MATH0
STAT.
Edward Bright
Mathematics Dept
r
A COURSE
OF
PURE MATHEMATICS
CAMBRIDGE UNIVERSITY PRESS
C. F. CLAY, MANAGER
LONDON : FETTER LANE, E.G. 4
NEW YORK : THE MACMILLAN CO.
BOMBAY \
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ALL RIGHTS RESERVED
A COURSE
OF
PURE MATHEMATICS
)
BY
G. H. HARDY, M.A., F.R.S.
FELLOW OF NEW COLLEGE
SAVILIAN PROFESSOR OF GEOMETRY IN THE UNIVERSITY
OF OXFORD
LATE FELLOW OF TRINITY COLLEGE, CAMBRIDGE
THIRD EDITION
Cambridge
at the University Press
1921
VU
First Edition 1908
Second Edition 1914
Edition 1921
PREFACE TO THE THIRD EDITION
NO extensive changes have been made in this edition. The mosfc
important are in 80-82, which I have rewritten in accord
ance with suggestions made by Mr S. Pollard.
The earlier editions contained no satisfactory account of the
genesis of the circular functions. I have made some attempt to
meet this objection in 158 and Appendix III. Appendix IV is also
an addition.
It is curious to note how the character of the criticisms I have
had to meet has changed. I was too meticulous and pedantic for
my pupils of fifteen years ago: I am altogether too popular for the
Trinity scholar of to-day. I need hardly say that I find such
criticisms very gratifying, as the best evidence that the book has
to some extent fulfilled the purpose with which it was written.
G. H. H.
August 1921
EXTRACT FROM THE PREFACE TO
THE SECOND EDITION
THE principal changes made in this edition are as follows.
I have inserted in Chapter I a sketch of Dedekind s theory
of real numbers, and a proof of Weierstrass s theorem concerning
points of condensation ; in Chapter IV an account of limits of
indetermination and the general principle of convergence ; in
Chapter V a proof of the Heine-Borel Theorem , Heine s theorem
concerning uniform continuity, and the fundamental theorem
concerning implicit functions; in Chapter VI some additional
matter concerning the integration of algebraical functions ; and
in Chapter VII a section on differentials. I have also rewritten
in a more general form the sections which deal with the defini
tion of the definite integral. In order to find space for these
insertions I have deleted a good deal of the analytical geometry
and formal trigonometry contained in Chapters II and III of
the first edition. These changes have naturally involved a
large number of minor alterations.
G. H. H.
October 1914
781474
EXTEACT FEOM THE PEEFACE TO THE
FIEST EDITION
book has been designed primarily for the use of first
year students at the Universities whose abilities reach or
approach something like what is usually described as scholarship
standard . I hope that it may be useful to other classes of
readers, but it is this class whose wants I have considered first.
It is in any case a book for mathematicians: I have nowhere
made any attempt to meet the needs of students of engineering
or indeed any class of students whose interests are not primarily
mathematical.
I regard the book as being really elementary. There are
plenty of hard examples (mainly at the ends of the chapters) : to
these I have added, wherever space permitted, an outline of the
solution. But I have done my best to avoid the inclusion of
anything that involves really difficult ideas. For instance, I make
no use of the principle of convergence : uniform convergence,
double series, infinite products, are never alluded to : and I prove
no general theorems whatever concerning the inversion of limit-
d*f d*f
operations I never even define 5-%- and =-4-. In the last two
cxdy dydx
chapters I have occasion once or twice to integrate a power-series,
but I have confined myself to the very simplest cases and given
a special discussion in each instance. Anyone who has read this
book will be in a position to read with profit Dr Bromwich s
Infinite Series, where a full and adequate discussion of all these
points will be found.
September 1908
CONTENTS
CHAPTER I
REAL VARIABLES
SECT. *AGE
1-2. Rational numbers . . . . . * -.-. , 1
3-7. Irrational numbers . . . -; * "~
8. Real numbers . 13
9. Relations of magnitude between real numbers . .15
10-11. Algebraical operations with real numbers .... 17
12. The number x/2 ..*... . 19
13-34. Quadratic surds ... . .. ,, . 19
15. The continuum ......... 23
16. The continuous real variable ; 26
17. Sections of the real numbers. Dedekind s Theorem . . 27
18. Points of condensation . . . .
19. Weierstrass s Theorem . V ." . . . . 30
Miscellaneous Examples 31
Decimals, 1. Gauss s Theorem, 6. Graphical solution of quadratic
equations, 20. Important inequalities, 32. Arithmetical and geometrical
means, 32. Schwarz s Inequality, 33. Cubic and other surds, 34.
Algebraical numbers, 36.
CHAPTER II
FUNCTIONS OF REAL VARIABLES
20. The idea of a function 38
21. The graphical representation of functions. Coordinates . 41
22. Polar coordinates . ........ djfc-
23. Polynomials . . . . ... &.
24-25. Rational functions 4 "
26-27. Algebraical functions ....- 4 9
28-29. Transcendental functions ... ...
30. Graphical solution of equations 58
31. Functions of two variables and their graphical repre
sentation ^
Vlll CONTENTS
SECT. PAG!*
32. Curves in a plane . . . . . . . 60
33. Loci in space . . . . . , . . 61
Miscellaneous Examples ...... 65
Trigonometrical functions, 53. Arithmetical functions, 55. Cylinders,
62. Contour maps, 62. Cones, 63. Surfaces of revolution, 63. Ruled
surfaces, 64. Geometrical constructions for irrational numbers, 66.
Quadrature of the circle, 68.
CHAPTER III
COMPLEX NUMBERS
34-38. Displacements * 69
39-42. Complex numbers -. .... . 78
43. The quadratic equation with real coefficients . . . 81
44. Argand s diagram . . . . . .. . . 84
45. de Moivre s Theorem . . . .? ., . ... . 86
46. Rational functions of a complex variable . .... . 88
47-49. Roots of complex numbers . . . . . * 98
Miscellaneous Examples . . . * . .... 101
Properties of a triangle, 90, 101. Equations with complex coefficients,
91. Coaxal circles, 93. Bilinear and other transformations, 94, 97, 104.
Cross ratios, 96. Condition that four points should be concyclic, 97.
Complex functions of a real variable, 97. Construction of regular polygons
by Euclidean methods, 100. Imaginary points and lines, 103.
CHAPTER IV
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
50. Functions of a positive integral variable . . . .106
51. Interpolation . 107
52. Finite and infinite classes 108
53-57. Properties possessed by a function of n for large values of n 109
58-61. Definition of a limit and other definitions . . . .116
62. Oscillating functions . 121
63-68. General theorems concerning limits 125
69-70. Steadily increasing or decreasing functions . . . l. il
71. Alternative proof of Weierstrass s Theorem . . . 1, 34<
72 The limit of x n 1 34
73. The limit of A +V .137
74. Some algebraical lemmas ...... . Ib S
75. The limit of n (#07-1). . 139
76-77. Infinite series . _ .140
78. The infinite geometrical series 143
1
CONTENTS IX
SECT. PAGE
79. The representation of functions of a continuous real variable
by means of limits - . . . . . . . 147
80. The bounds of a bounded aggregate ..... 149
81. The bounds of a bounded function . . . . 149
82. The limits of indetermination of a bounded function . . 150
83-84. The general principle of convergence . . . . .151
85-86. Limits of complex functions and series of complex terms . 153
87-88. Applications to z n and the geometrical series . . . 156
Miscellaneous Examples . . . . . . 157
Oscillation of sinn07r, 121, 123, 151. Limits of n k x n , %/x, f/n,
t , } a; n ,136, 139. Decimals, 143. Arithmetical series, 146. Harmonical
n! \n)
series, 147. Equation x n+l =f(x n ), 158. Expansions of rational functions,
159. Limit of a mean value, 160.
CHAPTER V
LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS
AND DISCONTINUOUS FUNCTIONS
89-92. Limits as #-s-oo or x-*~ co ...... 162
93-97. Limits as x+a ......... 165
98-99. Continuous functions of a real variable .... 174
100-104. Properties of continuous functions. Bounded functions.
The oscillation of a function in an interval . . 179
105-106. Sets of intervals on a line. The Heinc-Borel Theorem . 185 ^ *
107. Continuous functions of several variables .... 190
108-109. Implicit and inverse functions ...... 191 ,j^
Miscellaneous Examples . . . . . .194
Limits and continuity of polynomials and rational functions, 169, 176.
^m _ gin
Limit of - , 171. Orders of smallness and greatness, 172. Limit of
, 173. Infinity of a function, 177. Continuity of cos x and sinrr, 177*
Classification of discontinuities, 178.
CHAPTER VI
DERIVATIVES AND INTEGRALS
110-112. Derivatives . . ..... . . . 197
113. General rules for differentiation ...... 203
114. Derivatives of complex functions ..... 205
115. The notation of the differential calculus . . . . 205
116. Differentiation of polynomials ...... 207
117. Differentiation of rational functions ..... 209
118. Differentiation of algebraical functions . . .210.
CONTENTS
SECT.
PAGE
119. Differentiation of transcendental functions .... 212
120. Repeated differentiation .... 214
121. General theorems concerning derivatives. Rolle s Theorem 217
122-124. Maxima and minima 219
125-126. The Mean Value Theorem 226
127-128. Integration. The logarithmic function ... 228
129. Integration of polynomials ..... 232
130-131. Integration of rational functions . . k .... . 233
132-139. Integration of algebraical functions. Integration by
rationalisation. Integration by parts .... 236
140-144. Integration of transcendental functions ^ \ . .245
145. Areas of plane curves . .- . . . . 249
146. Lengths of plane curves .. . . * * , 251
Miscellaneous Examples .*..:.. .:".... 253
Derivative of x m , 201. Derivatives of cosz and siux, 201. Tangent
and normal to a curve, 201, 214. Multiple roots of equations, 208, 255
Rolle s Theorem for polynomials, 209. Leibniz Theorem, 215. Maxima
and minima of the quotient of two quadratics, 223, 256. Axes of a conic,
226. Lengths and areas in polar coordinates, 253. Differentiation of a
determinant, 254. Extensions of the Mean Value Theorem, 258. Formulae
of reduction, 259.
CHAPTER VII
ADDITIONAL THEOREMS IX THE DIFFERENTIAL AND INTEGRAL CALCULUS
147. Taylor s Theorem . 262
148. Taylor s Series ....... 266
149. Applications of Taylor s Theorem to maxima and minima . 268
150. Applications of Taylor s Theorem to the calculation of limits 268
151. The contact of plane curves . . . . . 270
152-154. Differentiation of functions of several variables . . . 274
155. Differentials f . . 280
156-161. Definite Integrals. Areas of curves ^ 283
162. Alternative proof of Taylor s Theorem .... 298
163. Application to the binomial series 299
164. Integrals of complex functions 299
Miscellaneous Examples . . . . . . 399
Newton s method of approximation to the roots of equations, 205.
Series for cosx and smx, 267. Binomial series, 267. Tangent to a curve,
272, 283, 303. Points of inflexion, 272. Curvature, 273, 302. Osculating
. I conies, 274, 302. Differentiation of implicit functions, 283. Fourier s
I integrals, 290, 294. The second mean value theorem, 296. Homogeneous
functions, 302. Euler s Theorem, 302. Jacobiaus, 303. Schwarz s in
equality for integrals, 300. Approximate values of definite integrals, 307. *
Simpson s Eule, 307.
) CONTENTS XI
CHAPTER VIII
THE CONVERGENCE OP INFINITE SERIES AND INFINITE INTEGRALS
SECT. PAGE
165-168. Series of positive terms. Cauchy s and d Alembert s tests
of convergence ...... . 308
169. Dirichlet s Theorem .313
170. Multiplication of series of positive terms . . . . 313
171-174. Further tests of convergence. Abel s Theorem. Maclaurin s
integral test . . . 315
175. The series Sn~ 8 . .319
* 176. Cauchy s condensation test 320
177-182. Infinite integrals . . . . . . . . .321
183. Series of positive and negative terms 335
184-185. Absolutely convergent series 336
-~ 186-187. Conditionally convergent series 338
188. Alternating series 340
189. Abel s and Dirichlet s tests of convergence . . . 342
190. Series of complex terms 344
191-194. Power series 345
195. Multiplication of series in general 349
Miscellaneous Examples 350
The series I,n k r n and allied series, 311. Transformation of infinite
integrals by substitution and integration by parts, 327, 328, 333. The
series Sa n cosn0, Sa n sinw0, 338, 343, 344. Alteration of the sum of a
series by rearrangement, 341. Logarithmic series, 348. Binomial series,
348, 349. Multiplication of conditionally convergent series, 350, 354.
Becurring series, 352. Difference equations, 353. Definite integrals, 355.
Schwarz s inequality for infinite integrals, 356.
CHAPTER IX
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS OF A REAL VARIABLE
196-197. The logarithmic function 357
198. The functional equation satisfied by log.T . . . . 360
199-201. The behaviour of log# as x tends to infinity or to zero . 3GO
202. The logarithmic scale of infinity . . . . . 362
203. The number e 363
204-206. The exponential function 364
207. The general power a* . .366
208. The exponential limit 368
209. The logarithmic limit 369
210. Common logarithms . . . . . . . . 369
211. Logarithmic tests of convergence. ..... 374
v
xii CONTENTS
SECT.
212. The exponential series . 373
213. The logarithmic series .381
214. The series for arc tan x 332
215. The binomial series . , . * . . 384
216. Alternative development of the theory . . , 386
Miscellaneous Examples . . . . . 337
Integrals containing the exponential function, 370. The hyperbolic
functions, 372. Integrals of certain algebraical functions, 373. Euler s
constant, 377, 389. Irrationality of e , 380. Approximation to surds by the
binomial theorem, 385. Irrationality of Iogi n, 387. Definite integrals, 393.
CHAPTER X
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL,
AND CIRCULAR FUNCTIONS
217-218. Functions of a complex variable . . . . . 395
219. Curvilinear integrals . . . . 4 . . ; . . 395
220. Definition of the logarithmic function .... 397
221. The values of the logarithmic function . . . . . ; . 399
222-224. The exponential function .. . . . . .. . 403
225-226. The general power a* . , 404
227-230. The trigonometrical and hyperbolic functions . . . 409
231. The connection between the logarithmic and inverse
trigonometrical functions 413
232. The exponential series 414
233. The series for coaz and sin z . . . . . . 416
234-235. The logarithmic series 417
236. The exponential limit 421
237. The binomial series 422
Miscellaneous Examples 425
The functional equation satisfied by Log 2, 402. The function e 3 , 407.
Logarithms to any base, 408. The inverse cosine, sine, aad tangent of a
complex number, 412. Trigonometrical series, 417, 420, 431. Boots of
transcendental equations, 425. Transformations, 426, 428. Stereographic
projection, 427. Mercator s projection, 428. Level curves, 429. Definite
integrals, 432.
APPENDIX I. The proof that every equation has a root . . . 433
APPENDIX II. A note on double limit problems 439
APPENDIX III. The circular functions 443
APPENDIX IV. The infinite in analysis and geometry . . . 445
CHAPTER I
REAL VARIABLES
1. Rational numbers. A fraction r=p/q, where p and q
are positive or negative integers, is called a rational number. We
can suppose (i) that p and q have no common factor, as if they
have a common factor we can divide each of them by it, and
(ii) that q is positive, since
pl(-q) = (-p)lq, (-p)/(-q)=p/q.
To the rational numbers thus denned we may add the rational
number obtained by taking p = 0.
We assume that the reader is familiar with the ordinary
arithmetical rules for the manipulation of rational numbers. The
examples which follow demand no knowledge beyond this.
Examples I. 1. Ifr and s are rational numbers, then r + s, r - *, rs, and
rjs are rational numbers, unless in the last case s = (when r/s is of course
meaningless).
2. If X, w, and n are positive rational numbers, and m > n, then
X(?n 2 -ft 2 ), 2Xwm, and \(m 2 + n 2 ) are positive rational numbers. Hence show
how to determine any number of right-angled triangles the lengths of all of
whose sides are rational.
3. Any terminated decimal represents a rational number whose denomi
nator contains no factors other than 2 or 5. Conversely, any such rational
number can be expressed, and in one way only, as a terminated decimal.
[The general theory of decimals will be considered in Ch. IV.]
4. The positive rational numbers may be arranged in the form of a simple
series as follows :
Show that plq is the [ (p + q - 1) (p + q - 2) + q]ih term of the series.
[In this series every rational number is repeated indefinitely. Thus 1
occurs as i, f ,,.... We can of course avoid this by omitting every number
H. 1
1
^ , , REAL VARIABLES [l
which has already occurred in a simpler form, but then the problem of deter
mining the precise position of pjq becomes more complicated.]
2. The representation of rational numbers by points
on a line. It is convenient, in many branches of mathematical
analysis, to make a good deal of use of geometrical illustrations.
The use of geometrical illustrations in this way does not, of
course, imply that analysis has any sort of dependence upon
geometry : they are illustrations and nothing more, and are em
ployed merely for the sake of clearness of exposition. This being
so, it is not necessary that we should attempt any logical analysis
of the ordinary notions of elementary geometry; we maybe content
to suppose, however far it may be from the truth, that we know
what they mean.
Assuming, then, that we know what is meant by a straight
line, a segment of a line, and the length of a segment, let us take
a straight line A, produced indefinitely in both directions, and a
segment A A l of any length. We call A the origin, or tJte point
0, and A l the point 1, and we regard these points as representing
the numbers and 1.
In order to obtain a point which shall represent a positive
rational number r=p/q, we choose the point A r such that
A A r being a stretch of the line extending in the same direction
along the line as A A l} a direction which we shall suppose to be
from left to right when, as in Fig. 1, the line is drawn horizontally
across the paper. In order to obtain a point to represent a
~ATT" ~AT~ A A x A,
Fig. 1.
negative rational number r = s, it is natural to regard length as
a magnitude capable of sign, positive if the length is measured in
one direction (that of A^A-^, an d negative if measured in the
other, so that AB = -BA ; and to take as the point representing
r the point A- s such that
1-3] REAL VARIABLES 3
We thus obtain a point A r on the line corresponding to every
rational value of r, positive or negative, and such that
and if, as is natural, we take A A 1 as our unit of length, and write
A A l = 1, then we have
A A r = r.
We shall call the points A r the rational points of the line.
3. Irrational numbers. If the reader will mark off on the
line all the points corresponding to the rational numbers whose
denominators are 1, 2, 3, ... in succession, he will readily convince
himself that he can cover the line with rational points as closely
as he likes. We can state this more precisely as follows : if we
take any segment EG on A, we can find as many rational points as
we please on BO.
Suppose, for example, that BC falls within the segment A! A.,.
It is evident that if we choose a positive integer k so that
k.BC>l ........................ (1),*
and divide A^A^ into k equal parts, then at least one of the points j
of division (say P) must fall inside BC, without coinciding with
either B or C. For if this were not so, BC would be entirely
included in one of the k parts into which A 1 A 2 has been divided,
which contradicts the supposition (1). But P obviously corre
sponds to a rational number whose denominator is k. Thus at
least one rational point P lies between B and C. But then we
can find another such point Q between B and P, another between
B and Q, and so on indefinitely ; i.e., as we asserted above, we can
find as many as we please. We may express this by saying that
BC includes infinitely many rational points.
The meaning of such phrases as * infinitely many or an infinity of\ in
such sentences as BC includes infinitely many rational points or there are
an infinity of rational points on C } or there are an infinity of positive
integers , will be considered more closely in Ch. IV. The assertion there are
an infinity of positive integers means given any positive integer n, however
large, we can find more than n positive integers . This is plainly true
* The assumption that this is possible is equivalent to the assumption of what
is known as the Axiom of Archimedes.
12
4 REAL VARIABLES [l
whatever n may be, e.g. for n = 100,000 or 100,000,000. The assertion means
exactly the same as we can find as many positive integers as we please .
The reader will easily convince himself of the truth of the follo\ving
assertion, which is substantially equivalent to what was proved in the second
paragraph of this section : given any rational number r, and any positive
integer ?i, we can find another rational number lying on either side of r and
differing from r by less than l/n. It is merely to express this differently to
say that we can find a rational number lying on either side of r and differing
from r by as little as we please. Again, given any two rational numbers-
r and s, we can interpolate between them a chain of rational numbers in
which any two consecutive terms differ by as little as we please, that is to
say by less than l/?i, where n is any positive integer assigned beforehand.
From these considerations the reader might be tempted to
infer that an adequate view of the nature of the line could be
obtained by imagining it to be formed simply by the rational
points which lie on it. And it is certainly the case that if we
imagine the line to be made up solely of the rational points,
and all other points (if there are any such) to be eliminated,
the figure which remained would possess most of the properties-
which common sense attributes to the straight line, and would,
to put the matter roughly, look and behave very much like
a line.
A little further consideration, however, shows that this view
would involve us in serious difficulties.
Let us look at the matter for a moment with the eye of
common sense, and consider some of the properties which we may
reasonably expect a straight line to possess if it is to satisfy the
idea which we have formed of it in elementary geometry.
The straight line must be composed of points, and any segment
of it by all the points which lie between its end points. With
any such segment must be associated a certain entity called its
length, which must be a quantity capable of numerical measure
ment in terms of any standard or unit length, and these lengths
must be capable of combination with one another, according to-
the ordinary rules of algebra, by means of addition or multipli
cation. Again, it must be possible to construct a line whose
length is the sum or product of any two given lengths. If the
length PQ t along a given line, is a, and the length QR, along
the same straight line, is b, the length PR must be a + 6.
3]
REAL VARIABLES
Moreover, if the lengths OP, OQ, along one straight line, are
1 and a, and the length OR along another straight line is b,
and if we determine the length OS by Euclid s construction (Euc.
VI. 12) for a fourth proportional to the lines OP, OQ, OR, this
length must be ab, the algebraical fourth proportional to 1, a, b.
And it is hardly necessary to remark that the sums and products
thus defined must obey the ordinary laws of algebra ; viz.
a + b = b + a, a + (b 4- c) = (a + b) + c,
ab = ba, a (be) = (ab) c, a (b + c) = ab + ac.
The lengths of our lines must also obey a number of obvious
laws concerning inequalities as well as equalities : thus if
A, B, C are three points lying along A from left to right, we must
have AB< AC, and so on. Moreover it must be possible, on our
fundamental line A, to find a point P such that A Q P is equal to
any segment whatever taken along A or along any other straight
line. All these properties of a line, and more, are involved in the
presuppositions of our elementary geometry.
Now it is very easy to see that the idea of a straight line as
composed of a series of points, each corresponding to a rational
number, cannot possibly satisfy all these requirements. There are
various elementary geometrical constructions, for example, which
purport to construct a length x such that x* = 2. For instance, we
M
Fig. 2.
may construct an isosceles right-angled triangle ABC such that
AB = AC=1. Then if BC=oc, # 2 = 2. Or we may determine
the length x by means of Euclid s construction (Euc. vi. 13) for
a mean proportional to 1 and 2, as indicated in the figure. Our
requirements therefore involve the existence of a length measured
by a number x, and a point P on A such that
6 EEAL VARIABLES [l
But it is easy to see that there is no rational number such that
its square is 2. In fact we may go further and say that there
is no rational number whose square is m/n, where m/n is any
positive fraction in its lowest terms, unless m and n are both
perfect squares.
For suppose, if possible, that
p having no factor in common with q, and m no factor in common
with n. Then np 2 = mq z . Every factor of <f must divide np 2 , and
as p and q have no common factor, every factor of (f must divide
n. Hence n = \q*, where X is an integer. But this involves
m = \p 2 : and as m and n have no common factor, X must be unity.
Thus in = p 2 , n q*, as was to be proved. In particular it follows,
by taking n = 1, that an integer cannot be the square of a rational
number, unless that rational number is itself integral.
It appears then that our requirements involve the existence of
a number x and a point P, not one of the rational points already
constructed, such that A P = x, # 2 = 2; and (as the reader will
remember from elementary algebra) we write x \/2.
The following alternative proof that no rational number can have its
square equal to 2 is interesting.
Suppose, if possible, that pjq is a positive fraction, in its lowest terms,
such that (p/20 2 = 2 or p 2 = 2q 2 . It is easy to see that this involves
(2g j) 2 =2(p- <?) 2 ; and so (2q p)/(p q) is another fraction having the
same property. But clearly q<p<2q, and so p-q<q. Hence there is
another fraction equal to pfq and having a smaller denominator, which
contradicts the assumption that p/q is in its lowest terms.
Examples II. 1. Show that no rational number can have its cube equal
to 2.
2. Prove generally that a rational fraction pjq in its lowest terms cannot
be the cube of a rational number unless p and q are both perfect cubes.
3. A more general proposition, which is due to Gauss and includes those
which precede as particular cases, is the following : an algebraical equation
with integral coefficients, cannot have a rational but non-integral root.
[For suppose that the equation has a root a/b, where a and b are integers
3, 4] REAL VARIABLES 7
without a common factor, and b is positive. Writing a/b for #, and multiply
ing by 6* 1 " 1 , we obtain
a fraction in its lowest terms equal to an integer, which is absurd. Thus 6
and the root is a. It is evident that a must be a divisor of p n .\
4. Show that if p n =l and neither of
is zero, then the equation cannot have a rational root.
5. Find the rational roots (if any) of
10 = 0.
[The roots can only be integral, and so "T, + 2, 5, +10 are the only
possibilities : whether these are roots can be determined by trial. It is clear
that we can in this way determine the rational roots of any such equation.]
4. Irrational numbers (continued). The result of our
geometrical representation of the rational numbers is therefore to
suggest the desirability of enlarging our conception of c number
by the introduction of further numbers of a new kind.
The same conclusion might have been reached without the use
of geometrical language. One of the central problems of algebra
is that of the solution of equations, such as
#2=1, a? = 2.
The first equation has the two rational roots 1 and 1. But,
if our conception of number is to be limited to the rational
numbers, we can only say that the second equation has no roots;
and the same is the case with such equations as ot? = 2, at = 7.
These facts are plainly sufficient to make some generalisation of
our idea of number desirable, if it should prove to be possible.
Let us consider more closely the equation # 2 = 2.
We have already seen that there is no rational number x which
satisfies this equation. The square of any rational number is
either less than or greater than 2. We can therefore divide the
rational numbers into two classes, one containing the numbers
whose squares are less than 2, and the other those whose squares
are greater than 2. We shall confine our attention to the -positive
rational numbers, and we shall call these two classes the class L, or
the lower class, or the left-hand class, and the class R, or the upper
8 REAL VARIABLES [l
class, or the right-hand class. It is obvious that every member of
R is greater than all the members of L. Moreover it is easy to
convince ourselves that we can find a member of the class L whose
square, though less than 2, differs from 2 by as little as we please,
and a member of R whose square, though greater than 2, also
differs from 2 by as little as we please. In fact, if we carry out
the ordinary arithmetical process for the extraction of the square
root of 2, we obtain a series of rational numbers, viz.
1, 1-4, 1-41. 1-414, 1-4142,...
whose squares
1, 1-96, 1-9881, 1-999396, 1-99996164,...
are all less than 2, but approach nearer and nearer to it ; and by
taking a sufficient number of the figures given by the process we
can obtain as close an approximation as we want. And if we
increase the last figure, in each of the approximations given above,
by unity, we obtain a series of rational numbers
2, 1-5, 1-42, 1-415, 1-4143,...
whose squares
4, 2-25, 2-0164, 2*002225, 2-00024449,...
are all greater than 2 but approximate to 2 as closely as we please.
The reasoning which precedes, although it will probably convince the
reader, is hardly of the precise character required by modern mathematics.
We can supply a formal proof as follows. In the first place, we can find
a member of L and a member of 21, differing by as little as we please. For
we saw in j^that, given any two rational numbers a and b, we can construct
a chain of rational numbers, of which a and b are the first and last, and in
which any two consecutive numbers differ by as little as we please. Let us
then take a member x of L and a member y of J?, and* interpolate between
them a chain of rational numbers of which x is the first and y the last, and
in which any two consecutive numbers differ by less than S, d being any
positive rational number as small as we please, such as 01 or -0001 or 000001.
In this chain there must be a last which belongs to L and a first which belongs
to R, and these two numbers differ by less than d.
We can now prove that an x can be found in L and a y in R such that
<Z x l and y 2 -2 are as small as we please, say less than 8. Substituting j
for 8 in the argument which precedes, we see that we can choose x and y so
that y x<\\ and we may plainly suppose that both x and y are less
than 2. Thus
4, 5] HEAL VARIABLES
and since ,?; 2 <2 and ^ 2 >2 it follows a fortiori that 2 x* and ?/ 2 2 are each
less than d.
It follows also that there can be no largest member of L or
smallest member of R. For if x is any member of Z, then # 2 < 2.
Suppose that # 2 = 2 S. Then we can find a member ^ of L
such that #! 2 differs from 2 by less than 8, and so # x 2 > # 2 or ^ > x.
Thus there are larger members of L than #; and as x is o^
member of L, it follows that no member of L can be larger than
all the rest. Hence L has no largest member, and similarly R has
no smallest.
5. Irrational numbers (continued}. We have thus divided
the positive rational numbers into two classes, L and R, such that
(i) every member of R is greater than every member of L, (ii) we
can find a member of L and a member of R whose difference is as
small as we please, (iii) L has no greatest and R no least member.
Our common-sense notion of the attributes of a straight line, the
requirements of our elementary geometry and our elementary
algebra, alike demand the existence of a number x greater than all
the members of L and less than all the members of R, and of
a corresponding point P on A such that P divides the points which
correspond to members of L from those which correspond to members
ofR.
L L L LL
RR R R R
.... | j 1 j I 1
A
r 1
Fig. 3.
Let us suppose for a moment that there is such a number x,
and that it may be operated upon in accordance with the laws of
algebra, so that, for example, # 2 has a definite meaning. Then # 2
cannot be either less than or greater than 2. For suppose, for
example, that so 2 is less than 2. Then it follows from what pre
cedes that we can find a positive rational number f such that f 2 lies
10 REAL VARIABLES [l
between # 2 and 2. That is to say, we can find a member of L
greater than x\ and this contradicts the supposition that # divides
the members of L from those of R. Thus x- cannot be less than
2, and similarly it cannot be greater than 2. We are therefore
driven to the conclusion that # 2 = 2, and that x is the number
which in algebra we denote by \/2. And of course this number
\/2 is not rational, for no rational number has its square equal to
2. It is the simplest example of what is called an irrational
number.
But the preceding argument may be applied to equations
other than x* 2, almost word for word ; for example to # 2 = N t
where N is any integer which is not a perfect square, or to
a; 3 = 3, a 3 = 7, a 4 = 23,
or, as we shall see later on, to x 3 = 3x + 8. We are thus led to
believe in the existence of irrational numbers x and points P on
A such that x satisfies equations such as these, even when these
lengths cannot (as \/2 can) be constructed by means of elementary
geometrical methods.
The reader will no doubt remember that in treatises on elementary algebra
the root of such an equation as xftn is denoted by fln or n 1 ^, and that a
meaning is attached to such symbols as
by means of the equations
And he will remember how, in virtue of these definitions, the laws of indices
such as
n r xn*=n r + 8 , (n r } 8 =n ri
are extended so as to cover the case in which r and 5 are any rational numbers
whatever.
The reader may now follow one or other of two alternative
courses. He may, if he pleases, be content to assume that
irrational numbers such as \/2, ^3, . . . exist and are amenable to
the algebraical laws with which he is familiar*. If he does this
he will be able to avoid the more abstract discussions of the next
few sections, and may pass on at once to 13 et seq.
If, on the other hand, he is not disposed to adopt so naive an
* This is the point of view which was adopted in the first edition of this book.
5, 6] REAL VARIABLES 11
attitude, he will be well advised to pay careful attention to the
sections which follow, in which these questions receive fuller
consideration *.
Examples III. 1. Find the difference between 2 and the squares of the
decimals given in 4 as approximations to x /2,
2. Find the differences between 2 and the squares of
r^
i, s, i, ti, u, n-
1 *
3. Show that if mjn is a good approximation to v/2, then (m-f-2)/(fl-f n) r
is a better one, and that the errors in the two cases are in opposite directions.
Apply this result to continue the series of approximations in the last
example.
.1
4. . If x and y are approximations to x/2, by defect and by excess respec
tively, and 2 - # 2 < 8, if- - 2 < S, then y-x<.
5. The equation # 2 =4 is satisfied by #=2. Examine how far the argu
ment of the preceding sections applies to this equation (writing 4 for 2
throughout). [If we define the classes Z, R as before, they do not include all
rational numbers. The rational number 2 is an exception, since 2 2 is neither
less than or greater than 4.]
6. Irrational numbers (continued}. In 4 we discussed
a special mode of division of the positive rational numbers x into
two classes, such that a? < 2- for the members of one class and
# 2 > 2 for those of the others. Such a mode of division is called a
section of the numbers in question. It is plain that we could
equally well construct a section in which the numbers of the two
classes were characterised by the inequalities a? < 2 and x 3 > 2, or
# 4 < 7 and X A > 7. Let us now attempt to state the principles
of the construction of such sections of the positive rational
numbers in quite general terms.
Suppose that P and Q stand for two properties which are
mutually exclusive and one of which must be possessed by every
positive rational number. Further, suppose that every such
number which possesses P is less than any such number which
possesses Q. Thus P might be the property x 2 < 2 and Q the
property x* > 2. Then we call the numbers which possess P the
lower or left-hand class L and those which possess Q the upper or
* In these sections I have borrowed freely from Appendix I of Bromwich s
Infinite Series.
12 REAL VARIABLES [l
right-hand class R. In general both classes will exist ; but it may
happen in special cases that one is non-existent and that every
number belongs to the other. This would obviously happen, for
example, if P (or Q) were the property of being rational, or of
being positive. For the present, however, we shaft confine
ourselves to cases in which both classes do exist ; and then it
follows, as in 4, that we can find a member of L and a member
of R whose difference is as small as we please.
In the particular case which we considered in 4, L had no
greatest member and R no least. This question of the existence
of greatest or least members of the classes is of the utmost im
portance. We observe first that it is impossible in any case that
L should have a greatest member and R a least. For if I were
the greatest member of L, and r the least of R, so that I < r, then
i (I + T) would be a positive rational number lying between I and
r, and so could belong neither to L nor to R ; and this contradicts
our assumption that every such number belongs to one class or to
the other. This being so, there are but three possibilities, which
are mutually exclusive. Either (i) L has a greatest member I, or
(ii) R has a least member r, or (iii) L has no greatest member and
R no least.
The section of 4 gives an example of the last possibility. An example
of the first is obtained by taking P to be a? 2 < 1 and Q to be *# 2 >1 ;
here Z=l. If P is # 2 < 1 and Q is x 2 > 1, we have an example of the
second possibility, with r\. It should be observed that we do not obtain
a section at all by taking P to be x* < 1 and Q to be # 2 > 1 ; for the special
number 1 escapes classification (cf. Ex. in. 5). y> 1)
7. Irrational numbers (continued). In the first two cases
we say that the section corresponds to a positive rational number
a, which is I in the one case and r in the other. Conversely, it is
clear that to any such number a corresponds a section which
we shall denote by a*. For we might take P and Q to be the
properties expressed by
x a, x > a
respectively, or by sc < a and x ^ a. In the first case a would be
the greatest member of L, and in the second case the least member
* It will be convenient to denote a section, corresponding to a rational number
denoted by an English letter, by the corresponding Greek letter.
6-8] REAL VARIABLES 13
of R. There are in fact just two sections corresponding to any
positive rational number. In order to avoid ambiguity we select
one of them ; let us select that in which the number itself belongs
to the upper class. In other words, let us agree that we will consider
A ~- " * "
only sections in which the lower class L has no greatest number.
There being this correspondence between the positive rational
numbers and the sections denned by means of them, it would be
perfectly legitimate, for mathematical purposes, to replace the
numbers by the sections, and to regard the symbols which occur
in our formulae as standing for the sections instead of for the
numbers. Thus, for example, a > a would mean the same as
a > a, if cc and a are the sections which correspond to a and a .
But when we have in this way substituted sections of rational
numbers for the rational numbers themselves, we are almost forced
to a generalisation of our number system. For there are sections
(such as that of 4) which do not correspond to any rational
number. The aggregate of sections is a larger aggregate than that
of the positive rational numbers; it includes sections corresponding
to all these numbers, and more besides. It is this fact which we
make the basis of our generalisation of the idea of number. We
accordingly frame the following definitions, which will however be
modified in the next section, and must therefore be regarded as
temporary and provisional.
A section of the positive rational numbers, in which both classes
exist and the lower class has no greatest member, is called a
positive real number.
A positive real number which does not correspond to a positive
rational number is called a positive irrational number.
8. Real numbers. We have confined ourselves so far to
certain sections of the positive rational numbers, which we have
agreed provisionally to call positive real numbers. Before we
frame our final definitions, we must alter our point of view a
little. We shall consider sections, or divisions into two classes,
not merely of the positive rational numbers, but of all rational
numbers, including zero. We may then repeat all that we have
said about sections of the positive rational numbers in 6, 7,
merely omitting the word positive occasionally.
s
REAL VARIABLES [l
DEFINITIONS. A section of the rational numbers, in which both
classes exist and the lower class has no greatest member, is called
a real number, or simply a number.
A real number which does not correspond to a rational number
is called an irrational number.
If the real number does correspond to a rational number, we
shall use the term rational as applying to the real number also.
The term rational number will, as a result of our definitions, be
ambiguous; it may mean the rational number of 1, or the corresponding
real number. If we say that > , we may be asserting either of two different
propositions, one a proposition of elementary arithmetic, the other a proposition
concerning sections of the rational numbers. Ambiguities of this kind are
common in mathematics, and are perfectly harmless, since the relations
between different propositions are exactly the same whichever interpretation
is attached to the propositions themselves. From i> and >j we can
infer > J ; the inference is in no way affected by any doubt as to whether
, J, and |- are arithmetical fractions or real numbers. Sometimes, of course,
the context in which (e.g.} occurs is sufficient to fix its interpretation.
When we say (see 9) that J< V(i), we must mean by the real number .
The reader should observe, moreover, that no particular logical importance
is to be attached to the precise form of definition of a real number 3 that we
have adopted. We defined a real number as being a section, i.e. a pair of
classes. We might equally well have defined it as being the lower, or the
upper, class ; indeed it would be easy to define an infinity of classes of
entities each of which would possess the properties of the class of real
numbers. What is essential in mathematics is that its symbols should be
capable of some interpretation ; generally they are capable of many, and
then, so far as mathematics is concerned, it does not matter which we adopt.
Mr Bertrand Russell has said that mathematics is the science in which
we do not know what we are talking about, and do not care whether what
we say about it is true , a remark which is expressed in the form of a
paradox but which in reality embodies a number of important truths. It
would take too long to analyse the meaning of Mr Russell s epigram in detail,
but one at any rate of its implications is this, that the symbols of mathe
matics are capable of varying interpretations, and that we are in general at
liberty to adopt whichever we prefer.
There are now three cases to distinguish. It may happen that
all negative rational numbers belong to the lower class and zero
and all positive rational numbers to the upper. We describe
this section as the real number zero. Or again it may happen
that the lower class includes some positive numbers. Such a section
8, 9] REAL VARIABLES 15
we describe as a positive real number. Finally it may happen
that some negative numbers belong to the upper class. Such
a section we describe as a negative real number*.
The difference between our present definition of a positive real number a
and that of 7 amounts to the addition to the lower class of zero and all the
negative rational numbers. An example of a negative real number is given
by taking the property P of 6 to be x + I<0 and Q to be #+1^0.
This section plainly corresponds to the negative rational number - 1. If we
took P to be # 3 < -2 and Q to be 3?> - 2, we should obtain a negative real
number which is not rational.
9. Relations of magnitude between real numbers. It
is plain that, now that we have extended our conception of
number, we are bound to make corresponding extensions of our
conceptions of equality, inequality, addition, multiplication, and so
on. We have to show that these ideas can be applied to the new
numbers, and that, when this extension of them is made, all the
ordinary laws of algebra retain their validity, so that we can
operate with real numbers in general in exactly the same way
as with the rational numbers of 1. To do all this systematically
would occupy a considerable space, and we shall be content to
indicate summarily how a more systematic discussion would
proceed.
We denote a real number by a Greek letter such as a, /3, 7, . . . ;
the rational numbers of its lower and upper classes by the corre
sponding English letters a, A ; 6, B\ c, C; .... The classes them
selves we denote by (a), (A), ..,
If a and /3 are two real numbers, there are three possibilities :
(i) every a is a b and every A&B\ in this case (a) is identical
with (b) and (A) with
* There are also sections in which every number belongs to the lower or to
the upper class. The reader may be tempted to ask why we do not regard these
sections also as defining numbers, which we might call the real numbers positive
and negative infinity.
There is no logical objection to such a procedure, but it proves to be incon
venient in practice. The most natural definitions of addition and multiplication do
not work in a satisfactory way. Moreover, for a beginner, the chief difficulty in the
elements of analysis is that of learning to attach precise senses to phrases containing
the word infinity ; and experience seems to show that he is likely to be confused by
any addition to their number.
16 REAL VARIABLES [i
(ii) every a is a b, but not all A s are B s ; in this case (a) is
a proper part of (6)*, and (B) a proper part of (A) ;
(iii) every A is a B, but not all a s are b s.
These three cases may be indicated graphically as in Fig. 4.
In case (i) we write a = ft, in case (ii) a < ft, and in case
(iii) a > ft. It is clear that, when
a and ft are both rational, these ? (i)
definitions agree with the ideas of
equality and inequality between + (ii)
rational numbers which we began
by taking for granted; and that ? 1 (iii)
any positive number is greater Fi g . 4.
than any negative number.
It will be convenient to define at this stage the negative a
ot a positive number a. If (a), (A) are the classes which consti
tute a, we can define another section of the rational numbers by
putting all numbers A in the lower class and all numbers a
in the upper. The real number thus defined, which is clearly
negative, we denote by a. Similarly we can define a when a
is negative or zero ; if a is negative, a is positive. It is plain
also that ( a) = a. Of the two numbers a and a one is always
positive (unless a = 0). The one which is positive we denote by
| a and call the modulus of a.
Examples IV. 1. Prove that = - 0.
2. Prove that /3 = a, j3<a, or /3>a according as a=ft a>ft or a</3.
3. If a = /3 and/3 = y, then a=y.
4. If a < ft /3<y, or a<ft /3 ^ y, then a<y.
5. Prove that /3 = a -/3< -a, or /3> a, according as a = /3, a<,3,
or a>/3.
6. Prove that a>0 if a is positive, and a<0 if a is negative.
7. Prove that a < a | .
8. Prove that 1< v/2 < v/3 < 2.
9. Prove that, if a and /3 are two different real numbers, we can always
find an infinity of rational numbers lying between a arid /3.
[All these results are immediate consequences of our definitions.]
* I.e. is included in but not identical with (&).
9, 10] REAL VARIABLES 17
10. Algebraical operations with real numbers. We now
proceed to define the meaning of the elementary algebraical opera
tions such as addition, as applied to real numbers in general.
(i) Addition. In order to define the sum of two numbers
a and ft, we consider the following two classes : (i) the class (c)
formed by all sums c = a + b, (ii) the class (C) formed by all sums
C = A+B. Plainly c < C in all cases.
Again, there cannot be more than one rational number which
does not belong either to (c) or to (C). For suppose there were
two, say r and s, and let s be the greater. Then both r and s
must be greater than every c and less than every (7; and so C c
cannot be less than s r. But
and we can choose a, b, A, B so that both A a and Bb
are as small as we like; and this plainly contradicts our
hypothesis.
If every rational number belongs to (c) or to (C), the classes (c),
(0) form a section of the rational numbers, that is to say, a number
7. If there is one which does not, we add it to (C). We have
now a section or real number 7, which must clearly be rational,
since it corresponds to the least member of (C). In any case
we call 7 the sum of a and ft, and write
7 = a + ft.
If both a and /3 are rational, they are the least members of the upper
classes (A) and (B}, In this case it is clear that a-f/3 is the least member
of (6*), so that our definition agrees with our previous ideas of addition.
(ii) Subtraction. We define a ft by the equation
The idea of subtraction accordingly presents no fresh difficulties.
Examples V. 1. Prove that a + ( - a) = 0.
2. Prove that a + 0=0 + a = a.
3. Prove that a + /3 = /3 + a. [This follows at once from the fact that the
classes (a+b) and (& + ), or (A+) and (B+A}, are the same, since, e.g.,
a-|-6 = 6-f a when a and b are rational.]
4. Prove that a + (0 + y ) = (a + 0) + y.
JI. 2
18 REAL VARIABLES [l
5. Prove that a - a = 0.
6. Prove that a - = - (/3 - a).
7. From the definition of subtraction, and Exs. 4, 1, and 2 above, it
follows that
We might therefore define the difference a-/3 = y by the equation -y+/3 = a.
8. Prove that a- (/3-y) = a- /3 + y.
9. Give a definition of subtraction which does not depend upon a previous
definition of addition. [To define y = a ft, form the classes (c), (C) for which
c = a-B,C=A-b. It is easy to show that this definition is equivalent to
that which we adopted in the text.]
10. Prove that
II" -|/3|j:g|a/3|<|
11. Algebraical operations with real numbers (con
tinued). (iii) Multiplication. When we come to multiplication,
it is most convenient to confine ourselves to positive numbers
(among which we may include 0) in the first instance, and to go
back for a moment to the sections of positive rational numbers
only which we considered in 4 7. We may then follow practi
cally the same road as in the case of addition, taking (c) to be (ab)
and (C) to be (AB). The argument is the same, except when we
are proving that all rational numbers with at most one -exception
must belong to (c) or (G). This depends, as in the case of addi
tion, on showing that we can choose a, A, b, and B so that (7 c is
as small as we please. Here we use the identity
G - c = A B - ab = (A - a) B + a (B - b).
Finally we include negative numbers within the scope of our
definition by agreeing that, if a and /3 are positive, then
(-a)/3 = -a& a(-/3) = -a, (-a) (-0) = a.
(iv) Division. In order to define division, we begin by de
fining the reciprocal I/a of a number a (other than zero). Con
fining ourselves in the first instance to positive numbers and
sections of positive rational numbers, we define the reciprocal of a
positive number a by means of the lower class (1 /A) and the upper
class (I/a). We then define the reciprocal of a negative number
a by the equation l/( )= (I/a). Finally we define a//3 by
the equation
10-13] REAL VARIABLES 19
We are then in a position to apply to all real numbers, rational
or irrational, the whole of the ideas and methods of elementary
algebra. Naturally we do not propose to carry out this task in
detail. It will be more profitable and more interesting to turn
our attention to some special, but particularly important, classes
of irrational numbers.
Examples VI. Prove the theorems expressed by the following
formulae :
OXa=0. 2. aXl = lXa = a. 3. ax(l/a)
4. a/3 = /3a. o. a (/3y) = (a/3) y. 6.
7. (a + /3) = + . 8.
12. The number */2. Let us now return for a moment to
the particular irrational number which we discussed in 4 5.
We there constructed a section by means of the inequalities
y? < 2, a? > 2. This was a section of the positive rational numbers
only ; but we replace it (as was explained in 8) by a section of
all the rational numbers. We denote the section or number thus
defined by the symbol \/2.
The classes by means of which the product of \/2 by itself is
defined are (i) (aa), where a and a are positive rational numbers
whose squares are less than 2, (ii) (A A ), where A and A are
positive rational numbers whose squares are greater than 2. These
classes exhaust all positive rational numbers save one, which can
only be 2 itself. Thus
Again
(- V2) 2 = (- V2) (- A/2) = V2 V2 = (V2) 2 = 2.
Thus the equation x* = 2 has the two roots \/2 and A/2. Similarly
we could discuss the equations # 2 = 3, x 3 = 7, ... and the corre
sponding irrational numbers V3, \/3, $7,....
13. Quadratic surds. A number of the form + \fa, where
a is a positive rational number which is not the square of another
rational number, is called a pure quadratic surd. A number of
the form a \/b, where a is rational, and ^/b is a pure quadratic
surd, is sometimes called a mixed quadratic surd.
29
-
20 REAL VARIABLES [l
The two numbers a<Jb are the roots of the quadratic equation
Conversely, the equation x 2 + 2px + q=0, where p and q are rational, and
p z -q>0, has as its roots the two quadratic surds -p*J(p 2 q).
The only kind of irrational numbers whose existence was
suggested by the geometrical considerations of 3 are these
quadratic surds, pure and mixed, and the more complicated
irrationals which may be expressed in a form involving the
repeated extraction of square roots, such as
V2 + V(2 + V2) + V{2 + V(2 + V2)).
It is easy to construct geometrically a line whose length is
equal to any number of this form, as the reader will easily see for
himself. That irrational numbers of these kinds only can be con
structed by Euclidean methods (i.e. by geometrical constructions
with ruler and compasses) is a point the proof of which must
be deferred for the present*. This property of quadratic surds
makes them especially interesting.
Examples VII. 1. Give geometrical constructions for
2. The quadratic equation cu, i2 + 26#+c has two real roots t if
b 2 -ac>0. Suppose a, b c rational. Nothing is lost by taking all three
to be integers, for we can multiply the equation by the least common
multiple of their denominators.
The reader will remember that the roots are {-,b^/(b 2 -ac)}/a. It is
easy to construct these lengths geometrically, first constructing *J(b 2 -ac).
A much more elegant, though less straightforward, construction is the
following.
* See Ch. II, Misc. Exs. 22.
f I.e. there are two values of x for which aa; 2 -f 2bx + c = 0. If 6 2 -ac<0 there
are no such values of x. The reader will remember that in books on elementary
algebra the equation is said to have two complex roots. The meaning to be
attached to this statement will be explained in Ch. III.
When 6 2 -rtc the equation has only one root. For the sake of uniformity
it is generally said in this case to have two equal roots, but this is a mere
convention.
13, 14]
REAL VARIABLES
21
Draw a circle of unit radius, a diameter PQ, and the tangents at the ends
of the diameters.
Q
Q
Fig. 5.
Take PP = Za/b and QQ = c/26, having regard to sign*. Join P Q ,
cutting the circle in M and N. Draw PM and PN, cutting QQ in X and Y.
Then QX and QY are the roots of the equation with their proper signsi.
The proof is simple and we leave it as an exercise to the reader.
Another, perhaps even simpler, construction is the following. Take a line
AB of unit length. Draw BC= -2b/a perpendicular to AB, and CD=c/a
perpendicular to BC and in the same direction as BA. On AD as diameter
describe a circle cutting BG in X and Y. Then BX and BY are the roots.
3. If ac is positive PP and QQ will be drawn in the same direction.
Verify that P Q will not meet the circle if 6 2 <c, while if b 2 = ac it will be
a tangent. Verify also that if 6 2 = ac the circle in the second construction
will touch BC.
4. Prove that
14. Some theorems concerning quadratic surds. Two
pure quadratic surds are said to be similar if they can be ex
pressed as rational multiples of the same surd, and otherwise to be
dissimilar. Thus
and so \/8, ^^f- are similar surds. On the other hand, if M and N
are integers which have no common factor, and neither of which
is a perfect square, >JM and *JN are dissimilar surds. For suppose,
if possible,
V-ir-* A
q V u
where all the letters denote integers.
* The figure is drawn to suit the case in which b and c have the same and a
the opposite sign. The reader should draw figures for other cases.
t I have taken this construction from Klein s Lemons sur certaines questions di
geometric elementaire (French translation by J. Griess, Paris, 1896).
22 REAL VARIABLES [l
Then ,JMN is evidently rational, and therefore (Ex. n. 3) \
integral. Thus MN = P 2 , where P is an integer. Let a, b, c, ...
be the prime factors of P, so that
where a, y3, y, . . . are positive integers. Then MN is divisible by
a 2 *, and therefore either (1) M is divisible by a 2a , or (2) N is
divisible by a 2 ", or (3) M and N are both divisible by a. The last
case may be ruled out, since M and N have no common factor.
This argument may be applied to each of the factors a 2a , b 2 ?, c 2y , . . . ,
so that M must be divisible by some of these factors and N by
the remainder. Thus
Jf-PA N-Pf,
where P^ denotes the product of some of the factors a 2a , b-P, c 2y , . . .
and P 2 2 the product of the rest. Hence M and N are both perfect
squares, which is contrary to our hypothesis.
THEOREM. If A, B, C, D are rational and
then either (i) A G, B = D or (ii) B and D are loth squares of
rational numbers.
For B- D is rational, and so is
If B is not equal to D (in which case it is obvious that A is also*
equal to (7), it follows that
is also rational. Hence \/B and */D are rational.
COROLLARY. // A + *JB = C+>JD, then A-
(unless *JB and *JD are both rational).
Examples VIII. 1. Prove ab initio that ^2 and x/3 are not similar
surds.
2. Prove that >Ja and *J(I/a), where a is rational, are similar surds
(unless both are rational).
3. If a and b are rational, then <Ja + *Jb cannot be rational unless Ja and
/& are rational. The same is true of *Ja Jb, unless a = b.
14, 15] REAL VARIABLES 23
4. If
then either (a) A = C and B=D, or (6) A = D and J3 = C,or (c) JA, *JB, ^<7,
^/^) are all rational or all similar surds. [Square the given equation and
apply the theorem above.]
5. Neither (a + Jb) 3 nor (a - */b} 3 can be rational unless Jb is rational.
6. Prove that if x=p + *Jq, where p and q are rational, then # m , where
in is any integer, can be expressed in the form P+QJq, where P and Q
are rational. For example,
Deduce that any polynomial in x with rational coefficients (i.e. any expression
of the form
where a , ... a n are rational numbers) can be expressed in the form
7. If a + v/&> where b is not a perfect square, is the root of an algebraical
equation with rational coefficients, then a-*Jb is another root of the same J
equation.
8. Express l!(p+Jq) in the form prescribed in Ex. 6. [Multiply
numerator and denominator by p - Jq.]
9. Deduce from Exs. 6 and 8 that any expression of the form G (x)IH (x\
where G(x] and H(x) are polynomials in x with rational coefficients, can be
expressed in the form P + QJq, where P and Q are rational.
10. If p, q, and p 2 -q are positive, we can express J(p+*Jq) in the form
t where
1 1 . Determine the conditions that it may be possible to express J(p^ + *Jq\
where p and q are rational, in the form V^ + Vy, where x and y are rational.
12. If a 2 - b is positive, the necessary and sufficient conditions that
should be rational are that a 2 -b and ^{a + N /(a 2 -6)j should both be squares
of rational numbers.
15. The continuum. The aggregate of all real numbers,
rational and irrational, is called the arithmetical continuum.
It is convenient to suppose that the straight line A of 2
is composed of points corresponding to all the numbers of the
arithmetical continuum, and of no others*. The points of the
* This supposition is merely a hypothesis adopted (i) because it suffices for the
purposes of our geometry and (ii) because it provides us with convenient geometrical
illustrations of analytical processes. As we use geometrical language only for
purposes of illustration, it is not part of our business to study the foundations
of geometry.
24 REAL VARIABLES [l
line, the aggregate of which may be said to constitute the linear
continuum, then supply us with a convenient image of the
arithmetical continuum.
We have considered in some detail the chief properties of a
few classes of real numbers, such, for example, as rational numbers
or quadratic surds. We add a few further examples to show how
very special these particular classes of numbers are, and how, to
put it roughly, they comprise only a minute fraction of the infinite
variety of numbers which constitute the continuum.
(i) Let us consider a more complicated surd expression such as
Our argument for supposing that the expression for z has a meaning might be
as follows. We first show, as in 12, that there is a number ?y = x /15 such that
2/ 2 =15, and we can then, as in 10, define the numbers 4 + ^/15, 4- v /15.
Now consider the equation in z l ,
The right-hand side of this equation is not rational : but exactly the same
reasoning which leads us to suppose that there is a real number x such that
#3=2 (or any other rational number) also leads us to the conclusion that there
is a number z l such that 2 1 3 =4+ v /li3. We thus define ^ = ^(4+^15), and
similarly we can define 2 2 =4/(4 ^15) ; and then, as in 10, we define zz^z^.
r v v< > -> i + -v c i . > * * ^ *
Now it is easy to verify that
2 3 = 3, + 8> . \ $?*
V And we might have given a direct proof of the existence of a unique number
\z Buch that z 3 =3z+8. It is easy to see that there cannot be two such
numbers. For if z i 3 = 3z 1 + 8 and 2 2 3 = 3%-f8, we find on subtracting and
dividing by Zi~z 2 that Zi 2 +z 1 z 2 +z 2 2 =3. But if z l and z 2 are positive ^ 3 >8,
z 2 3 >8 and therefore Zj>2, 2 2 >2, Zi 2 + z 1 z 2 + z 2 2 > 12, and so the equation
just found is impossible. And it is easy to see that neither z l nor z. 2 can
be negative. For if zi is negative and equal to -r, is positive and
3 -3+8 = 0, or 3- 2 = 8/. Hence 3- 2 >0, and so <2. But then
8/>4, and so 8/ cannot be equal to 3- 2 , which is less than 3.
Hence there is at most one z such that z 3 = 32 + 8. And it cannot be
rational. For any rational root of this equation must be integral and a
factor of 8 (Ex. n. 3), and it is easy to verify that no one of 1, 2, 4, 8 is a root.
Thus z 3 = 3-2+ 8 has at most one root and that root, if it exists, is positive
and not rational. We can now divide the positive rational numbers x into
two classes Z, A according as x 3 < 3x + 8 or x 3 > &v + 8. It is easy to see that
if a? > 3. + 8 and y is any number greater than #, then also y 3 > 3y + 8. For
suppose if possible y 3 <3?/-|-8. Then since # 3 >3# + 8 we obtain on sub
tracting y 3 -tf 3 <3(y-#), or y 2 + xy + x 2 < 3, which is impossible; for y is
15] REAL VARIABLES 25
positive and #>2 (since ^ 3 >8). Similarly we can show that i
and y < x then also y z < 3y -f 8.
Finally, it is evident that the classes L and It both exist ; and they form
a section of the positive rational numbers or positive real number z which
satisfies the equation z 3 = 3z + 8. The reader who knows how to solve cubic
equations by Cardan s method will be able to obtain the explicit expression of
z directly from the equation.
(ii) The direct argument applied above to the equation
y? 3x + 8 could be applied (though the application would be
a little more difficult) to the equation
a? = x + 16,
and would lead us to the conclusion that a unique positive real
number exists which satisfies this equation. In this case, how
ever, it is not possible to obtain a simple explicit expression
for x composed of any combination of surds. It can in fact
be proved (though the proof is difficult) that it is generally
impossible to find such an expression for the root of an equation
of higher degree than 4. Thus, besides irrational numbers which
can be expressed as pure or mixed quadratic or other surds, or
combinations of such surds, there are others which are roots of
algebraical equations but cannot be so expressed. It is only in
very special cases that such expressions can be found.
(iii) - But even when we have added to our list of irrational
numbers roots of equations (such as x? = cc-\- 16) which cannot be
explicitly expressed as surds, we have not exhausted the different
kinds of irrational numbers contained in the continuum. Let us
draw a circle whose diameter is equal to A A l} i.e. to unity. It is
natural to suppose* that the circumference of such a circle has a
length capable of numerical measurement. This length is usually
denoted by TT. And it has been shown f (though the proof is un
fortunately long and difficult) that this number TT is not the
root of any algebraical equation with integral coefficients, such,
for example, as
7T 2 = ??, 7T 3 = H, 7T 5 = 7T + U,
* A proof will be found in Ch. VII.
f See Hobson s Trigonometry (3rd edition), pp. 305 et seq., or the same writer s
Squaring the Circle (Cambridge, 1913).
26 REAL VARIABLES [l
where n is an integer. In this way it is possible to define a
number which is not rational nor yet belongs to any of the classes
of irrational numbers which we have so far considered. And this
number TT is no isolated or exceptional case. Any number of other
examples can be constructed. In fact it is only special classes of
irrational numbers which are roots of equations of this kind, just
as it is only a still smaller class which can be expressed by means
of surds.
16. The continuous real variable. The real numbers
may be regarded from two points of view. We may think of
them as an aggregate, the arithmetical continuum defined in
the preceding section, or individually. And when we think of
them individually, we may think either of a particular specified
number (such as 1, -J, \/2, or TT) or we may think of any number,
an unspecified number, the number x. This last is our point of
view when we make such assertions as ( x is a number , l x is the
measure of a length , # may be rational or irrational , The x
which occurs in propositions such as these is called the continuous
real variable : and the individual numbers are called the values of
the variable.
A variable , however, need not necessarily be continuous.
Instead of considering the aggregate of all real numbers, we
might consider some partial aggregate contained in the former
aggregate, such as the aggregate of rational numbers, or the
aggregate of positive integers. Let us take the last case. Then
in statements about any positive integer, or an unspecified positive
integer, such as ( n is either odd or even , n is called the variable,
a positive integral variable, and the individual positive integers
are its values.
Naturally a and n are only examples of variables, the
variable whose field of variation is formed by all the real
numbers, and that whose field is formed by the positive integers.
These are the most important examples, but we have often to
consider other cases. In the theory of decimals, for instance, we
may denote by x any figure in the expression of any number as a
decimal. Then # is a variable, but a variable which has only ten
different values, viz. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The reader should
15-17] REAL VARIABLES 27
think of other examples of variables with different fields of varia
tion. He will find interesting examples in ordinary life : policeman
X) the driver of cab x, the year x, the xth day of the week. The
values of these variables are naturally not numbers.
17. Sections of the real numbers. In 4 7 we con
sidered sections of the rational numbers, i.e. modes of division of
the rational numbers (or of the positive rational numbers only)
into two classes L and R possessing the following characteristic
properties:
(i) that every number of the type considered belongs to one
and only one of the two classes ;
(ii) that both classes exist ;
(iii) that any member of L is less than any member of R.
It is plainly possible to apply the same idea to the aggregate
of all real numbers, and the process is, as the reader will find in
later chapters, of very great importance.
Let us then suppose * that P and Q are two properties which
are mutually exclusive, and one of which is possessed by every
real number. Further let us suppose that any number which
possesses P is less than any which possesses Q. We call the
numbers which possess P the lower or left-hand class L, and
those which possess Q the upper or right-hand class R.
Thus P might be x < N /2 and Q be x > ^2. II is important to observe
that a pair of properties which suffice to define a section of the rational
numbers may not suffice to define one of the -real numbers. This is so, for
example, with the pair c x < v /2 and x > s /2 or (if we confine ourselves
to positive numbers) with # 2 < 2 and * x 2 > 2 . Every rational number
possesses one or other of the properties, but not every real number, since in
either case v/2 escapes classification.
There are now two possibilities!. Either L has a greatest
member I, or R has a least member r, Both of these events
* The discussion which follows is in many ways similar to that of 6. We
have not attempted to avoid a certain amount of repetition. The idea of a section,
first brought into prominence in Dedekind s famous pamphlet Stctigkett und
irrationale Zahlen, is one which can, and indeed must, be grasped by every reader .
of this book, even if he be one of those who prefer to omit the discussion of the j
notion of an irrational number contained in 6 12.
t There were three in 6. b / ^
28 HEAL VARIABLES [l
cannot occur. For if L had a greatest member I, and R a least
member r, the number \(l-\-r) would be greater than all members
of L and less than all members of R, and so could not belong to
either class. On the other hand one event must occur*.
For let LI and Rj_ denote the classes formed from L and R by
taking only the rational members of L and R. Then the classes
L^ and R 1 form a section of the rational numbers. There are now
two cases to distinguish.
It may happen that L l has a greatest member a. In this case
a must be also the greatest member of L. For if not, we could find
a greater, say (S. There are rational numbers lying between a and
ft, and these, being less than ft, belong to L, and therefore to L^
and this is plainly a contradiction. Hence a is the greatest
member of L.
On the other hand it may happen that L l has no greatest
member. In this case the section of the rational numbers formed
by L 1 and R! is a real number a. This number a must belong
to L or to R. If it belongs to L we can shew, precisely as before,
that it is the greatest member of L , and similarly, if it belongs
to R, it is the least member of R.
Thus in any case either L has a greatest member or R a
least. Any section of the real numbers therefore corresponds to
a real number in the sense in which a section of the rational
numbers sometimes, but not always, corresponds to a rational
number. This conclusion is of very great importance ; for it shows
that the consideration of sections of all the real numbers does not
lead to any further generalisation of our idea of number. Starting
from the rational numbers, we found that the idea of a section of
the rational numbers led us to a new conception of a number, that
of a real number, more general than that of a rational number;
and it might have been expected that the idea of a section of the
real numbers would have led us to a conception more general still.
The discussion which precedes shows that this is not the case, and
that the aggregate of real numbers, or the continuum, has a kind
of completeness which the aggregate of the rational numbers
lacked, a completeness which is expressed in technical language
by saying that the continuum is closed.
* This was not the case in 6.
17, 18] REAL VARIABLES 29
The result which we have just proved may be stated as follows:
Dedekind s Theorem. If the real numbers are divided into
two classes L and R in such a way that
(i) every number belongs to one or other of the two classes,
(ii) each class contains at least one number,
(iii) any member of L is less than any member of R,
then there is a number a, which has the property that all the numbers
less than it belong to L and all the numbers greater than it to R.
The number a. itself may belong to either class.
In applications we have often to consider sections not of all numbers but
of all those contained in an interval (0, y\ that is to say of all numbers
x such that /3 x ^ y. A section of such numbers is of course a division of
them into two classes possessing the properties (i), (ii), and (iii). Such
a section may be converted into a section of all numbers by adding to L all
numbers less than /3 and to R all numbers greater than y. It is clear that
the conclusion stated in Dedekind s Theorem still holds if we substitute the
real numbers of the interval (ft y} for the real numbers , and that the
number a in this case satisfies the inequalities /3 <a^y.
18. Points of accumulation. A system of real numbers, OP
of the points on a straight line corresponding to them, denned in
any way whatever, is called an aggregate or set of numbers or
points. The set might consist, for example, of all the positive
integers, or of all the rational points.
It is most convenient here to use the language of geometry*.
Suppose then that we are given a set of points, which we will
denote by S. Take any point f , which may or may not belong to S.
Then there are two possibilities. Either (i) it is possible to choose
a positive number 8 so that the interval (f S, % + S) does not con
tain any point of S, other than f itself f, or (ii) this is not possible.
Suppose, for example, that S consists of the points corresponding to all
the positive integers. If is itself a positive integer, we can take S to be any
number less than 1, and (i) will be true; or, if is halfway between two
positive integers, we can take 8 to be any number less than \. On the other
hand, if S consists of all the rational points, then, whatever the value of ,
(ii) is true ; for any interval whatever contains an infinity of rational points.
* The reader will hardly require to be reminded that this course is adopted
solely for reasons of linguistic convenience.
t This clause is of course unnecessary if does not itself belong to S.
30 REAL VARIABLES [l
Let us suppose that (ii) is true. Then any interval (f 8, f 4- S),
however small its length, contains at least one point fj which
belongs to 8 and does not coincide with ; and this whether f
itself be a member of S or not. In this case we shall say that f is
a point of accumulation of 8. It is easy to see that the interval
(?~8, f+S) must contain, nojj_merely one, but infinitely many
points of 8. For, when we have determined f 1} we can take an
interval (f 81, f + 81) surrounding f but not reaching as far as f lt
But this interval also must contain a point, say f a , which is a
member of $ and does not coincide with f. Obviously we may
repeat this argument, with f 2 in the place of f j ; and so on
indefinitely. In this way we can determine as many points
\
as we please, all belonging to S, and all lying inside the interval
A point of accumulation of $ may or may not be itself a point
of S. The examples which follow illustrate the various possibilities.
Examples IX. 1. If S consists of the points corresponding to the
positive integers, or all the integers, there are no points of accumulation.
2. If S consists of all the rational points, every point of the line is a
point of accumulation.
3. If S consists of the points 1, J, , ..., there is one point of accumula
tion, viz. the origin.
4. If S consists of all the positive rational points, the points of accumula
tion are the origin and all positive points of the line.
1 19. Weierstrass s Theorem. The general theory of sets
r S of points is of the utmost interest and importance in the higher
branches of analysis ; but it is for the most part too difficult to be
included in a book such as this. There is however one funda
mental theorem which is easily deduced from Dedekind s Theorem
and which we shall require later.
THEOREM. If a set S contains infinitely many points, and is
entirely situated in an interval (a, j3), then at least one point of the
interval is a point of accumulation of 8.
We divide the points of the line A into two classes in the
following manner. The point P belongs to L if there are an
18, 19] REAL VARIABLES 31
infinity of points of S to the right of P, and to R in the contrary
case. Then it is evident that conditions (i) and (iii) of Dedekind s
Theorem are satisfied ; and since a belongs to L and /3 to R,
condition (ii) is satisfied also.
Hence there is a point f such that, however small be 8, f S
belongs to L and f+S to R, so that the interval (f-5, +)
contains an infinity of points of S. Hence f is a point of accumu
lation of S.
This point may of course coincide with a or /3, as for instance when a=0,
$ = 1, and S consists of the points 1, |, ^, .... In this case is the sole
point of accumulation.
MISCELLANEOUS EXAMPLES ON CHAPTER I.
1. What are the conditions that ax + by + cz=0, (1) for all values of
a, y, z\ (2) for all values of x, y, z subject to ax+py+yz=0 , (3) for all
values of x, y, z subject to both ax+$y + yz = Q and
2. Any positive rational number can be expressed in one and only one
way in the form
< * 1+ r J 2 + rTT3~ f " + i7273~7^I
where aj, 2 -"j % are integers, and
Oga lt 0< 2 <2, 0^ 3 <3, ...0<a fc <.
3. Any positive rational number can be expressed in one and one way
only 7 as a simple continued fraction
where a 1} 2 > are positive integers, of which the first only may be zero.
[Accounts of the theory of such continued fractions will be found in text
books of algebra. For further information as to modes of representation of
rational and irrational numbers, see Hobson, Theory of Functions of a Real
Variable, pp. 4549.]
4. Find the rational roots (if any) of 9# 3 - &v 2 -f 1 5x - 10 = 0. *
5. A line AB is divided at C in aurea sectione (Euc. II. 11) i.e. so that
AB . AC=C 2 . Show that the ratio A CjAB is irrational.
[A direct geometrical proof will be found in Bromwich s Infinite Series,
143, p. 363.]
6. A is irrational. In what circumstances can - i . where a, b. c. d
cA + d
are rational, be rational?
J
32 REAL VARIABLES [l
7. Some elementary inequalities. In what follows a 1? a 2 , ... de
note positive numbers (including zero) and jt?, #, ... positive integers. Since
ajP-a 2 P and !- 2 have the same sign, we have (af - a 2 *>) (a^ - a 2 ) >0, or
........................ (1),
an inequality which may also be written in the form
+ 2 p + g > /V +
2 =\ 2
By repeated application of this formula we obtain
and in particular > .............................. (4)>
When ;? = 2=1 in (1), or p = 2 in (4), the inequalities are merely different
forms of the inequality af+af^Za&z, which expresses the fact that the
arithmetic mean of two positive numbers is not less than their geometric
mean.
8. Generalisations for n numbers. If we write down the \n(n-l}
inequalities of the type (1) which can be formed with n numbers a lt a 2 ,..., ,
and add the results, we obtain the inequality
........................ (6).
Hence we can deduce an obvious extension of (3) which the reader may
formulate for himself, and in particular the inequality
(7).
9. The general form of the theorem concerning the arithmetic and
geometric means. An inequality of a slightly different character is
that which asserts that the arithmetic mean of a lt a 2 , ..., a n is not less
than their geometric mean. Suppose that a r and a ? are the greatest and
least of the a s (if there are several greatest or least a s we may choose any
of them indifferently), and let O be their geometric mean. We may suppose
> 0, as the truth of the proposition is obvious when G= 0. If now we replace
a r and a s by
we do not alter the value of the geometric mean ; and, since
/ + a 8 -a r -a s = (a r - G) (a, - (?)/ < 0,
we certainly do not increase the arithmetic mean.
It is clear that we may repeat this argument until we have replaced each
of a l , 2 , ..., a n by G; at most n repetitions will be necessary. As the final
value of the arithmetic mean is G, the initial value cannot have been less.
REAL VARIABLES 33
10. Schwarz s inequality. Suppose that a lt 2 , ..., a n and 6 1? 6 2 , ..., b n
are any two sets of numbers positive or negative. It is easy to verify the
identity
(2a A) 2 = 2 r 2 2a. 2 - 2 (a r b 8 - a 8 b r )\
where r and s assume the values 1, 2, ..., n. It follows that
an inequality usually known as Schwarz s (though due originally to Cauchy).
11. If a lf a 2 , ..., a n are all positive, and * n = a 1 + o 2 + . .. + , then
(Math. Trip. 1909.)
12. If !, 2 , ..., n and 6 15 b%, ..., 6 n are two sets of positive numbers,
arranged in descending order of magnitude, then
13. If a, 6, c, ... & and ^4, Z?, (7, ... K are two sets of numbers, and all of
the first set are positive, then
lies between the algebraically least and great ast of A, /?, ..., K.
14. If Jp t Jq are dissimilar surds, and a+b *Jp+c */q+d J(pq}**0,
where a, b, c, d are rational, then a = 0, 6 = 0, c = 0, d=Q.
[Express *Jp in the form M+ N Jq t where M and N are rational, and apply
the theorem of 14.]
1 5. Show that if a *JZ + b ^/3 + c J5 = 0, where a, 6, c are rational numbers,
then a = 0, 6 = 0, c=0.
16. Any polynomial in Jp and V^ with rational coefficients (i.e. any
sum of a finite number of terms of the form A (Jp} m (<Jq} n ^ where m and n
are integers, and A rational), can be expressed in the form
a + b Jp + c V q + d >Jpq,
where a, 6, c, d are rational.
17. Express**,- ~f , ,-, where a, b, etc. are rational, in the form
a + e -
where A, S, C> D are rational.
[Evidently
Jq _ (a + b Jp + c Jq) (d+e Jp-fjq) __ a
where a, /3, etc. are rational numbers which can easily be found. The required
n. 3
REAL VARIABLES [l
reduction may now be easily completed by multiplication of numerator and
denominator by e-Vjo. For example, prove that
18. If a, b, x, y are rational numbers such that
then either (i) x=a t y=b or (ii) l-ab and \-xy are squares of rational
numbers. (Math. Trip. 1903.)
19. If all the values of x and y given by
ax 2 + Zhxy + by 2 = 1, a x* + 2k xy + b y 2 = I
(where a, A, b, a , h , b are rational) are rational, then
(h - A )2 _ ( a - a ) (6 - 6 ), (a& - a &) 2 + 4 (ah - a h] (bh 1 - b h]
are both squares of rational numbers. (Math. Trip. 1899.)
20. Show that N /2 and N /3 are cubic functions of N ^2 + x /3, with rational
coefficients, and that ^2 ^6 + 3 is the ratio of two linear functions of
x /2 + .v/3. (Math. Trip. 1905.)
21. The expression
is equal to 2m if 2m 2 > a > m 2 , and to 2 v /(a - wi 2 ) if a > 2wi 2 .
22. Show that any polynomial in #2, with rational coefficients, can be
expressed in the form
where a, 6, c are rational.
More generally, if p is any rational number, any polynomial m ^p with
rational coefficients can be expressed in the form
where a , j, ... are rational and a = %p. For any such polynomial is of the
form
where the 6 s are rational. If Jc^ m 1, this is already of the form required. If
Jc>m- 1, let a r be any power of a higher than the (m l)th. Then r=\m + s,
where X is an integer and 0^s< m- 1 ; and a r =a xm+s =p x a s . Hence we can
get rid of all powers of a higher than the (m l)th.
23. Express (4/2 -I) 6 and (#2-l)/(#2 + l) in the form
where a, b, c are rational. [Multiply numerator and denominator of the
second expression by ^4- 4/2 + 1.]
24. If
where a, 5, c are rational, then a=0, 5 = 0, c = 0.
REAL VARIABLES 35
[Let y = f/2. Then # 3 = 2 and
Hence 2c 2 + 26+a 3 =0 or
Multiplying these two quadratic equations by a and c and subtracting,
we obtain (a6-2c 2 )y+a 2 -26c=0, or y= - (a 2 - 26c)/(a6 - 2c 2 ), a rational
number, which is impossible. The only alternative is that ab 2c 2 =0,
a 2 - 26e = 0.
Hence a6 = 2c 2 , a 4 =46 2 c 2 . If neither a nor b is zero, we can divide the
second equation by the first, which gives a 3 =26 3 : and this is impossible,
since ^/2 cannot be equal to the rational number a/6. Hence a6 = 0, c = 0,
and it follows from the original equation that a, 6, and c are all zero.
As a corollary, if a+&^2=Hc<v/4 = d+e^/2+/>/4, then a = d, b = e, c=f.
It may be proved, more generally, that if
p not being a perfect mth power, then a =ai = ... = a m _ 1 = j but the proof is
less simple.]
25. If A + $B=C+j/D, then either A = C, B=D, or and D are both
cubes of rational numbers.
26. If %A + $B + $C= 0, then either one of A, B, C is zero, and the other
two equal and opposite, or JfA t j/B, $C are rational multiples of the same
surd j/X.
27. Find rational numbers a, 3 such that
28. If (a-Z)3)6>o, then
3f 963 + a /a-6*\ sf 96 + ^
//a-6*\\ s/f
v v"36-)/ + v r~
is rational. [Each of the numbers under a cube root is of the form
where a and /3 are rational.]
29. If a = Z/p, any polynomial in a is the root of an equation of degree n,
with rational coefficients.
[We can express the polynomial (x say) in the form
where l it m ly ... are rational, as in Ex. 22.
29
36 REAL VARIABLES
Similarly x 2 =l 2 + m 2 a+... +r 2 a (n ~ 1) ,
Hence L v x + L%x 2 + . . . + L n x n = A,
where A is the determinant
mi ... i
m n ...r n
and Zi, Z 2 , ... the minors of l lt 1 2 , ....]
30. Apply this process to x=p+Jq, and deduce the theorem of 14.
31. Show that y = a + bp 1 3 + cp 2/3 satisfies the equation
# 3 - 3a/+ 3y (a 2 - 6cjo) - a 3 - 6 3 p - c 3 ^ 2 + Zabcp = 0.
32. Algebraical numbers. We have seen that some irrational numbers
(such as x/2) are roots of equations of the type
a^x n + a^x n ~ l + . . . + a n = 0,
\ where , a^ ..., a n are integers. Such irrational numbers are called alge-
} braical numbers: all other irrational numbers, such as TT ( 15), are called
transcendental numbers. Show that if x is an algebraical number, then so are
kx t where k is any rational number, and x m n , where m and n are any integers.
33. If x and y are algebraical numbers, then so are x-\-y>x-y,xy and x\y.
[We have equations a$x m + aiX m ~ 1 + . . . + a m = 0,
where the a s and 6 s are integers. Write x+y=z,y = z-x in the second,
and eliminate x. We thus get an equation of similar form
satisfied by z. Similarly for the other cases.]
34. If a n + a 1 #*- 1 + ...+a n = 0,
where a , a 1} ..., a n are any algebraical numbers, then x is an algebraical
number. [We have n + l equations of the type
a 0>r a r mr + ai tr a r m r~ l + ...+a mrir = (r = 0, 1, ..., ri),
in which the coefficients , r> a i, r> are integers Eliminate a . a lt ..., a n
between these and the original equation for x.}
35. Apply this process to the equation x 2 - 2x V2 + V3 0.
[The result is ^-
REAL VARIABLES 37
36. Find equations, with rational coefficients, satisfied by
37. If # 3 = x + 1 , then # 3n = a H x + b n + cjx, where
38. If # G +^ 5 -2. 4 -.r 3 +tf 2 + l=0 and y = A 4 -^ 2 4-^- 1, then y satisfies
a quadratic equation with rational coefficients. (Afat/i. Trip. 1903.)
[It will be found that y 1 + y + 1 = 0.]
CHAPTER II
FUNCTIONS OF HEAL VARIABLES
20. The idea of a function. Suppose that x and y are
two continuous real variables, which we may suppose to be repre
sented geometrically by distances A^P = x, B^Q^y measured
from fixed points A 0) B along two straight lines A, M. And
let us suppose that the positions of the points P and Q are not
independent, but connected by a relation which we can imagine
to be expressed as a relation between x and y: so that, when
P and x are known, Q and y are also known. We might,
for example, suppose that y = x, or y=2x, or |#, or # 2 + l. In
all of these cases the value of x determines that of y. Or
again, we might suppose that the relation between x and y is
given, not by means of an explicit formula for y in terms of as,
but by means of a geometrical construction which enables us to
determine Q when P is known.
In these circumstances y is said to be a function of x. This
notion of functional dependence of one variable upon another is
perhaps the most important in the whole range of higher mathe
matics. In order to enable the reader to be certain that he
understands it clearly, we shall, in this chapter, illustrate it by
means of a large number of examples.
But before we proceed to do this, we must point out that
the simple examples of functions mentioned above possess three
characteristics which are by no means involved in the general
idea of a function, viz.:
(1) y is determined for every value of x\
(2) to each value of x for which y is given corresponds one
and only one value ofy,
(3) the relation between x and y is expressed by means of
an analytical formula, from which the value of y corresponding to
a given value of x can be calculated by direct substitution of the
latter.
20] FUNCTIONS OF REAL VARIABLES 39
It is indeed the case that these particular characteristics are
possessed by many of the most important functions. But the con
sideration of the following examples will make it clear that they
are by no means essential to a function. All that is essential is
that there should be some relation between x and y such that to
some values of x at any rate correspond values of y.
Examples X. 1. Let y =x or 2# or \x or x 2 + 1 Nothing further need
be said at present about cases such as these.
2. Let #=0 whatever be the value of x. Then y is a function of x, for we
can give x any value, and the corresponding value of y (viz. 0) is known. In
this case the functional relation makes the same value of y correspond to all
values of x. The same would be true were y equal to 1 or - \ or V 2 instead
of 0. Such a function of x is called a constant.
3. Let y = x. Then if x is positive this equation defines two values of y
corresponding to each value of #, viz. Jx. If #=0, y0. Hence to the
particular value 0- of x corresponds one and only one value of y. But if x is
negative there is no value of y which satisfies the equation. That is to say,
the function y is not defined for negative values of x. This function therefore
possesses the characteristic (3), but neither (1) nor (2).
4. Consider a volume of gas maintained at a constant temperature and
contained in a cylinder closed by a sliding piston*.
Let A be the area of the cross section of the piston and W its weight.
The gas, held in a state of compression by the piston, exerts a certain pressure
p per unit of area on the piston, which balances the weight TF, so that
W=A Po .
Let v be the volume of the gas when the system is thus in equilibrium.
If additional weight is placed upon the piston the latter is forced downwards.
The volume (v) of the gas diminishes ; the pressure (p) which it exerts
upon unit area of the piston increases. Boyle s experimental law asserts that
the product of p and v is very nearly constant, a correspondence which, if
exact, would be represented by an equation of the type
pv=a (i),
where a is a number which can be determined approximately by experiment.
Boyle s law, however, only gives a reasonable approximation to the facts
provided the gas is not compressed too much. When v is decreased and p
increased beyond a certain point, the relation between them is no longer
expressed with tolerable exactness by the equation (i). It is known that a
* I borrow this instructive example from Prof. H. S. Carslaw s Introduction to
the Calculus.
40 FUNCTIONS OF HEAL VARIABLES [ll
much better approximation to the true relation can then be found by means
of what is known as van der Waals law , expressed by the equation
(ii),
where a, /3, y are numbers which can also be determined approximately by
experiment.
Of course the two equations, even taken together, do not give anything
like a complete account of the relation between p and v. This relation is no
doubt in reality much more complicated, and its form changes, as v varies,
from a for ^ nearly equivalent to (i) to a form nearly equivalent to (ii). But,
from a mathematical point of view, there is nothing to prevent us from con
templating an ideal state of things in which, for all values of v not less than
a certain value V, (i) would be exactly true, and (ii) exactly true for all
values of v less than V. And then we might regard the two equations as
together denning p as a function of . It is an example of a function which
for some values of v is denned by one formula and for other values of v is
denned by another.
This function possesses the characteristic (2) . to any value of v only one
value of p corresponds : but it does not possess (1). For p is not denned as
a function of v for negative values of vj a negative volume means
nothing, and so negative values of v do not present themselves for considera
tion at all.
5. Suppose that a perfectly elastic ball is dropped (without rotation)
from a height \gr l on to a fixed horizontal plane, and rebounds continually.
The ordinary formulae of elementary dynamics, with which the reader is
probably familiar, show that h = gt 2 if <t <r, h=\g (2r-t) 2 if rt <3r, and
generally
if (2n- l)r<<(2w + l)T, h being the depth of the ball, at time t, below its
original position. Obviously A is a function of t which is only denned for
positive values of t.
** 6. Suppose that y is denned as being the largest prime factor of x. This
is an instance of a definition which only applies to a particular class of values
of x, viz. integral values. * The largest prime factor of J 3 *- or of N /2 or of TT
means nothing, and so our defining relation fails to define for such values of x
as these. Thus this function does not possess the characteristic (1). It does
possess (2), but not (3), as there is no simple formula which expresses y in
terms of x.
7. Let y be defined as the denominator of x when x is expressed in its
lowest terms. This is an example of a function which is defined if and only
if x is rational. Thus y = 7 if x - 11/7 : but y is not defined for # =^2, the
denominator of ^2 being a meaningless form of words.
20, 21]
FUNCTIONS OF REAL VARIABLES
41
8. Let y be defined as the height in inches of policeman Cx, in the
Metropolitan Police, at 5.30 p.m. on 8 Aug. 1907. Then y is defined for a
certain number of integral values of x, viz. 1, 2, ... , N, where N is the total
number of policemen in division C at that particular moment of time.
21. The graphical representation of functions. Sup
pose that the variable y is a function of the variable x. It will
generally be open to us also to regard x as a function of y, in virtue
of the functional relation between x and y. But for the present we
shall look at this relation from the first point of view. We shall
then call x the independent variable and y the dependent variable ,
and, when the particular form of the functional relation is not
specified, we shall express it by writing
y-/()
(or F (x}, <j) (x), i/r (x), . . . , as the case may be).
The nature of particular functions may, in very many cases, be
illustrated and made easily intelligible as follows. Draw two lines
X, Y at right angles to one another
and produced indefinitely in both direc
tions. We can represent values of x
and y by distances measured from
along the lines OX, OY respectively,
regard being paid, of course, to sign,
and the positive directions of measure
ment being those indicated by arrows
in Fig. 6.
Let a be any value of x for which
y is defined and has (let us suppose)
the single value b. Take OA = a,
OB = b, and complete the rectangle
Fig.
OAPB. Imagine the point P marked on the diagram. This
marking of the point P may be regarded as showing that the
value of y for x = a is b.
If to the value a of x correspond several values of y (say
b, b , b"}, we have, instead of the single point P, a number of
points P, P , P".
We shall call P the point (a, b) ; a and b the coordinates of P
referred to the axes OX, OY ; a the abscissa, b the ordinate of P ,
OX and Y the axis of x and the axis of y, or together the
42 FUNCTIONS OF REAL VARIABLES [ll
axes of coordinates, and the origin of coordinates, or simply
the origin.
Let us now suppose that for all values a of x for which y is
defined, the value b (or values b, b , b", ...) of y, and the corre
sponding point P (or points P, P t P", ...), have been determined.
We call the aggregate of all these points the graph of the
function y.
To take a very simple example, suppose that y is defined as
a function of x by the equation
Ax + By + C = ........................ (1),
where A, B, C are any fixed numbers*. Then y is a function of x
which possesses all the characteristics (1), (2), (3) of 20. It is
easy to show that the graph of y is a straight line. The reader is
in all probability familiar with one or other of the various proofs
of this proposition which are given in text-books of Analytical
Geometry.
We shall sometimes use another mode of expression. We
shall say that when x and y vary in such a way that equation (1)
is always true, the locus of the point (x, y) is a straight line, and
we shall call (1) the equation of the Zocws, and say that the equation
represents the locus. This use of the terms locus , equation of
the locus is quite general, and may be applied whenever the
relation between x and y is capable of being represented by an
analytical formula.
The equation Ax + By + G = is the general equation of the first
degree, for Ax + By + C is the most general polynomial in x and y
which does not involve any terms of degree higher than the first
in x and y. Hence the general equation of the first degree repre
sents a straight line. It is equally easy to prove the converse
proposition that the equation of any straight line is of the first
degree.
We may mention a few further examples of interesting geo
metrical loci defined by equations. An equation of the form
* If Z? = 0, y does not occur in the equation. We must then regard y as a
function of x defined for one value only of x, viz. x - C/A, and then having all
values.
21, 22] FUNCTIONS OF REAL VARIABLES 43
or # 2 + 2/ 2 + 2Gx+ 2%-f (7 = 0,
where G 2 + P 2 C > 0, represents a circle. The equation
(the general equation of the second degree) represents, assuming
that the coefficients satisfy certain inequalities, a conic section,
i.e. an ellipse, parabola, or hyperbola. For further discussion of
these loci we must refer to books on Analytical Geometry.
22. Polar coordinates. In what precedes we have determined
the position of P by the lengths of its coordinates OM=x, MP = y.
If OP = r and MOP = 0, 6 being an
angle between and 2?r (measured in
the positive direction), it is evident that
x = r cos 0, y r sin 6,
r = \/(# 2 + 2/ 2 ), cos 6 : sin 6 : 1 : : x : y : r,
and that the position of P is equally well
determined by a knowledge of r and 6. o * M
We call r and 6 the polar coordinates Fi g- 7.
of P. The former, it should be observed, is essentially positive*.
If P moves on a locus there will be some relation between r
and 6, say r =/(0) or = F(r). This we call the polar equation
of the locus. The polar equation may be deduced from the (x, y)
equation (or vice versa) by means of the formulae above.
Thus the polar equation of a straight line is of the form
rcos(# a)=p,
where p and a are constants. The equation r = 2a cos 6 represents
a circle passing through the origin ; and the general equation of
a circle is of the form
r 2 + c 2 - 2rc cos (6 - a) = A 2 ,
where A, c, and a are constants.
* Polar coordinates are sometimes defined so that r may be positive or negative.
In this case two pairs of coordinates e.g. (1,0) and (-1, TT) correspond to the
same point. The distinction between the two systems may be illustrated by means
of the equation lfr = l -ecos0, where Z>0, e>l. According to our definitions r
must be positive and therefore cos0<l/e: the equation represents one branch only
of a hyperbola, the other having the equation -l\r = \-e cos 0. With the system
of coordinates which admits negative values of r, the equation represents the whole
hyperbola.
44
FUNCTIONS OF KEAL VARIABLES
[n
23. Further examples of functions and their graphical
representation. The examples which follow will give the
reader a better notion of the infinite variety of possible types of
functions.
A. Polynomials. A polynomial in x is a function of the
form
a Q x m + a^- 1 + ...+ a m ,
where a , a 1} ..., a m are constants. The simplest polynomials are
the simple powers y x, a?, a?, . . ., x m , .... The graph of the function
x m is of two distinct types, according as m is even or odd.
First let m = 2. Then three points on the graph are (0, 0),
(1, 1), ( 1, 1). Any number of additional points on the graph
may be found by assigning other special values to x\ thus the
values
tf = i 2, 3,-i, -2, 3
give
= J, 4, 9,
4,9.
(0,0)
Fig. 8.
If the reader will plot off a fair number of points on the graph, he
will be led to conjecture that the
form of the graph is something
like that shown in Fig. 8. If
he draws a curve through the
special points which he has proved
to lie on the graph and then tests
its accuracy by giving x new
values, and calculating the cor
responding values of y, he will
find that they lie as near to the curve as it is reasonable to expect,
when the inevitable inaccuracies of drawing are considered. The
curve is of course a parabola.
There is, however, one fundamental question which we cannot
answer adequately at present. The reader has no doubt some
notion as to what is meant by a continuous curve j _a_c_urve without
breaks or jumps ; such a curve, in fact, as is roughly represented
in Fig. 8. The question is whether the graph of the function
y = # 2 is in fact such a curve. This cannot be proved by merely
23]
FUNCTIONS OF REAL VARIABLES
45
constructing any number of isolated points on the curve, although
the more such points we construct the more probable it will
appear.
This question cannot be discussed properly until Ch. V. In
that chapter we shall consider in detail what our common sense
idea of continuity really means, and how we can prove that such
graphs as the one now considered, and others which we shall
consider later on in this chapter, are really continuous curves.
For the present the reader may be content to draw his curves as
common sense dictates.
It is easy to see that the curve y = # 2 is everywhere convex to the axis of x.
Let PO, PI (Fig. 8) be the points (x Qt x\ (a?i, V)- Then the coordinates of
a point on the chord P Q Pi are # X# +/i#i, y=X# 2 -f-/*#i 2 > where X and p. are
positive numbers whose sum is 1. And
so that the chord lies entirely above the curve.
The curve y = x 4 is similar to y = x* in general appearance, but
flatter near 0, and steeper beyond the points A, A (Fig. 9),
and y x m , where m is even and greater than 4, is still more so,
As m gets larger and larger the flatness and steepness grow
more and more pronounced, until the curve is practically indis
tinguishable from the thick line in the figure.
ly=x*
: A/=.2
x
III
A
*0v
J
o
Fig. 9. Fig. 10.
The reader should next consider the curves given by y=x m y
when m is odd. The fundamental difference between the two
cases is that whereas when m is even ( x) m = x m , so that the
curve is symmetrical about OF, when m is odd ( x) m = x m , so
46 FUNCTIONS OF REAL VARIABLES [ll
that y is negative when x is negative. Fig. 10 shows the curves
y = x, y = a?, and the form to which y x m approximates for
larger odd values of m
It is now easy to see how (theoretically at any rate) the graph
of any polynomial may be constructed. In the first place, from
the graph of y = x m we can at once derive that of Cx m , where C is
a constant, by multiplying the ordinate of every point of the
curve by C. And if we know the graphs of f(x) and F (#), we
can find that of f(x) + F(x) by taking the ordinate of every point
to be the sum of the ordinates of the corresponding points on the
two original curves.
The drawing of graphs of polynomials is however so much
facilitated by the use of more advanced methods, which will be
explained later on, that we shall not pursue the subject further
here.
Examples XI. 1. Trace the curves 3/=7# 4 , y=3# 5 , y=x l .
[The reader should draw the curves carefully, and all three should be
drawn in one figure*. He will then realise how rapidly the higher powers
of x increase, as x gets larger and larger, and will see that, in such a
polynomial as
(or even # 10 -f SO^+VOCte*), it is the first term which is of really preponderant
importance when x is fairly large. Thus even when #=4, a? 10 > 1,000,000,
while 30^< 35,000 and 700^? 4 < 180,000; while if #=10 the preponderance
of the first term is still more marked.]
2. Compare the relative magnitudes of x 12 , 1,000,000^, 1,000,000,000,000^
when #=1, 10, 100, etc.
[The reader should make up a number of examples of this type for himself.
This idea of the relative rate of growth of different functions of x is one with
which we shall often be concerned in the following chapters.]
3. Draw the graph of ax* + Zbx + c
[Here y {(ac - 6 2 )/a} =a {x + (bja}} 2 . If we take new axes parallel to the
old and passing through the point x= b/a, y = (ac b 2 )/a, the new equation
isy = ax 2 . The curve is a parabola.]
4. Trace the curves y =#3 -3# + l, y=# 2 (#-l), y=x(x-\} z .
* It will be found convenient to take the scale of measurement along the axis
of y a good deal smaller than that along the axis of x, in order to prevent the
figure becoming of an awkward size.
23, 24] FUNCTIONS OF REAL VARIABLES 47
24. B. Rational Functions. The class of functions which
ranks next to that of polynomials in simplicity and importance
is that of rational functions. A rational function is the quotient
of one polynomial by another : thus if P (x), Q (x) are polynomials,
we may denote the general rational function by
In the particular case when Q (x} reduces to unity or any other
constant (i.e. does not involve x), R (x) reduces to a polynomial :
thus the class of rational functions includes that of polynomials
as a sub-class. The following points concerning the definition
should be noticed.
(1) We usually suppose that P (x} and Q (x) have no common factor x +a
or xv + axv~ l + bxv-* + ...+k, all such factors being removed by division.
(2) It should however be observed that this removal of common factors
does as a rule change the function. Consider for example the function #/#,
which is a rational function. On removing the common factor x we obtain
1/1 = 1. But the original function is not always equal to 1 : it is equal to 1
only so long as # =f=0. If # = it takes the form 0/0, which is meaningless.
Thus the function xjx is equal to 1 if x^Q and is undefined when #=0. It
therefore differs from the function 1, which is always equal to 1.
(3) Such a function as
may be reduced, by the ordinary rules of algebra, to the form
which is a rational function of the standard form. But here again it must be
noticed that the reduction is not always legitimate. In order to calculate the
value of a function for a given value of x we must substitute the value for x
in the function in the form in which it is given. In the case of this function
the values x= 1, 1, 0, 2 all lead to a meaningless expression, and so the
function is not defined for these values. The same is true of the reduced
form, so far as the values - 1 and 1 are concerned. But x = and x = 2 give
the value 0. Thus once more the two functions are not the same.
(4) But, as appears from the particular example considered under (3),
there will generally be a certain number of values of x for which the function
is not defined even when it has been reduced to a rational function of the
standard form. These are the values of x (if any) for which the de
nominator vanishes. Thus (^ 2 -7)/(^ 2 -3^ + 2) is not defined when # = 1
or 2.
48 FUNCTIONS OF REAL VARIABLES [ll
(5) Generally we agree, in dealing with expressions such as those con
sidered in (2) and (3), to disregard the exceptional values of x for which such
processes of simplification as were used there are illegitimate, and to reduce
our function to the standard form of rational function. The reader will
easily verify that (on this understanding) the sum, product, or quotient of
two rational functions may themselves be reduced to rational functions of
the standard type. And generally a rational function of a rational function
is itself a rational function: i.e. if in z = P(y)IQ(y}> where P and Q are
polynomials, we substitute y PiWIQiW, we obtain on simplification an
equation of the form z = P 2 (x)/Q 2 (x ).
(6) It is in no way presupposed in the definition of a rational function
that the constants which occur as coefficients should be rational numbers.
The word rational has reference solely to the way in which the variable x
appears in the function. Thus
is a rational function
The use of the word rational arises as follows. The rational function
P (x)IQ(x] may be generated from x by a finite number of operations upon
a, including^on^mjiltiplic^tipn of x by itself or a constant, ajddition of terms
thus obtained, and division of one function, obtained by such multiplications
and additions, by another. In so far as the variable x is concerned, this pro
cedure is very much like that by which all rational numbers can be obtained
from unity, a procedure exemplified in the equation
3~ 1+1+1
Again, any function which can be deduced from x by the elementary
operations mentioned above, using at each stage of the process functions
which have already been obtained from x in the same way, can be reduced to
the standard type of rational function. The most general kind of function
which can be obtained in this way is sufficiently illustrated by the example
2
which can obviously be reduced to the standard type of rational function.
25. The drawing of graphs of rational functions, even more
than that of polynomials, is immensely facilitated by the use of
methods depending upon the differential calculus. We shall
therefore content ourselves at present with a very few examples.
Examples XII. 1. Draw the graphs ofy=l/x t y= I/a 8 , y = I/a 3 , . . . ,
[The figures show the graphs of the first two curves. It should be
observed that, since 1/0, I/O 2 , ... are meaningless expressions, these functions
are not defined for # = 0.]
24-26]
FUNCTIONS OF REAL VARIABLES
49
2. Trace y =#+(!/#), #-(!/*), # 2 + (l/# 2 ), # 2 -(l/# 2 ) and
taking various values, positive and negative, for a and b.
3. Trace
4. Trace y = \l(x-a)(x-b\ !/(#-) (x-b) (x-c\ where a<b<c.
5. Sketch the general form assumed by the curves y = \lx m as m
becomes larger and larger, considering separately the cases in which m is
odd or even.
Fig. 11.
Fig. 12.
26. C. Explicit Algebraical Functions. The next im
portant class of functions is that of explicit algebraical functions.
These are functions which can be generated from # by a finite
number of operations such as those used in generating rational
functions, ^gether^jwSB) a finite number of operations of root ]]
extraction. Thus
-7T
are explicit algebraical functions, and so is x m l n (i.e. yat m ) f where m
and n are any integers.
It should be noticed that there is an ambiguity of notation
involved in such an equation as y = Jx. We have, up to the
present, regarded (e.g.) \/2 as denoting the positive square root
of 2, and it would be natural to denote by \/#, where x is any
n. 4
50 FUNCTIONS OF HEAL VARIABLES [ll
positive number, the positive square root of a?, in which case
y ^Jx would be a one-valued function of x. It is however
often more^ convenient to regard *Jx as standing for the two-valued
function whose two values are the positive and negative square
roots of x.
The reader will observe that, when this course is adopted, the
function \jx differs fundamentally from rational functions in two
respects. In the first place a rational function is always defined
for all values of x with a certain number of isolated exceptions.
But \jx is undefined for a whole range of values of x (i.e. all
negative values). Secondly the function, when x has a value
for which it is defined, has generally two values of opposite signs.
The function tyx, on the other hand, is one-valued and defined
for all values of x.
Examples XIII. 1. V{(#- )(&-#)}, where a<b, is defined only for
a < x b. If a<x<b it has two values : if x = a or b only one, viz. 0.
2. Consider similarly
3, Trace the curves y*=x, y 2 =&, y 2 =x 3 .
4. Draw the graphs of the functions y = *j(<#-x i \ y=bj{l - (# 2 /a 2 )}.
27. D. Implicit Algebraical Functions. It is easy to
verify that if
J
or if y = V#
then 2/ 4 - (4i/ 2 + 4y + 1) a? = 0.
Each of these equations may be expressed in the form
y + E iy -^r...+R m =Q .................. (1),
where R 1} R 2 , ..., R m are rational functions of x\ and the reader
will easily verify that, if y is any one of the functions considered
in the last set of examples, y satisfies an equation of this form.
26, 27] FUNCTIONS OF REAL VARIABLES 51
It is naturally suggested that the same is true of any explicit
algebraic function. And this is in fact true, and indeed not
difficult to prove, though we shall not delay to write out a formal
proof here. An example should make clear to the reader the lines
on which such a proof would proceed. Let
x + \/x
Then we have the equations
_x -\-u-\-v-\-w
y x u + v w
u^ x, v* = x + u, w 3 = 1 + oo,
and we have only to eliminate u t v, w between these equations in
order to obtain an equation of the form desired.
We are therefore led to give the following definition : a, function
yf( x ) will be said to be an algebraical function of x if it is the
root of an equation such as (1), i.e. the root of an equation of the
m th degree in y, whose coefficients are rational functions of x. There
is plainly no loss of generality in supposing the first coefficient to
be unity.
This class of functions includes all the explicit algebraical
functions considered in 26. But it also includes other functions
which cannot be expressed as explicit algebraical functions. For
it is known that in general such an equation as (1) cannot be
solved explicitly for y in terms of x, when m is greater than 4,
though such a solution is always possible if m = 1, 2, 3, or 4 and
in special cases for higher values of m.
The definition of an algebraical function should be compared
with that of an algebraical number given in the last chapter
(Misc. Exs. 32). \- * (,
Examples XIV. 1. If m = l, y is a rational function.
2. If m = 2, the equation is f + R^ + R 2 = 0, so that
This function is defined for all values of x for which R? >4/? 2 . It has two
values if R 1 2 >4:R 2 and one if .# 1 2 =4 J ft 2 .
If 7/1 = 3 or 4, we can use the methods explained in treatises on Algebra for
the solution of cubic and biquadratic equations. But as a rule the process is
complicated and the results inconvenient in form, and we can generally study
the properties of the function better by means of the original equation.
" 42
52 FUNCTIONS OF HEAL VARIABLES [ll
3. Consider the functions denned by the equations
in each case obtaining y as an explicit function of x t and stating for what
values of x it is denned.
4. Find algebraical equations, with coefficients rational in #, satisfied by
each of the functions
5. Consider the equation y*=x z .
[Here y*=x. If a? is positive, y=\/x\ if negative, ?/ = V ( - # ). Thus the
function has two values for all values of x save #=0.]
6. An algebraical function of an algebraical function of x is itself an
algebraical function of x.
[For we have
where *+, (^)^-i +...+,$;
Eliminating we find an equation of the form
Here all the capital letters denote rational functions.]
7. An example should perhaps be given of an algebraical function which
cannot be expressed in an explicit algebraical form. Such an example is the
function y defined by the equation
y-y-tf = 0.
But the proof that we cannot find an explicit algebraical expression for y in
terms of x is difficult, and cannot be attempted here.
28. Transcendental functions. All functions of x which
are not rational or even algebraical are called transcendental
functions. This class of functions, being defined in so purely
negative a manner, naturally includes an infinite variety of \vhole
kinds of functions of varying degrees of simplicity and importance.
Among these we can at present distinguish two kinds which are
particularly interesting.
E. The direct and inverse trigonometrical or circular
functions. These are the sine and cosine functions of elementary
trigonometry, and their inverses, and the functions derived from
them. We may assume provisionally that the reader is familiar
with their most important properties *.
* The definitions of the circular functions given in elementary trigonometry pre
suppose that any sector of a circle has associated with it a definite number called its
area. How this assumption is justified will appear in Ch. VII.
27, 28]
