second 
term is 1, in the third term is 2, in the fourth term is 3, in the 
fifth term is 4. If n denotes the number of terms, what must 
be the coefficient of d in the last term ? From what we have 
observed it must be one less than the number of the term, that is, 
n — 1. Hence we have the formula for the nth term, 

I = a +(n - l)d. (A) 

BXBBCISBS 



170. 1. Find the 10th term in the arithmetical series, 2, 7, 
12, 17, . . . 

Here a = 2, d = 6, w = 10. 

Substitute these values in (A), Z = 2 +(10 - 1)6, 

« = 47. 

Hence the 10th term is 47, as may be verified by writing down all the 
terms to the 10th term. 

2. Find the 16th term of the series —2, —4, —6, — 8, • • •. 

3. Find the 12th term of the series 1, 1^, 2, 2^, 3, • • .. 

4. Find the 20th term of the series 5, 5 -\-2x, 5 4- 4 «, 

6. Find the 24th term of the series V2, V2 4. 1, V2 -h 2, 
V2+3, .... 



160 . ELEMENTARY ALGEBRA 

6. Find the (n — l)th term of the series 3, 6, 9, 12, 15, • • •. 
Find the expression for any given term (the nth term) of 

the following series of numbers ; . 

7. 3, 6, 7, 9, . . .. 10. I, V^, IVV, • • •. 

8. 7, 12, 17, 22, . . .. 11. 66, 59, 62, • • .. 

9. 13, 17|, 22, 26^, . . .. 12. 2.4, 2.1, 1.8, • • .. 

13. Find the (n — 2)th term of the series 5, 1, — 3, — 7, • • -. 

14. A bullet is fired vertically upward so that at the end of 
the first second it has a velocity of 200 ft. per sec, at the end 
of the second second a velocity of 168 ft. per sec, at the end 
of the third second a velocity of 136 ft. per. sec, and so on. 
Compute the velocity at the end of the sixth second, at the end 
of the tenth second. Interpret the second answer. 

15. A body falling from rest falls 16 ft. during the first 
second, 48 ft. during the second, 80 ft. during the third, 112 
ft. during the fourth, and so on. How far will it fall during 
the ninth second ? 

ARITHMETICAL MEANS 

171. The arithmetical means between two numbers are num- 
bers which, together with the two given numbers as first and 
last terms, form an arithmetical series. 

If the two given numbers are 5 and 50, then 14, 23, 32, 41 
are four arithmetical means, because 5, 14, 23, 32, 41, 50 is an 
arithmetical series. 

EXBSBCI8BS 

172. 1. Insert six arithmetical means between 7 and 63. 

The six terms to be found, and the given numbers 7 and 63, will make 
8 terms. 

We have n = 8, a = 7, Z = 63. Substitute in (-4), 

63 = 7+(8-l)d, 
63=7 + 7d, 
d=8. 

Hence the required series is 7, 15, 23, 31, 39, 47, 55, 63. 



SERIES AND LIMITS 161 

2. Insert 5 arithmetical means between 6 and 72. 

3. Insert 4 arithmetical means between 7 and — 23. 
4.' Insert 8 arithmetical means between 7 and S^, 

5. Insert 3 arithmetical means between x and y, 

6. Insert 7 arithmetical means between V7 and 10V7. 

7. Insert an arithmetic mean between 100 and 133. * 

SUM OF AN ARITHMETICAL SERIES 

173. The sum of the terms of an arithmetical series can 
always be found by writing down all the terms, and adding 
them. But, if the number of terms is great, this operation is 
quite laborious. We proceed to derive a formula by which the 
sum of a large number of terms may be computed with less 
labor. 

Observing that, if the last term is I, the term immediately 
preceding may be written I — d, the term before this I — 2dy 
and so on, we may indicate the sum of the series thus, 

^ = a+(a + d) + (a-h2d)+ ... +(Z-2d)-h(Z- d)+L (1) 

Reverse the order of the terms in the right side of (1), 
^=:Z+(Z-d)4.(?-2d)-h ... -h(a + 2d)+(a + (f)+a. (2) 

Adding (1) and (2), 

In (3) there are as many parentheses (a + 1) as there are 
terms in the series ; hence. 



162 ELEMENTARY ALGEBRA 

BXBBCI8BS 

174. 1. Find the sum of 20 terms of the series 99, 103, 107, 
111, .... 

Hero n = 20, a = 09, d = 4. 

Henoe, I = a + (fi - 1)<2 = 99 + 19 x 4 = 175, 
and ^ = |(a+Z) = y(99+176) = 10x274=2740. 

The roqaired sum is 2740. 

2. Find the sum of 25 terms of the series — 60, — 46, 
- 40, • . . . 

3. Find the sum of 18 terms of the series 11, 8, 6, 2, • • • . 

4. Find the sum of the first 200 integers, 1, 2, 3, • • • . 

6. Find the sum of all the integers between --60 and 76, 
excluding —50 and 75. 

6. Find the sum of 15 terms in 1^, If, 2, 2f, • • •. 

7. In formula (J3) substitute for I its value as given in 
formula {A) and derive a formula for S which does not con- 
tain L 

8. Find the sum of the first one hundred odd numbers. 

9. Find the sum of the first one hundred even numbers. 

10. How many of the integers 1, 2, 3, • • • must be added to 
yield the sum 55 ? 

Use the formula ^9 = - (2 a +(n - l)(l), derived in Ex. 7. Do both 
answers satisfy the conditions of the problem ? 

11. How many terms of the series 5, 4, 3, • • . are necessary 
to yield a sum 9 ? 

Do both answers satisfy the conditions of the problem ? 

12. The second term of an arithmetical series is 11, the 
fifth term is 20 ; find the 14th term. 

13. The first week a store was opened the expenses ex- 
ceeded the income by % 52.25. The second week the loss was 



SERIES AND LIMITS 163 

$41.75. If the improvement in the trade continued at the 
same rate, how much profit was made in 24 weeks ? 

14. A man enters an office at a salary of $ 1200, which is 
increased annually $ 75. How much will the firm pay him 
during 18 years ? 

15. The three formulas found below give the salaries offered 
by three companies to men entering their employ. S is the 
monthly salary in dollars earned after a given number of 
years (n). Calculate which company after 20 years' employ 
will give the highest salary. How much does a man earn in 
20 years in each case ? 

(a) 5'=100 4-|w, (b) /S' = 95 4-4n, (c) S=SS+^n. 

GEOMETRICAL SERIES 

176. The series 3, 6, 12, 24, is not arithmetical ; the dif- 
ference between successive terms is not the same. The 
successive terms are formed in accordance with a diflerent law. 
It is readily seen that any term after the first is derived from 
the preceding one by multiplying by 2. Such a series is called 
a geometHcal series or a geometrical progression. 

A geometric series is a succession of numbers in which each 
number after the first, when divided by the preceding number, 
always gives the same quotient. 

The quotient is called the common ratio. 

EXERCISES 

176. Which of the following series are geometrical and 
which arithmetical? 

1. 7, 35, 175, .... 5. 2, - 4, 8, - 16, • . .. 

2. 6, 12, 18, .... 6. 2, 1, 0, - 1, • • •. 

3. 2, 8, 32, .... 7. V2, 1, V|, . • -. 

4. 2, 8, 32, 40, .... 8. 2i, 2i, 2f , . . .. 



164 ELEMENTARY ALGEBRA 

LAST TERM OF A GEOMETRICAL SERIES 

177. If the first term of a geometrical series is a, the com- 
mon ratio is r, the number of terms n, then an expression for 
/, the last term, may be obtained by inspecting the terms in 
the general geometrical series, 

a, ar, ar^, ar*, ar^^ • • •• 

We observe that in the second term, the exponent of r is 1, in 
the third term it is 2, in the fourth term it is 3, in the fifth term 
it is 4. Evidently, in the nth term, the exponent of r is n— 1. 

Hence we have the formula for the last term of a geomet- 
rical progression, ^ ^ ^^-i ^q^ 

BXBBOISBS 

178. 1. Find the seventh term in the geometrical series, 
16, 32, 64, . . .. 

Here a = 16, r = 2, n = 7. 

Hence, I = ar^^ = 16(2)^-1 = 16 x 64 = 1024. 

The seventh term is 1024. This result may be verified easily by writ- 
ing down the first seven terms of the series. 

2. Find the 8th term of the series 6, 2^, 1 J, • • •. 

3. Find the 6th term of the series 130, 390, 1170, • • .. 

4. Find the 9th term of 3, - 6, 12, - 24, . • .. 

5. Find the 8th term of 3, — 1, i, — |, • • •. 

6. Indicate the 15th term of the series — -y 1, -— -, •••. 

V2 6 

GEOMETRICAL MEANS 

179. The geometrical means between two numbers are 
numbers which, together with the two given numbers as first 
and last terms, form a geometrical series. 

If the two given numbers are 8 and ^, then 4, 2, 1 are three 
geometrical means, because 8, 4, 2, 1, ^ is a geometric series. 



SERIES AND LIMITS 165 

BXBBCISBS 

180. 1. Insert four geometrical means between 5 and 160. 

The two given numbers and the four means make together 6 terms. 
We have a = 6, 1 = 160, n = 6. We must find r. 

By(C), l = ar^\ 

160 = 6 »*. 

r = 2. 
Hence the required geometrical series is 5, 10, 20, 40, 80, 160. 

2. Insert six geometrical means between 10 and — 1280. 
Solve r^ = - 128 by trial. 

3. Insert four geometrical means between 3 a and 96 a*. 

4. Insert one geometrical mean between 133 and 1197. 
What is the geometrical mean between two numbers ? What 
is the arithmetical mean between two numbers ? 

6. Two numbers differ by 6, and their arithmetical mean 
exceeds their geometric mean by 1. Find the numbers. 

SUM OF A GEOMETRICAL SERIES 

181. The sum of the tirst n terms of a geometrical series 
may be indicated thus, 

S = a-^ar-^ar^-\ f- aV"-^ -f ar^"^ + ar^-K (1) 

Multiply both sides of (1) by r, 

r/S = ar 4- ar* 4- a^^ 4- • • • 4- a^*"* 4- ar""* 4- ar*'. (2) 

Subtract (1) from (2) and observe that all the terms dis- 
appear in subtraction, except a and ai**. We obtain, 

rS — S = ar^ — a 
S(r - 1) = a(r-- 1) 

r — 1 



166 ELEMENTARY ALGEBRA 

BXBBOISB8 

182. 1. Find the sum of six terms of the geometrical series 
11, 22, 44, ... . 

Here a = 11, n = 6, r = 2. 

Hence. /? = -<'!^rL!l = yC^.rJl = ll(?lrill= n x 63 = 603. 

r-1 2-1 1 

2. Find the sum of seven terms of 1, ^, ^, • . • . 

3. Find the sum of six terms of 1, — V2, 2, — 2 V2, • • • . 

4. Find the sum of five terms of a?, — ay', ajy*, .... 

6. Show that formula ((7) may be written 8 = ^\ "~ — ^ • 

1 — r 

For what values of r is this form more convenient than (C) ? 

CL ^— 1*1 

6. Show that (0) may be written S = 

1 — r 

7. What will $1000 amount to in four years, interest 3 %, 
compounded annually ? 

8. What will $ 600 amount to in two years at 4 ^ annual 
interest, compounded semiannually ? 

9. What will $ 700 amount to in eight years at 4 % annual 
interest, compounded semiannually ? 

We simplify thia computation by the use of logarithms. 
The amount is given by the expression x = ^700 (1.02)". 

log 1.02 = 0.0086, 
log (1.02)w = 16 X 0.0086 = 0.1376 

log 700 = 2.8451 
logaj = 2.9827 
flc = f 961. 

This answer is only approximate. Had we used a six-place table of 
logarithms, instead of the four-place table, the answer would have come 
out 9 960.06, correct to the nearest cent. 



SERIES AND LIMITS 167 

10. What sum of money, at 3 % interest, compounded 
annually, will amount to $ 1000 in 20 years ? 

Let Jjjl X be the sum, in 20 years this amounts to $x(1.03)*>. 
Hence the equation, x(l.OS)^ = 1000 

1000 







(1.03)20 


Log 1000 




= 3.0000 


Log 1.03 


= 0.0128. 




Log (1.03)«> 


= 20 X 0.0128 


= 0.2660 
loga; = 2.7440, 
x = $664.60. 


The answer, 


correct to the nearest cent, is $ 663.68. 



11. Wliat sum of money, at 4 % interest, compounded 
annually, will amount to $ 695 in 10 years ? 

12. A sum of $ 497 draws compound interest at 3^ % for 7 
years. Find its amount. 

13. If $ 1 could have drawn 6 % compound interest during 
the past 500 years, what would be its amount now ? 

14. Owing to the introduction of automobiles the number of 
horses in a town decreased in five years from 560 to 336. 
Assuming a constant annual rate of decrease, find the number 
after four more years. 

15. Determine the present value of $ 1000 due in 5 years, 
on the supposition that money can be invested so as to yield 
3 % interest when compounded annually. 

16. A man undertook to pay $ 100 to a charity one year, 
$ 90 the next year, ^ of the latter sum the third year, and so 
on, until his death. He died after making 21 payments. 
What was the total of his gifts ? 

Use logarithms in the computation. 

17. Another man pledged $ 175 and promised to increase 
the amount by ^ every year. What is the total of his dona- 
tions after 15 years ? 



168 ELEMENTARY ALGEBRA 

18. A man saved $ 250 every year and invested it in a busi- 
ness that brought him 4 % per annum compound interest. He 
made these investments f6r six consecutive years. What was 
the total sum standing to his credit immediately after he paid 
in his last $250? 

19. A man's business is increasing at a uniform rate, and so 
rapidly that his income is doubled in two years. What is 
the rate of increase or growth-factor f 

If at starting his business was 9 a, it was 9 or at the end of the first 
year, and 9 ar^ at the end of the second. We have then aH = 2 a, or 
r = \/2. The growth-factor = >/2. 

20. What is the annual rate of increase, or annual growth- 
factor, if income increased uniformly and doubled in the course 
of four years ? 

21. In 4 years the population of a town increased from 
2000 to 2662. Determine the growth-factor on the supposition 
that the rate of increase was uniform. 

22. A young tree grew in 6 years from a height of 32 in. to 
a height of 20 ft. 3 in. What constant annual growth-factor 
would account for this increase ? 

23. The population of a mining town, now 10,000, is falling 
off at the rate of 1 % per annum. What will the population 
be 2 years from now ? 

24. Between the ages 9^ and 14^ the average height of boys 
increases from 60 to 60 inches. Find how tall a boy would be 
at 10|, 11|, 12^, and 13| years of age, on the supposition that 
the growth-factor was the same for these years. Compare 
your results with the statistical averages which give the 
heights as 51.9, 53.6, 55.4, 57.5 inches, respectively. 

Use logarithms. 



SERIES AND LIMITS 169 



INFINITE GEOMETRICAL SERIES 

183. An important kind of series arises when the number 
of terms in it is no longer restricted to a fixed finite number, 
but is permitted to increase without end. 

Consider the series 1, ^, J, \, ^. 

It has five terms. Imagine now that the number of terms 
is much larger, and that this number is steadily growing. We 
obtain a series which may be written, 

1» h h h^y^f-hy-^ infinity. (1) 

The number of terms is taken larger than any large finite 
number that we may name. 

We say that the series is infinite. The characteristic prop- 
erty of it is that it has no last term. Whenever we try to 

seize upon any one term, say ■— -, as being the last one, we can 
always go one step farther and write down the term — ^ which 

comes after it. Considerations of this sort compel us to accept 
the conclusion that there is no last term in such a series. 

Another noticeable feature about the series (1) is that its 
terms decrease toward the right; that is, the terms farther 
from the starting point are always the smaller. The nth term 

may be written 

By taking n sufficiently large, we may make this nth term 
smaller than any small number which we may name. For 
instance, the nth. term will become smaller than -j-J^, if we 

1 11 

take n = 8. In this case — - becomes — = — -, which is less 

2»-i 27 128' 

thauy^. 

If you name some smaller number, say y^T^, then again a 

value of n can be chosen, so that the nth term becomes smaller 

than it. Take, n = 11, then h. = hxh = :^y^l= ^ 



2w 2' 2» 128 8 1024' 



170 ELEMENTARY ALGEBRA 

which is less than y^Vir* ^^^ ^^"^7 name a still smaller fraction, 
and again a value can be assigned to n in the expression for the 
nth term, which makes that nth term less than that small fraction. 
No matter how small a fraction is written down, we can always 
find some term in series (1) which is smaller than that fraction. 

In such a case we say that the term — -^ approojohea zero 

as a limitf as n increases wUhoiU end. 

SUM OF AN INFINITE GEOMETRICAL SERIES 
184. We proceed now to find the sum of the infinite series 
IH 1 1 [-•••H : -h • • • to infinity. Examine formula 

(C) ; we see that 

o __ a(r* — 1) _ a(l — r *) __ a — ar^ __ a^ ar* 



r — 1 1 — r 1 — r 1 — r 1 — r 

We know that this series gives the sum correctly for any 
finite and fixed number of terms. What is the sum when the 
series becomes infinite ? 

In the infinite series 1, -J., J, i., . . . we have r = J^, a = 1. 
Other geometrical series may be written, but in all of them 
that we shall consider now, we shall assume that the ratio r is 
numerically less than 1. In that case the power r* becomes 
smaller for larger values ofn and approaches the limit zero as n 
increases without end. 

That this is so, may become clearer by a second illustration. 
Let r = I, then r^ = ^, r" = ^^y, r* = \^, r* = ^^^, and so on. 
The diminution in the value of the power is not as rapid now 
as in the case, r = ^ ; nevertheless, a value of the integer n can 
always be found, such that r" becomes less than any small frac- 
tion previously named. To make r" less than y^itj ^® need 
only take n = 12. Since 

-jj 64 .f 12 «8 -^ 64x64 4096 

r = , we get r^^ = r^ *t^ = 



729' ° 729x729 531441 



SERIES AND LIMITS 171 

In this fraction the numerator is seen to be less than the 
one-hundredth part of the denominator. 

185. What happens to the fraction , when the integer 

1 — r 

n LQcreases without end, and r is numerically less than 1 ? 
Bear in mind that a, the first term of the series, is a fixed 
number ; r is also a fixed number, hence the denominator 1 — r 
is a fixed number. Only the factor r" changes, as n increases 
without end. And we have seen that, under these circum- 
stances, r* approaches the limit zero. Hence the numerator 
ar** also approaches the limit zero. Hence, the entire fraction, 

, approaches the limit zero as n increases without end. 



1-r 

The series, 

S = a -\' ar -\- aj^ -{- ...4- ar"""^ + . . . 

which for a fixed, finite number of terms n has the sum 

a ar^ 



aS = 



1 — r 1 —r 



has the sum, 8 = - , (D) 

1 — r 

when the series becomes infinite, so that n increases without 
end, provided that r is numerically less than 1. 

From what we have said it appears that by the " sum " of 
an infinite series we mean the value which the sum of the first 
n terms of the series approaches as a limit, when n increases 
without end. 

When the common ratio r of the geometrical series is 7iot 
numerically less than 1, the above reasoning does not apply. 
In that case r" does not approach the limit as w increases 
without end. In the series 1, 2, 4, 8, • • • r = 2, and ?•* = 2" ; 2" 
becomes greater and greater as n increases. Thus 2* = 4, 
2* = 8, 2* = 16, etc. It is seen that 2» increases without limit 



172 ELEMENTARY ALGEBRA 

when n increases. Hence the fraction does not ap- 

1 — r 
proach the limit zero. 

When in an infinite geometrical series the ratio r is numeri- 
cally less than 1, so that the sum of the first n terms ap- 
proaches or converges to as a limit, the series is said to 

1 — r 
be convergent. 

Convergent aeries play a very important part, both in theo- 
retical and in applied mathematics. 



BXBROISBS 

186. 1. Find the sum of the infinite geometrical series, 

^2 4 8 ^2"-^ 

Here a = 1, r = J. 

Hence ,8^ = _?_ = -J_ = 2. 
1-r 1-J 

The Bum of the series is 2. 

To convince himself of the correctness of this result the student may 
find the sum of 8, 4, 6, or more terms and see that the smn gets very 
close to 2, the closer when more terms are added. 

Let 8n mean the sum of the first n terms, then we see that 82 = 1.5, 
8z = 1.76, 84 = 1.876, 8s = 1.9376, 8^ = 1.96875. By actual addition we 
can approach to the limit 2 closer and closer, but we cannot reach it, 
because we can perform this addition for only a finite number of terms. 

2. Find the sum of the infinite geometrical series 1, |^, |^, 

ift^? • • •• 

3. Find the sum of the infinite geometrical series 1, ^, 

4. Find the sum of the infinite geometrical series 2, — J, 

h — T^7) • • •• 

6. Find the sum of the infinite geometrical series 1, — , -, 
11 2* 2 

^'2-^'-- 



SERIES AND LIMITS 173 

6. Find a common fraction which expresses the exact 
value of the repeating decimal, .252525 • • •. 

The repeating fraction may be written as an infinite geometrical 
series thus, 

.2625 ... = tW + tAW + Tirt^inj + -• 
Here o = fW* »• = nflftnr ■*• ^ = rir- 

Hence ^ = _«_ = .JL= 25,^^ ^26 ^^ 

l_r 1-tJ^ 100 100 99' 

Find the common fraction which is the exact value of each 
of the following repeating decimals : 

7. .333.... 8. .151515.... 9. 7.363636.... 
10. .555... 11. .325325. ... 12. 7.8212121.... 

13. A man undertook to pay $500 to a charity one year, 
$ 450 the next year, -^ of the latter sum the third year, and 
so on, until his death. What was the outside limit of the 
expectation of this charity? If the man died after making 
15 donations, how much did his total payments fall short of 
that sum ? 

Use logarithms in performing the computation. 

THEORY OF LIMITS 

187. In deriving the sum of an infinite converging geomet- 
rical series we touched upon the theory of limits which plays 
a fundamental r51e in higher mathematics. The ideas which 
we brought out suggest the following definition of the limit 
of a varying number. 

2%e limit of a variable is a fixed number which the variable 
approaches in siich a way that the difference between the variable 
and its limit becomes and remains numericaUy less than any num- 
ber, however smallj but never becomes zero. 

Thus, in fi; = 1+1 + 1+...+ ^ 



2 4 ^2*»-i' 

the variable is 8n^ which increases when the integer n increases. 



174 ELEMENTARY ALGEBRA 

The limit of 8^ is the number 2. 

As n increases without end, the variable Sn approaches 2 and may be 
made to approach it so closely that 2 — i9» is less than any previously 
assigned small number different from zero, such as ^ithm ^^ looiooc 

A variable of some importance is the fraction -, when n 

n 

takes in succession the values 1, 2, 3, 4, • • •. It is seen that 

the variable - becomes smaller and smaller and approaches 

n 

the limit zero. Under the same conditions the fraction -, 

n 

where a is any fixed number, will likewise approach the limit 
as the integer n increases without end. 

Quite different is the behavior of -, when a is a fixed num- 

X 

ber different from zero, and x takes in succession the values 
1, ^, i, i, • • •. For simplicity, let a = 1. 

Then, for a; = 1, - = 1, for a; =i, - = 8, 

X S X 

fora: = i, ^ = 2, for a: = 1, ^= 16, 

2 a? 16 a? 

for a? = - , - = 4, for a; = — - , - = 32, 

4' a? ' 32' a? ' 

and so on. The smaUer the denominator, the larger the value 
of the fraction. 

We let the variable x approach the limit zero in such a way 
that it does not reach zero. We stipulate that x shall not be- 
come zero for the reason that we cannot divide by zero, as was 
explained in a previous chapter. If, then, x is permitted to 
approach as a limit, without reaching its limit, then the 

fraction - increases in value without limit. That is, we can 
x 1 

select a value for a;, such that - will be greater than any pre- 
viously assigned large number. 



SERIES AND LIMITS 175 

For instaoice, select the number 1000 ; we can take x so small 
that - is greater than 1000. Let « = — - = 



X ° 210 2*^.2^ 32.32 

= , then - = 1024, which is more than 1000. In the same 

1024' X ' 

way, X can be so chosen, that - exceeds 10,000, or any other 

X 

fixed number. Hence the theorem, 

As the variable x is made to approach the limit zero, without 
rea>ching zero, - increases so as to exceed any large number that 

X 

may be previously assigned. 

This theorem is sometimes condensed in this manner : 
As X approaches zero, - approaches infinity. 

X 

A still further compression of the theorem consists in the 
use of the symbolism 

This symbolism is objectionable, because it may convey the 
idea that division by is permissible. The reader must re- 
member that these syjnbols are intended merely as a con- 
venient abbreviation for the theorem given above. 



EXEBCISBS 

a I 35 
188. 1. What value does the fraction — -^— - approach, when 

b — X 

a and b are fixed numbers (b ^ 0) and cc is a variable approach- 
ing the limit zero ? 

Under these conditions the numerator a + ^ approaches a, the denomi- 
nator h — x approaches 6 . 

Hence the fraction approaches -• 

h 



176 ELEMENTARY ALGEBRA 

2. What value does the fraction — approach, when x in- 
creases without limit ? a + x 

The reasoning is easier, if we first divide numerator and denominator 
by X, so that x occurs in the denominators of the minor fractions. 



«-l 



We obtain, 



a — x^x 



a-^x ^ A.I 



X 



Since a is a fixed number, and x increases without limit, the fraction 
- approaclies the limit zero, hence, the numerator - ~ 1 approaches — 1 

X X 

as a limit ; the denominator - -f 1 approaches + 1 as a limit ; the com- 
ae 

plex fraction approaches -^^— or — 1 as a limit. 

What is the limiting value of the following fractions when 
X approaches zero as a limit ? 

' 4 — 6aj' ' 1— «* * 2 — aj*' 

*• ~ • o. — -. o. 



aj4-6 S-haj* aj*-f2a:*-i-3 

What is the limiting value of the following fractions when 
the variable x increases without limit ? 

X l-|-2aj a — ar 

10. _?_. 12. «±^. 14. ^+£±«. 

x + 6 c + dx x^ — x-i-b 



SUPPLEMENT 177 



SUPPLEMSNT 

THE HIGHEST COMMON FACTOR BY THE METHOD OF 

DIVISION 

189. The method of finding the h. c. f. of two polynomials 
not easily factored is similar to the method for finding the h. e. f . 
of two numbers in arithmetic, neither of which can be easily 
factored. The method was first suggested by Euclid (300 B.C.). 

BXAMPIiB 

Find the h. c. f . of 1343 and 3002. 

Id43)3002[2 
268.6 
316) 1348 [4 
1264 
79)316 [4 
316 
Process : Divide 3002 by 1343. 

Take the remainder 316 as a new divisor, and 1348 as a new dividend. 
Take the next remainder 79 as a new divisor, and 316 as a new dividend. 
The next remainder is zero. 
The last divisor 79 is the h. c. f . of 1343 and 3002. 

Proof: Let ^ = 1343, i? = 316. 

B = 3002, Q = 2. 
We have, as always. 

Dividend = Divisor x Quotient + Remainder. 
B = AxQ +B, 
or B = B-AQ 

From the last equation we see that any factor of both A and B 
must be also a factor of B — AQ^ and therefore a factor of the re- 
mainder B. Hence, all common factors of A and B are common factors 
of A and B, 

Instead of finding the h. c. f. of A and B, we may therefore find the 
h. c. f . of A and B ; the latter process involves smaller numbers. 

Repeating the above process, divide A by B. Let Q' be the new quo- 
tient 4, and B' be the new remainder 79. As before, all common factors 
of A and B are factors of B' ; for, A = Q'B + B' and B' = A— Q'B. 



178 ELEMENTARY ALGEBRA 

Hence, the problem reduces itself farther, to finding the h.c. f. of B 
and B' ; that is, to finding the h. c. f. of 816 and 79. As 816 = 79 x 4, it 
follows that 79 is the h. c. f . required. 

This process may be used in finding the h. c.f. of poly- 
nomials which cannot be easily factored. Numerical factors 
common to the polynomials are easily detected by inspec- 
tion. Such numerical common factors, as well as other 
common factors that can be found by inspection, should be 
removed at once. 

BXAMPLB 

Find the h.c.f. of 2iB»-22aj-12 and 4iB»-|-16aj«-|-16rc-|-12. 

Process : 2 x« - 22 « - 12 = 2(a* - 11 a:- 6). 

4 x» + 16 a;a + 16 a + 12 = 4(x» + 4 a;2 + 4 x + 8). 

The numerical factor 2 is common to the polynomials. 

x»-llx-6)a* + 4a;2+ 4x + 8Ll 
z« -11 a; -6 

4a:«-|-16x + 9)4x»-44a;-24 |a;-16 



-16x2- 


63x- 


24 

4 


-60x2« 


212 X- 


96 


-60x2- 


225 X- 


136 



18 13x-h39 



X + 3)4 x2 + 16 X + 9 |4x4-8 
4x2+ 12x 

3x + 9 
3x + 9 



The h.c.f. =2(x + 8). 



To avoid fractions any expression may be multiplied or divided by 
any number which is not a factor of the other. In the example above, 
when 4 x2 in the first remainder will not be contained in x^ a whole num- 
ber of times, the expression x'— llx — 6 is multiplied by 4. Again 
when 4 x2 is not contained in — 16 x2 a whole number of times, the 



SUPPLEMENT 179 

expression — 15 x^ — 63 2c — 24 is multiplied by 4. In the remainder 
13 X + 30, the factor 13 may be discarded because the expression 
4 a;2 _|_ 15 x -j- does not have a factor 13. It would be wrong to multiply 
4 x^ + 16 X + 9 by 13, because 13 is a factor of 13 x + 30. 

Continue to divide until the remainder is of a lower degree than the 
divisor. Then take the remainder for a new divisor and the previous 
divisor for a new dividend and proceed as before. 

The 1. c. m. of two expressions not easily factored may be found by 
first finding their h. c. f . The 1. c. m. required will then be the product 
of either expression and the quotient obtained by dividing the other ex- 
pression by the h.c. f. It is seldom that resort to this long process be- 
comes necessary. 

EXERCISES 
190. Find h. c. f. of : 

1. 2a2 + a — 3and4a'-|-8a2 — a — 6. 

2. 2a^ — 5a;2 — 2aj + 2and2aj» — 7a;2 + 9aj — 3. 

4. a* — a' + 2 a2 — a 4- 1 and a* + a' -|- 2 a2 + a + 1. 

5. 8 am«-|-24 am^-}- 22 am-\-6 a and 6 m'+13 m2 + 8 m + 1. 

6. 2a:3-10aj2-14aj + 70anda^-3aj2-7a;-15. 

7. S xlh/—10 a^y-\-7 xh/—2 xy and 6 a^y --11 oiih/ -^8 ah/— 2 xh^. 

8. 4a» - 4a2 - 5a + 3 and 10a2- 19a + 6. 

The h. c. f . of three or more expressions may be found as follows : Find 
the h. c. f. of two of them ; then find the h. c. f. of this result and another 
of the given expressions, and so on. The last h. c. f . is the one required. 

9. 2a^-2aj2_2aj-4, 3iB»-6a;2+9aj-18,and8aj»-12a;2 
-4aj-8. 

10. a^x — 6 a2aj -f- 11 ox — 6 aj, cj^x — 9 a2a; -|- 26 ox — 24 aj, and 
a«aj2 — 8 a2aj2 4. 19 aaj2 — 12 x^. 



180 ELEMENTARY ALGEBRA 

SOLUTIONS OF QUADRATIC EQUATIONS 

191. In previous solutions of the quadratic equation ax^ 4- 
6a; 4- c = 0, obtained by completing the square, the first step 
was to divide both sides by a. There are other methods of 
procedure which possess certain advantages. If a, 6, c are 
integers and it is desired to avoid fractions in the process of 
completing the square, the Hindu method of completing the 
square may be used. It is as follows : 

Given ox* + ftx = — c. 

Multiply both sides by /our times the coefficient ofx^^ or 4 a. 

We obtain 4 a^c^ -\- 4 abx = - 4 ac. 

Divide 4 abx by twice the square root of 4 a'x' ; this gives 5. 
Add the square of 6 to both sides, 

4a^x^ + 4abx + fe^ = 52 _ 4 <|c. 

Take the square root of both sides, 

2ax + b = ± y/h^ - 4 oc. 



Transpose h and divide by 2 a, x ^ - & J:>/"&^ - 4ac ^ 

2a 

Sometimes fractions may be avoided by the use of a smaller 
factor than 4 a. Consider the equation, 

3a? -h 4a; = 8. 

Make the first term a perfect square by multiplying by 3, 

9a;2+12x = 24. 
Divide 12 x by twice the square root of 9 x^ ; this gives 2. 
Add the square of 2 to both sides, 

9«3 + i2a;-|-4 = 28. _ 
Then 8a; + 2 =±^28, _ 

-2 J:>/28 
""- 3 ' 

-2±2V7. 
3 

Solve by completing the square without introducing frac- 
tions : 

1. 2a«- 13a; 4-10 = 0. 3. 4a;* + 6aj- 41 =0. 

2. 5a;2-h7a;-13 = 0. 4. 7a;2 - 14a; - 3 = 0. 



SUPPLEMENT 181 

MATHEMATICAL INDUCTION AND PROOF OF THE 

BINOMIAL THEOREM 

192. There is an important method of reasoning in mathe- 
matics, called mathematieal induction, which we shall use in 
proving the binomial formula. The method will be grasped 
more readily, if we give a non-mathematical illustration of it. 

Non 'mathematical Illustration, 

For the sake of argument, suppose it established as true 

(1) That there was once an Indian named Hiawatha* 

(2) That every Indian named Hiawatha (if ever such an Indian ex- 
isted) had a grown son named Hiawatha^ 

From these two propositions certain conclusions can be drawn. 

By (1) we know that an Indian named Hiawatha once existed. 

By (2) we know that he had a grown son named Hiawatha. Desig- 
nate him Hiawatha II. 

By (2) we know that Hiawatha II had himself a grown son named 
Hiawatha. Designate him Hiawatha III. 

By (2) we know that Hiawatha III had himself a grown son named 
Hiawatha. And so on. 

Thus the inference is drawn from propositions (1) and (2) that there 
existed an unbroken, never-ending line of descent of Indians named 
Hiawatha. 

Mathematical Illustration 

It is to be proved that tf^e sum of the first n positive integers i« ^ (n -h 1). 

I. We see by trial that 1 + 2 = J(2 + 1), 

1 -h 2 + 3 = i(3 -h 1). 

This shows that the theorem is correct for the particular cases n = 2 
and n = 8. 

II. We establish the conditional proposition, that if the theorem is 
true for some value of n, say n = m, then the theorem must be true for 
n = m + l. 

If the theorem is true for » = m, we have 

l-|-2 + 3 + .-.+m = ^(w+l). 

Add (m -I- 1) to both sides, 

1 + 2 + 3-1- ... +m+(w+l) = ^(m+l)+(w + l). 



182 ELEMENTARY ALGEBRA 

Combine terms on the right side, 

1 + 2 + 3 + . . . + m + (m + 1)= 5L±i([w, + 1]+ 1). 

As this result is the original theorem for n = m + 1, we have shown 
that the theorem is true forn = m + 1, provided it is true for n = m. 
Next we apply the reasoning called mathematical induction : 
By I, the formula 

1 + 2 + 3 + . .. + n = |(n + l) 

is true for n = 3. 

By II it follows that, being true for n=3, it must be true also for n=4. 

By II it follows that, being true for n = 4, it must be true also for 
n = 6. And so on for n = 6, 7, 8, • • •. Consequently, the theorem is 
true for any positive integral value of n, no matter how great it may be. 
Thus, the theorem is established. 

Note. Very often a beginner fails to i)erceive the need of such an 
involved argument. If a theorem is true for certain special cases, he may 
conclude at once, without further argument, that it is true for all cases. 
But such jumping at conclusions is dangerous. The relation a^ = 2 a 
holds for a = 0, also for a = 2 ; is it true generally ? Is it true for any 
other value of a ? 

PROOF OF THE BINOMIAL THEOREM WHEN THE EXPO- 
NENT OF THE BINOMIAL IS A POSITIVE INTEGER 

193. By actual multiplication it is found that 

(a ± 6)*= a?±2db-\-h\ 

{a ± by= a« ± 3a«6 -+ 3a6» ± b\ 

(a ± by= a* ± 4 a'6 4- 6 a'fe* ± 4a&» +- b\ 

By inspection we find that these products follow the follow- 
ing laws : 

I. The first term is a raised to the same power as that of 
the binomial. In each succeeding term the exponent of a 
decreases by 1. 



SUPPLEMENT 183 

II. The factor b does not appear in the first term ; the ex- 
ponent of 6 in the second term is 1 and increases by 1 in each 
succeeding term. 

III. The coefficient of any term after the first is found by 
multiplying the coefficient of the preceding term by the ex- 
ponent of a in that term, and divided by one more than the 
exponent of 6. , 

IV. If the binomial is a + b, the signs of the product are 
all plus ; if the binomial is a — &, the signs are alternately -f- 
and — . 

V. The number of terms is one more than the exponent of 
the binomial. 

VI. Each term is of the same degree as the binomial. 

If we use the exponent n, these six laws are embodied in the 
following formula : * 

(a + by = a» -h na^'^b + ^l^^^lDa^-^^^ 

_^ n(«-l)(«-2) ^^y + . • . + naly-^+ b". (1) 

Thus far, this formula has been shown to be true only for the 
particular cases n = 2, 3, or 4. (Substitute 4 for n and show 
that the product of (a + by is obtained.) Is formula (1) true 
for all positive integral values of n ? Were we to attempt to 
prove its generality by actual multiplication, we would soon 
weary ; the task would be impossible. By mathematical in- 
duction the proof is short. 

Let us establish the conditional proposition that if formula 
(1) holds for, say n = m, it must hold also for m +1. In (1) 
write m in place of n. Then multiply both sides of the re- 
sulting equation by (a + b). We obtain, 

* By the notation 2 1 we mean 2 x 1 » by 3 ! we mean 3x2x1; generally, 
nl=»n(n-l)(n — 2) ...3.2.1. WecaUn! "factorial n." 



184 ELEMENTARY ALGEBRA 

(a + 6)« = a* + ma^'^b+ 5*l5Lziiia«»-a6a + • • • + mab"^^ + ft*, (2) 

(a + 6)= a-}- h 

(a + 6)"+i 

2J 

^1 « I 

+ (m+l)a6« + &«+i. (8) 

The operation given above should be studied very carefully. Several 
steps require explanation. We have multiplied both sides of equation 
(2) by a + 2). Examine every step carefully. In the right sides we first 
multiplied by a, then by b. The dots stand for terms that could not all 
be written down, because the number of terms is m + 1, and m is not 

given in Hindu-Arabic numerals. The term *»(t»— l) q,>-25> which is 

2! 

obtained when we multiply by 5, is not written down and is among the 
terms represented by dots. The term ^(^~ ^/ qaftm-i ig obtained by 

multiplying by b the term ^(^~ ^^ q2fc«>-a which in (2) lies among the 

terms represented by dots. 

Be sure to verify the addition of the partial products. 

We observe that (3) is the same in form as (1) ; that is, if 
in (1) we write m -f- 1 for w, we obtain (3). 

By the above multiplication we have proved that the for- 
mula (1) is true for n = m -f- 1, provided it is true for n = m. 

Now we proceed to the argument by mathematical induction : 

Formula (1) is known from actual multiplication to be true 
for n = 4. But we have shown that if (1) is true for w = 4, it 
must be true for n = 5. 

Again, if (1) is true for n = 5, it must be true for n:=6. 
And so on, for n = 6, 7, 8, • • • . 

Hence formula (1) is true for any positive integral value of 
the exponent. 



SUPPLEMENT 185 

BXEBCISBS 

194. 1. Expand (a + 3 y)\ 

In formula (1) substitute a = a;, 6 = (3y), n = 6. We obtain, 

Simplifying, 
(«+3y)«= a^ + 18a^y+186a:*yH540a:»y8+1215a;2y*+1468icye+729 y«. 

2. Expand (2p — qy. 

In formula (1) substitute a = 2p, 6 =(— g), n = 6. 
(2i>- g)6 =(2p)6 + 5(2p)*(- g) + ^ (2p)»(-g)« 

Simplifying, 

(2p - g)6 = 32 j)6 - SOp^g + 80i)«g« - 40p2ga + I0|)gf4 - g6. 

Expand : 
3. (a-26)». 7. (2i> 4-3^)5. 11. (a + 6)'. 

*• («-iy)*- 8. (V2a-6y. 12. (a + V^6)». 

6. (3aj-|2/. _ / , 

/I Y 9. (V2a;+^/2y)«. 13. (a -V- 16)*. 

\m "" **/ ' 10. (a - h)\ 14. ( V^ «+ V^ y)*. 

195. It can be proved that, under certain limitations, tlie 
Binomial Formula is applicable to cases in which the exponent 
w of (a + 6)** is not a positive integer, but is a negative integer, 
or a positive or negative fraction. In such cases the binomial 
expansion becomes an infinite series. Such an infinite series 
can be used for purposes of computation only when it is con- 
vergent; that is, only when the sum of the first r terms of the 
series approaches a finite constant as a limit, as r increases 
without end. When a > 6, the expansion, expressed in ascend- 
ing powers of 6, is always a convergent series. Omitting all 
proofs, we proceed to applications. 



186 ELEMENTARY ALGEBRA 

1. Find approximatiiely the cube root of 27^. 

This can be obtained by the regular process of extracting the cube root, 
or by the use of logarithms. A third method is by means of the binomial 
theorem. 

We notice that (27.2)> is a little over 8. Take 27-2 = 27 + .2, where 
27 is a perfect cube. Then by the binomial formula, taking 

a = 27, 6 = .2, n = 1, 
(27 + .2)* =(27)1 + l.(27)-*(.2) + Ki.Ilil(27)**(.2)2 + . . . 

Here .2 is small when compared with 27. Hence the series is convergent. 

In the expansion, .2 occurs to the first power in the second term, to 
the second power in the third term, and to higher powers in the terms 
foUowing which are not put down. 

The simplified value of the second term is + ^\^ = .007407, of the 
third term is — ^^^ = — .000018. 

If we take only the first two terms in the expansion, and neglect all the 
rest, we obtain an approximate value for the cube root, 8.007407. 

If we take the first three terms, the more accurate value, 3.007889, is 
obtained. 

If in a calculation, account is taken of a number h which is small com- 
pared with the other numbers involved, but the square and higher powers 
of h are discarded, then the calculation is said to be carried to the fir^t 
approximation. 

If both the first and second powers of b are used, but no higher powers of 
&, then the calculation is carried to the second approximation. And so on. 

In the above example, where h = .2, the first approximation to the 
cube root of 27} is 8.007407 ; the second approximation is 8.007889. 

EXBBCISBS 

196. Find to a first approximation the values of the follow- 
ing expressions in which the second number is much smaller 
than the first : 

1. (32^)*. 4. (a* 4- 6)*. 

2. (63)* =(64-1)*. 6. Va« + 2 c. 

3. V99.=V100-1. 6. -^/IpTq, 

7. (a«-36)*. 8. (9m«-n)^. 9. ■\/a^—-\/a. 



SUPPLEMENT 187 

10. A brass cube 1 inch each way is heated until its edges 
are 1.003 inches long. Compute to the first approximation the 
area of each face and the volume of the cube. 

11. A metal cube 2 inches each way is cooled so that its 
volume is reduced to 7.998 cubic inches. What is now the 
length of an edge ? 

12. Compute to a second approximation the amount after 
10 years, of $ 575 at 2 %, interest compounded annually. 

Evaluate 676 (1 -h «02)^<> by retaining the first and second powers of 
.02 in the binomial expansion, but discarding higher powers. Does the 
first approximation give results differing from those due to simple 
interest ? 

13. In measuring the diameter of a sphere whose diameter 
is actually 5", an error of 1 % is probable. What is the probable 
error, computed to a first approximation, of the volume derived 
from the inaccurate diameter ? 

The volume of a sphere = | irt*. 
Take r» = \(6 + .06)». 

14. How great an error is made by assuming 

1-a^ 



1 + a 



= 1 — a + a* — a' + a*. 



when a = ^ ? What is the ratio of the error to the whole of 
the fraction ? 

16. Proceed on the assumption that the earth is a sphere, 
the radius of which is 4000 miles. On the supposition that its 
radius was at one time 10 miles greater than at present, calcu- 
late to the first approximation, and also to the second approxi- 
mation, the amount that it has lost during the contraction, 
(1) in superficial area, (2) in volume. 

[The superficial area = 4 irr^.] 



188 



ELEMENTARY ALGEBRA 



197. A Table of Squares and Cubes, Square Boots and Cube 

Roots of Numbers from 1 to 200, 









Square 


• 

Cubs 








Sqvars 


Cubs 


8QVAXB8 


CVBSt 


No. 

— * — 
1 


K0OT8 


Roots 

ft 


SQVAKn 


CVBM 


No. 
61 


Roots 


Roots 


1 


1 


1.000 


1.000 


2,601 


132,651 


7.141 


3.708 


4 


8 


2 


1.414 


1.260 


2,704 


140,608 


62 


7.211 


3.733 


9 


27 


8 


,M^.?32 


1.442 


2,809 


148,877 


68 


7.280 


3.756 


16 


64 


4 


2.000 


1.587 


2,916 


167,464 


64 


7.348 


3.780 


26 


125 


6 


2.236 


1.710 


3,026 


166,376 


66 


7.416 


3.803 


36 


216 


6 


2.449 


1.817 


3,136 


176,616 


66 


7.483 


3.826 


49 


343 


7 


2.646 


1.913 


3,249 


186,193 


67 


7.660 


3.849 


64 


612 


8 


2.828 


2.000 


3,364 


195,112 


68 


7.616 


3.871 


81 


729 


9 


3.000 


2.080 


3,481 


206,379 


69 


7.681 


3.893 


100 


1,000 


10 


3.162 


2.164 


3,600 


216,000 


60 


7.746 


3.915 


121 


1,331 


11 


3.317 


2.224 


3,721 


226,981 


61 


7.810 


3.936 


144 


1,728 


12 


3.464 


2.289 


f 3,844 


238,328 


62 


7.874 


3.958 


169 


2,197 


18 


3.606 < 


3.351 


• 3,969 


260,047 


68 


7.937 


3.979 


196 


2,744 


14 


3.742 


2.410 


4,096 


262,144 


64 


8.000 


4.000 


225 


3,376 


16 


3.873 


2.466 


4,226 


274,625 


66 


8.062 


4.021 


266 


4,096 


16 


4.000 


2.620 


4,356 


287,496 


66 


8.124 


4.041 


289 


4,913 


17 


4.123 


2.671 


4,489 


300,763 


67 


8.185 


4.061 


324 


6,832 


18 


4.243 


2.621 


4,624 


314,432 


68 


8.246 


4.062 


361 


6,859 


19 


4.359 


2.668 


4,761 


328,609 


69 


8.307 


4.102 


400 


8,000 


20 


4.472 


2.714 


4,900 


343,000 


70 


8.367 


4.121 


441 


9,261 ' 21 


4.583 


2.769 


6,041 


357,911 


71 


8.426 


4.141 


484 


10,648 22 


4.690 


2.802 


6,184 


373,248 


72 


8.486 


4.160 


629 


12,167 


28 


4.796 


2.844 


5,329 


389,017 


78 


8.644 


4.179 


676 


13,824 


24 


4.899 


2.884 


6,476 


405,224 


74 


8.602 


4.198 


625 


15,625 


26 


6.000 


2.924 


5,626 


421,876 


76 


8.660 


4217 


676 


17,676 


26 


6.099 


2.962 


5,776 


438,976 


76 


8.718 


4.236 


729 


19,683 


27 


6.196 


3.000 


6,929 


466,533 


77 


8.776 


4.254 


784 


21,952 28 


6.292 


3.037 


6.084 


474,562 


78 


8.832 


4.273 


841 


24,389 ; 29 


6.385 


3.072 


6,241 


493,039 


79 


8.888 


4.291 


900 


27,000 80 


6.477 


3.107 


6,400 


612,000 


80 


0.9S% 


4.309 


961 


29,791 81 


5.568 


3.141 


6,561 


631,441 


81 


9.000 


4.327 


1,024 


32,768 82 


5.657 


3.175 


6,724 


651,368 


82 


9.056 


4.344 


1,089 


35,937 88 


6.746 


3.208 


6,889 


671,787 


88 


9.110 


4.362 


1,156 


39,304 


84 


6.831 


3.240 


7,056 


692,704 


84 


9.165 


4.380 


1,225 


42,875 


86 


6.916 


3.271 


7,226 


614,125 


86 


9.219 


4.397 


1,296 


46,656 861 


6.000 


3.302 


7,396 


636,056 


86 


9.274 


4.414 


1,369 


50,653 


87 


6.083 


3.332 


7,669 


658,503 


87 


9.327 


4.431 


1,444 


54,872 


88 


6.164 


3.362 


7,744 


681,472 


88 


9.381 


4.448 


1,621 


59,319 


89 


6.245 


3.391 


7,921 


704,969 


89 


9.434 


4.465 


1,600 


64,000 


40 


6.325 


3.420 


8,100 


729,000 


90 


9.487 


4.481 


1,681 


68,921 


41 


6.403 


3.448 


8,281 


753,571 


91 


9.539 


4.498 


1,764 


74,088 


42 


6.481 


3.476 


8,464 


778,688 


92 


9.691 


4.614 


1,849 


79,507 


48 


6.657 


3.503 


8,649 


804,367 


98 


8.644 


4.631 


1,936 


85,184 


44 


6.633 


3.530 


8,836 


830,584 


94 


9.695 


4.647 


2,025 


91,125 


46 


6.708 


3.557 


9,025 


857,375 


96 


9.747 


4.563 


2,116 


97,336 , 46 


6.782 


3.683 


9,216 


884,736 


96 


9.798 


4.679 


2,209 


103,823 


47 


6.856 


3.609 


9,409 


912,673 


97 


9.849 


4.595 


2,304 


110,692 


48 


6.928 


3.634 


9,604 


941,192 


98 


9.900 


4.610 


2,401 


117,649 49 


7.000 


3.659 


9,801 


970,299 


90 


9.960 


4.626 


2,500 


125,000 60 


7.071 


3.684 


10,000 


1,000,000 


100 


10.000 


4.642 



SUPPLEMENT 



S4IM1B 


Cdbh 


Ko. 


Root* 


Etwm 


IKIFIBES 


ClTBia 


No. 


8OT.« 

Roots 


Roan 


10,201 


1 " 01 





10,050 


*.667 






1 1 


12.288 


O.320 


10,<HH 


1 OS 





10.100 


4.672 






1 2 


12.329 


6.337 


10.609 


1 27 





10.149 


4.688 






S 


12.369 


0.348 


10,816 


1 64 





10.198 


4.703 






4 


12.410 


6.360 


11,025 


1 25 





10.247 


4.718 






ISS 


12.500 


5.372 


11236 


1 16 





10.296 


4.733 






1 B 


12.490 


6.383 


lt.449 


1 43 





10.344 


4.747 






1 7 


12.530 


5.395 


i: a 


1 12 





10.392 


4.763 






1 8 


12.670 


6.406 


I )i 


I 39 


' 


10.440 


4.77V 






I 8 


12.610 


5.418 


i: » 


1 00 


10 


10.488 


4.791 






160 12.649 


6.429 


i: a 


1, , 31 


1 


10.536 


4.806 






1 1 . 12.689 


6.440 


11 H 


1,404,928 


1 


10.683 


4.820 






2 


12.728 


5.461 


i: 19 


1,442,897 


1 


:o.&to 


4.830 






3 


12.767 


5.483 




l,4Ht,544 


1 


10.677 


4.849 






84 


a 


5.474 


i: » 


1,020,870 


1 


10,724 








es 


5.486 


i; 16 


1,660,896 


1 


10.770 


4^877 






66 


12.884 


5.496 


i: » 


1,601,613 


1 


10.817 


4.891 






7 


12.923 


6J107 


i; 14 


1,643,032 


1 


10.863 








8 


12.961 


5.018 


H 11 


1.685,169 


1 


10.909 


4^919 






eg 


13.000 


6.029 


u « 


1,728,000 


£0 


10.954 


4.932 









13.038 


5.040 


U 11 


1,771,061 


2 


IIJWO 


4.M6 






71 


13.077 


6.660 




1B1B848 




11.IM0 


4.960 






7S 


13.116 


5.661 


U 19 


1,W«B«T 


8 


11.091 


4.973 






73 


13.103 


6.572 


11 '6 


1,! 4 


24 


11.136 


4.987 






4 


13.191 


5.583 


U » 


i; s 


U 


11.180 


6.000 






7fi 


13229 


5.593 


11 '6 


2,1 6 


s 


11.220 


0.013 






6 


13.266 


5.604 


If !9 


2/ 3 


2 


11.269 


6.027 






7 


13.304 


5.615 


1« H 


2,< a 


8 


11.314 


6.040 






78 


13,342 


0.820 


It H 


2, 8 


29 


11.368 


0.053 






79 


13.379 


6.636 


1( 10 


2 


S 


11.402 


6.066 






80 


13,416 


5.646 


1^ 11 


2 1 


S 


11.446 








81 


13454 


5.607 


n » 


2.1 8 


s 


11.489 


6:092 






82 


13.491 


5.667 


n 19 


2r 7 


3 


11.533 


0.104 






183 




6.677 


n i6 


2,' 4 


8 


11.576 


6.117 






184 


13!s6fl 


0.688 


If !9 


Z 5 


8 


11.619 


5.130 






186 


13.601 


0.698 


It e 


2J 6 


S 


11.662 


6.143 






186 




6.708 


It 19 


3.. 8 


3 


11.705 


0.155 






187 


13,870 


0.71S 


11 4 


2,1 2 


S 


11.747 


6.168 






188 


13.711 


6.729 


1! 11 


2) g 


3 


11.71« 


0.180 






89 


13.748 


0.739 


11 10 


2.' 


4 


11,832 


5.192 


™...0 




190 


13.784 


5.749 


li a 


2.: ;l 


4 


11.874 


0.206 


36,481 




191 


13,820 


5.769 


a M 


2; e 


4 


11.916 


6.217 


36,864 




1 2 


13.866 


6.769 


20,419 


2,' 1 


4 


11.968 


0.229 


37.249 




193 


13.892 


5.779 


20,736 


2; 4 


4 


12.000 


5.241 


37,636 




194 


13.928 


5.789 


21,026 


3.1 


4 


12.042 


5.2H 


38,025 




9S 


13.964 




21,316 


3 4i 


4 


12.083 


5.266 


38.416 




96 


14.000 


5.809 


21800 


3; 3 


4 


12.124 


0.278 


38,809 




97 


14.036 


5.819 


21,904 


3,; S 


4 


12.1H6 


5.290 


39,204 




98 


14.071 


0.828 


22.201 


3,. 9 


4 


12.207 


5.301 


30,601 




9 


14.107 


5.838 


22,600 


3;. 


S 


12.247 


5.313 


40;000 




800 


14.1*2 


6.848 



REVIEW EXERCISES 

(Sblbotbd from Collbob Entbanob Examinations) 

FACTORING 

198. 1. Factor 6 a* + 10a6 — 4&«, aj» + 125, 1 — a? — a^ + a^, 
a^ + a?* + 1. Univ, of State of N.Y. 

2. Find the prime factors of 

(a) (a?- ««)» + (« - !)• + (1 - x)K 

(6) (2aj + a — ft)* - (aj - a + by. Mass. InstUute of Tech. 

8. Factor a? — 2 mas -|- wi* — n', 

a^-^^ — 11 aj»+^"» -h 18 ajy»* Univ. of Pa. 

4. (a) Resolve the following into their prime factors : 

(1) (aj«-y«)«-3^. 

(2) 10aj»-7aj-6. 

(b) Find the h. c. f . and the 1. c. m. of 

aj»-3aj*4-a5-3, 

aj' — Sa^ — a5-f-3. Colunibia. 

6. Find the h. c. f . : »* — y*, 

a?* -h 2«y — 3 y*. Cornell. 

6. Factor the following expressions : 

(a) ai — 6i, 

(5) aj*yV — aJ*2 — yH; -h 1, 

(c) 16(a; -f- y)* — (2 a; — y)*- Jfoun^ Holyoke. 

190 



REVIEW EXERCISES 



191 



FRACTIONS 



a 



199. 1. Simplify tj- 



a 



fe' + 



cb 



b 



Harvard. 



2, Simplify the expression 



ic-hy — 



aj-hy- 



xy 



x-\-yl 



a^ — y^ 



Cornell. 



3. Divide /^g^ziJ^^ ^4-y Wa^ + y^^^±y.\ 

\a?-y^ Q^ — xyJ Xx — y xy — yy 



Sheffield. 



4. Simplify the expression 

A y\A ab-¥ \ a* ^ a--b 
\ ay\ a* Ja^ + b^ ' a^-^b^' 



Mass. Inst. Tech, 



6. Find the product of 



Vasaar. 



12 3- 

6. If m = T » n = -, p = — ;—p^f what is the value of 



m 



+ 



a + l' a + 2'^ a4-3 



+ 



1 — m 1 — w 1 —p 



Univ. of Pa. 



EQUATIONS 

200. 1. Solve ^ + ^±^ = ?. 

aj 4- X -{- a 2 



Tale. 



2, Solve for x : — z=. H — ^=3 = Va — b. Univ. of Pa. 



Va — X -y/x — b 



192 ELEMENTARY ALGEBRA 

3. Solve for a; : ?i^-?-6=s Cornell 

x-1 a?-|-2aj-3 



4. (a) Solve for a? : V2 a? - 3 a + V3 a? — 2 a = 3Va. 
(b) Solve for m: 

1-- = — —.4- ^" . Univ. of Pa. 



6. Solve H = 0. Princeton. 

a? 4- 1 05 — 1 a? — 3 a? — 5 



6. Solve the equation V2 x -}- 5 ^ V6 — as = 1. 

Uwiv. of Pa. 

7. Solve 3 aJ' - 11 aj = 70. CTniv. o/ /Seoie o/ N. T. 

8. Solve the quadratic equation 

aj» — 1.6 a? 4- 0.3 = 0. Harvard. 

9. If d is one of the roots of the equation aa? + 6aj -|- c = 0, 
find the other root. Univ. of Pa. 

10. Solve — ^ 4- - = 3. Univ. of State ofN. T. 

x-\- 1 X 



11. ■ Solve Vaj-f- a -f- Vi 4- -y/x — a = 0. 

Uhry. o/ State of N. Y. 

12. (i) Solve for t: {t- 2)(t 4- 3) = (3 « 4- 4)(2 - t). 
(ii) Solve for x : 

■^ H -^ = . -. Univ. of Pa. 



V^T2 V3aj-2 V3a:»4-4a;-4* 



REVIEW EXERCISES 



193 



RADICALS 
201. 1. Find the square root of 

af^^2ix^ — x^-\-Sa^-\-2x + l. Univ, of State of N.T. 



2. 



3. 



Univ, of State of N, Y. 



Solve for x and y the equations 

Vx-^-y = a H- 6, 
* — y = (« — ft) Va? + y. 



C/niv. of Pa, 



4. Simplify (a? -f 1)' — x-y/x H- 1, V— m'n • V— mn', 



i 1 
6*^ 






C/niv. o/ Pa, 



5. Simplify the product of 

(ayx~^)^y (bxy~^y, and (y*a"'6~*)*. Princeton. 



6. Solve the simultaneous equations 

x-i-h2y-i^i, 

7. Simplify (a) Ve - V20. (6) i^v^Tl + o:^ 

8. Find the value of a^6^, if 

a = a?^y » and 6 = ^ «"^y*, 



FoZe. 



Cornell, 



and reduce the result to a form having only positive exponents. 

Haroard. 

9. Simplify — — ^- — ^— , and compute the value of the 

fraction to two decimal places. Yale. 

o 



194 



ELEMENTARY ALGEBRA 



SYSTEMS OF LINEAR EQUATIONS 

. 1. Solve = a, - + - = 0, = c. 

X y y ^ z X 

Univ. of State of N.T. 

2. Solve the equations, 2 a + 5 y = 85, 

2^4-52 = 103, 
2 2 -I- 5 a? = 57. Vassar. 



3. Solve the following set of equations : 

« -H y = - 1. 
aj-h3y + 22 = -4. 
05 — y + 42 = 5. 



Cornell. 



4. Solve the equations : 

7a?-h6 ig^ 5g-13 8y-x 

11 "^^ 2 5 ' 

3(3iB + 4) = 10y-15. 



Mass, Inst. Tech. 



6. Draw the lines represented by the equations 

3 a— 2y= 13 and 2a;-|-5y= — 4, 

and find by algebra the coordinates of the point where they in- 
tersect. Univ. of Col. 

6. Solve the simultaneous equations : 

a + a 
and verify your results. Harvard. 



SYSTEM OF EQUATIONS, ONE OR BOTH QUADRATIC 
203. 1. Solve the system : 

aJ* + y' + «y = 54^, 



2. Solve 



a? — y = 2, 



a^ — a^ = 30. 



Univ. of Pa. 



Univ. of State of N. T. 



REVIEW EXERCISES 195 

3. Solve for t and u: - H — = 74, = 2. Univ. of Pa. 

4. Solve the equations 

2 a? — 3 y = 0. Columbia. 

5. Plot the following two equations, and find from the graphs 
the approximate value of their common solutions : 

a^ -f y2 = 26. 
4 a;* -f- 9 y* = 144. ColunMa. 

6. Solve the following pair of equations for x and y : 

a' + y' = 4, 

a? = (1 H- V2)y - 2. Oomc/Z. 



7. Solve 



a; — y = 2, 



and sketch the graphs. 

Mass. Inst. Tech. 



BINOMIAL THEOREM 

204. 1. Write in the simplest form the last three terms of 
the expansion of (4 a' — o^x^y. Tale. 

2. Expand (1 — V2)*^ by the Binomial Theorem. 

Univ. of Pa. 

3. Write out by the binomial theorem the first ^ve terms of 
/^2 a - — y . Univ. of State of N. T. 

4. Find the seventh term of f a + - ) • Columbia. 

\ ay 

6. In the expansion of ( 2a?-f-^--] the ratio of the fourth 
term to the fifth is 2 : 1. Find a?, Princeton, 



196 ELEMENTARY ALGEBRA 

PROBLEMS 

205. 1. Two cars of equal speed leave A and B, 20 mi. 
apart, at different times. Just as the cars pass each other an 
accident reduces the power and their speed is decreased 10 mi. 
per hour. One car makes the journey from A to B in 56 min., 
and the other from B to A in 72 min. What is their common 
speed ? Tale. 

2. The sum of two numbers is 13, and the sum of their 
cubes is 910. Find the smaller number, correct to the second 
decimal place. Vassar. 

3. A man arranges to pay a debt of $ 3600 in 40 monthly 
payments which form an A. P. After paying 30 of them he 
still owes ^ of his debt. What was his first payment ? 

Princeton. 

4. A page is to have a margin of 1 inch, and is to contain 
35 square inches of printing. How large must the page be, if 
the length is to exceed the width by 2 inches ? Mount Holyoke. 

5. Insert 10 arithmetical means between 10 and 61, and 
find the sum of the entire series. Univ. of Pa. 

6. A man bought a number of cattle which cost him in all 
$672. If each head had cost him $4 less, he would have 
been able to buy 3 more. How many did he buy and at what 
price? Univ. of Pa. 

7. A boy is 5 years older than his sister and ^ as old as 
his father ; the sum of the ages of all three is 51. Find the 
age of the father. Univ, of State of N. T. 

8. Find two consecutive numbers whose product is 306. 

Univ. of State of N. T. 

9. A man engaged to work a days on these conditions : for 
each day he worked he was to receive b cents, and for each day 
he was idle he was to forfeit c cents. At the end of a days he 
received d cents. How many days was he idle ? Univ. of Pa. 



REVIEW EXERCISES 197 

10. For what values of m will the roots of the equation 
(m -h |)a?2 — 2(m + l)aj -f 2 = be equal ? 

Form the equation whose roots are | and ^. Univ. of Pa. 

11. If a number of two digits be divided by the product of 
its digits, the quotient will be 6. If 9 be added to the number, 
the sum will be equal to the number obtained by interchanging 
the digits. What is the number ? Univ. of Pa. 

12. A and B each shoot thirty arrows at a target. B makes 
twice as many hits as A, and A makes three times as many 
misses as B. Find the number of hits and misses of each. 

Univ. of Cat. 

13. The sides of a triangle are a, h, c. Calculate the radii 
of the three circles having the vertices as centers, each being 
tangent externally to the other two. Harvard. 

14. The force P necessary to lift a weight IT by means of a 
certain machine is given by the formula 

P=a-h6Tr, 

where a and h are constants depending on the amount of fric- 
tion in the machine. If a force of 7 pounds will raise a weight 
of 20 pounds, and a force of 13 pounds will raise a weight of 
60 pounds, what force is necessary to raise a weight of 40 
pounds ? 

(First determine the constants a and h.) Harvard. 

16. Find the sum of 8 terms of the progression 

6 + 3^ -h 2| -f .... Halyard. 

16. How many terms must be taken in the series 2, 5, 8, 11, 
• • ., so that the sum shall be 345 ? Mass. Inst. Tech. 



INDEX 



Numbers refer to sections. 



Absolute, term, 70. 

value of a number, 1. 
Addition, 2. 

checking, 2. 

fractions, 68. 

radicals, 154. 
Algebraic expression, 1. 
Antecedent, 98. 
Antilogarithm, 119. 
Approximations, 195. 
Arithmetical mean, 171. 
Arithmetical operations, order of, 
Arithmetical series, 167. 
Associative laws,' 76. 

Base, 107. 
Binomial, 1. 

surds, 158. 

theorem, 84, 85, 192, 193. 
Braces, 6. 
Brackets, 6. 

Characteristic, 112. 
Checking, addition, 2. 

division, 9. 

equation, 20. 

multiplication, 8. 

subtraction, 3. 
Coefficient, 1. 

detached, 8. 
Commutative laws, 76. 
Complex numbers, 151, 164. 
Consequent, 98. 
Constants, 140. 
Contact, point of, 138. 
Convergent series, 195. 
Coordinates, 24. 

Degree, of equation, 19. 

of term, 8. 
Determinants, 87. 



Discriminant, 136. 
Dissimilar terms, 1. 
Division, 9. 

by zero, 94. 

checking, 9. 

of fractions, 62. 

of radicals, 157. 
Double root, 138. 

Elimination, by addition, 29. 

by comparison, 29. 
5. by substitution, 29. 

Equation, 19. 

axioms used in, 20. 

complete quadratic, 70. 

containing fractions, 95. 

cubic, 19. 

degree of, 19. 

double root, 138. 

exponential, 127. 

general quadratic, 70. 

graph of linear, 22. 

graph of quadratic, 138, 

impossible, 160. 

incomplete quadratic, 70. 

indeterminate, 22. 

involving parentheses, 21. 

irrational, 160. 

linear, 19, 22. 

of condition, 19. 

of identity, 19. 

quadratic, 19, 70, 131. 

quadratic form, 132, 162. 

quartic, 19. 

radical, 160, 162. 

root of, 19, 146. 
Equations, equivalent, 26, 27. 

homogeneous, 144. 

inconsistent, 27. 

independent, 24, 27. 

linear, 87. 

199 



200 



INDEX 



Equations — continued 

one linear, one quadratic, 139. 

simultaneous linear, 22. 

simultaneous quadratic, 139. 

solution by quadratics, 146. 

symmetrical, 143. 
Exponents, 1, 149, 150. 

fractional, 35. 

negative, 37. 

positive integral, 32. 

sero, 36. 
Extraneous roots, 95, 160. 
Extremes, 98. 

Factor, 1. 

H. C. F., 66, 189. 

imaginary, 82. 

irrational, 82. 

literal, 1. 

prime, 45. 

rational, 45. 

rationaUxing, 152, 157. 

theorem, 78. 
Factoring, 45, 79. 
Formulas, 31, 75, 97. 100, 101, 102, 

105, 126. 
Fractions, 58. 

addition, 68. 

complex, 64. 

division, 62. 

multiplication, 62. 

reduction of, 58. 

subtraction, 68. 

value of, 60. 
Fu{iction, 100, 101, 103, 138. 

Geometrical mean, 179. 
Geometrical series, 175. 

infinite, 183, 184, 185, 195. 
Graphs, 103, 106. 

identical, 26. 

of complex numbers, 164. 

of linear equation, 22. 

of quadratic equation, 138. 

of simultaneous linear equations, 
24. 

parallel, 27. 

variation shown by, 103, 105. 

H. C. F.. 66, 189. 



Hindu method, 191. 
Historical Notes, 77, 130, 164. 
Homogeneous expressions, 8, 144. 

Imaginary, factors, 82. 

numbers, 138, 151, 152, 156. 
Inconsistent equations, 27. 
Integral expression, 45. 
Interpolation, 119. 
Intersection, 138. 
Involution, 39. 
Is not equal to, 76, note. 

Laws of Algebra, 76. 
L. C D., 68. 
L. C. M., 66. 
Limits. 167, 185, 187. 
Logarithmic, curve, 108, 110. 

table, 117. 
Logarithms, 107, 117, 123. 

Mantissa, 112. 

Mathematical Induction, 192. 
Mean proportional, 98. 
Means, 98. 
Monomial, 1. 
Multiplication, 8. 

by detached coefficients, 8. 

checking, 8. 

fractions, 62. 

radicals, 156. 

type forms, 17. 

Number, absolute value, 1. 
Numbers, complex, 151, li4. 

imaginary, 138. 151, 166. 

irrational, 39, 151. 

negative,. 151. 

positive, 151. 

rational, 45, 151. 

real, 151, 165. 

Order of fundamental operations, 5. 
Origin, 164. 

Parentheses, 6, 12. 

equations, 21. 

insertion of, 14. 

removal of, 12. 
Polynomial, 1. 



INDEX 



201 



Portraits, de Morgan, 77. 

Wallis, frontispiece. 
Power, 1, 39. 

law of signs, 33. 
Principal roots, 38, 39, 160. 
Problems, practical, 31, 93, 97, 104, 

105, 126. 129. 
Products, special, 17. 
Proportion, 98. 

Quadratic^ equation, 19, 70. 

complete, 70, 73. 

general, 70. 

graph of, 138. 

incomplete, 70. 

solved by completing square, 73, 
191. 

solved by factoring, 73. 

solved by formula, 73. 

nature of roots, 136. 

relation of roots and coefficients, 
134. 
Quaternions, 76. 

Radicals, 152. 

addition, 154. 

division, 152^ 

multiplication, 156. 

rationalizing, 152, 153. 

simplifjring, 152. 

subtraction, 154. 
Radicand, 152. 
Ratio, 98. 

Rational integral expression, 45, 78. 
Remainder theorem, 78. 
Review exercises, 198. 
Root, double, 138. 



of equation, 19. 
square, 41, 43, 158. 
Roots, 39. 
extraneous, 95, 160. 
nature of, 136. 
principal, 38, 39, 160. 
relation between roots and coeffi- 
cient, 134. 

Series, arithmetical, 167. 

convergent, 185, 195. 

geometrical, 195. 

infinite, 183, 195. 
Similar terms, 1. 
Simultaneous, equations, 22, 139. 

homogeneous, 144. 

sjrmmetrical, 143. 
Square root,_41. 

of a+2V6, 158. 
of fractions, 43. 
of numbers, 43. 
tables of, 197. 
Subtraction, 3. 
checking, 38. 
fractions, 68. 
radicals, 154. 

Tables of square roots, 197. 
Terms, dissimilar, 1. 

similar, 1. 
Trinomial, 1. 

Variables, 101, 138, 140. 
Variation, 102, 103. 
Vinculum, 6. 

Zero, division by, 94. 
exponent, 36, 



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