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Title: A Course of Pure Mathematics
Third Edition
Author: G. H. (Godfrey Harold) Hardy
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A COURSE
OF
PURE MATHEMATICS
CAMBRIDGE UNIVERSITY PRESS
C. F. CLAY, Manager
LONDON: FETTER LANE, E.C. 4
NEW YORK : THE MACMILLAN CO.
BOMBAY
CALCUTTA
MADRAS



MACMILLAN AND CO., Ltd.
TORONTO : THE MACMILLAN CO. OF
CANADA, Ltd.
TOKYO : MARUZEN-KABUSHIKI-KAISHA
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
First Edition 1908
Second Edition 1914
Third Edition 1921
PREFACE TO THE THIRD EDITION
No extensive changes have been made in this edition. The most impor-
tant are in §§ 80–82, which I have rewritten in accordance 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 in-
serted in Chapter I a sketch of Dedekind’s theory of real numbers, and a
proof of Weierstrass’s theorem concerning points of condensation; in Chap-
ter 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 con-

cerning implicit functions; in Chapter VI some additional matter concern-
ing 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 definition of the definite integral. In order to find
space for these insertions I have deleted a good deal of the analytical ge-
ometry 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
EXTRACT FROM THE PREFACE TO THE FIRST
EDITION
This 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-
operations—I never even define
∂
2
f

∂x ∂y
and
∂
2
f
∂y ∂x
. In the last two 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. PAGE
1–2. Rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
3–7. Irrational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
8. Real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
9. Relations of magnitude between real numbers . . . . . . . . . . . . . . . . . 16
10–11. Algebraical operations with real numbers . . . . . . . . . . . . . . . . . . . . . 18
12. The number
√
2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
13–14. Quadratic surds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
15. The continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
16. The continuous real variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
17. Sections of the real numbers. Dedekind’s Theorem . . . . . . . . . . . . 30
18. Points of condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

19. Weierstrass’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Decimals, 1. Gauss’s Theorem, 6. Graphical solution of quadratic
equations, 22. Important inequalities, 35. Arithmetical and geomet-
rical means, 35. Schwarz’s Inequality, 36. Cubic and other surds, 38.
Algebraical numbers, 41.
CHAPTER II
FUNCTIONS OF REAL VARIABLES
20. The idea of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
21. The graphical representation of functions. Coordinates . . . . . . . . 46
22. Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
23. Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
24–25. Rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
26–27. Algebraical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
28–29. Transcendental functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
30. Graphical solution of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
CONTENTS viii
SECT. PAGE
31. Functions of two variables and their graphical representation . . 68
32. Curves in a plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
33. Loci in space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Trigonometrical functions, 60. Arithmetical functions, 63. Cylinders, 72.
Contour maps, 72. Cones, 73. Surfaces of revolution, 73. Ruled sur-
faces, 74. Geometrical constructions for irrational numbers, 77. Quadra-
ture of the circle, 79.
CHAPTER III
COMPLEX NUMBERS
34–38. Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
39–42. Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

43. The quadratic equation with real coefficients . . . . . . . . . . . . . . . . . . 96
44. Argand’s diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
45. De Moivre’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
46. Rational functions of a complex variable . . . . . . . . . . . . . . . . . . . . . . 104
47–49. Roots of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Properties of a triangle, 106, 121. Equations with complex coeffi-
cients, 107. Coaxal circles, 110. Bilinear and other transforma-
tions, 111, 116, 125. Cross ratios, 114. Condition that four points
should be concyclic, 116. Complex functions of a real variable, 116.
Construction of regular polygons by Euclidean methods, 120. Imaginary
points and lines, 124.
CHAPTER IV
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
50. Functions of a positive integral variable . . . . . . . . . . . . . . . . . . . . . . . 128
51. Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
52. Finite and infinite classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
CONTENTS ix
SECT. PAGE
53–57. Properties possessed by a function of n for large values of n . . . 131
58–61. Definition of a limit and other definitions . . . . . . . . . . . . . . . . . . . . . 138
62. Oscillating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
63–68. General theorems concerning limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
69–70. Steadily increasing or decreasing functions . . . . . . . . . . . . . . . . . . . . 157
71. Alternative proof of Weierstrass’s Theorem . . . . . . . . . . . . . . . . . . . 159
72. The limit of x
n
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
73. The limit of


1 +
1
n

n
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
74. Some algebraical lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
75. The limit of n(
n
√
x −1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
76–77. Infinite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
78. The infinite geometrical series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
79. The representation of functions of a continuous real variable by
means of limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
80. The bounds of a bounded aggregate . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
81. The bounds of a bounded function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
82. The limits of indetermination of a bounded function . . . . . . . . . . 180
83–84. The general principle of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 183
85–86. Limits of complex functions and series of complex terms . . . . . . 185
87–88. Applications to z
n
and the geometrical series . . . . . . . . . . . . . . . . . 188
Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
Oscillation of sin nθπ, 144, 146, 181. Limits of n
k
x
n
,
n

√
x,
n
√
n,
n
√
n!,
x
n
n!
,

m
n

x
n
, 162, 166. Decimals, 171. Arithmetical series, 175. Harmonical
series, 176. Equation x
n+1
= f(x
n
), 190. Expansions of rational func-
tions, 191. Limit of a mean value, 193.
CHAPTER V
LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND
DISCONTINUOUS FUNCTIONS
89–92. Limits as x → ∞ or x → −∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
CONTENTS x

SECT. PAGE
93–97. Limits as x → a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
98–99. Continuous functions of a real variable . . . . . . . . . . . . . . . . . . . . . . . . 210
100–104. Properties of continuous functions. Bounded functions. The
oscillation of a function in an interval . . . . . . . . . . . . . . . . . . . . 215
105–106. Sets of intervals on a line. The Heine-Borel Theorem . . . . . . . . . . 223
107. Continuous functions of several variables . . . . . . . . . . . . . . . . . . . . . . 228
108–109. Implicit and inverse functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
Limits and continuity of polynomials and rational functions, 204, 212.
Limit of
x
m
− a
m
x −a
, 206. Orders of smallness and greatness, 207. Limit of
sin x
x
, 209. Infinity of a function, 213. Continuity of cos x and sin x, 213.
Classification of discontinuities, 214.
CHAPTER VI
DERIVATIVES AND INTEGRALS
110–112. Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
113. General rules for differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
114. Derivatives of complex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
115. The notation of the differential calculus . . . . . . . . . . . . . . . . . . . . . . . 246
116. Differentiation of polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
117. Differentiation of rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
118. Differentiation of algebraical functions . . . . . . . . . . . . . . . . . . . . . . . . 253

119. Differentiation of transcendental functions . . . . . . . . . . . . . . . . . . . . 255
120. Repeated differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
121. General theorems concerning derivatives. Rolle’s Theorem . . . . 262
122–124. Maxima and minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
125–126. The Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
127–128. Integration. The logarithmic function . . . . . . . . . . . . . . . . . . . . . . . . . 277
129. Integration of polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
130–131. Integration of rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
CONTENTS xi
SECT. PAGE
132–139. Integration of algebraical functions. Integration by rationalisa-
tion. Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
140–144. Integration of transcendental functions . . . . . . . . . . . . . . . . . . . . . . . . 298
145. Areas of plane curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
146. Lengths of plane curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
Derivative of x
m
, 241. Derivatives of cos x and sin x, 241. Tangent and
normal to a curve, 241, 257. Multiple roots of equations, 249, 309. Rolle’s
Theorem for polynomials, 251. Leibniz’ Theorem, 259. Maxima and min-
ima of the quotient of two quadratics, 269, 310. Axes of a conic, 273.
Lengths and areas in polar coordinates, 307. Differentiation of a deter-
minant, 308. Extensions of the Mean Value Theorem, 313. Formulae of
reduction, 314.
CHAPTER VII
ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND INTEGRAL
CALCULUS
147. Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
148. Taylor’s Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

149. Applications of Taylor’s Theorem to maxima and minima . . . . . 326
150. Applications of Taylor’s Theorem to the calculation of limits . . 327
151. The contact of plane curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
152–154. Differentiation of functions of several variables . . . . . . . . . . . . . . . . 335
155. Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
156–161. Definite Integrals. Areas of curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
162. Alternative proof of Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 367
163. Application to the binomial series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
164. Integrals of complex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
Newton’s method of approximation to the roots of equations, 322. Se-
ries for cos x and sin x, 325. Binomial series, 325. Tangent to a curve,
331, 346, 374. Points of inflexion, 331. Curvature, 333, 372. Osculating
CONTENTS xii
conics, 334, 372. Differentiation of implicit functions, 346. Fourier’s inte-
grals, 355, 360. The second mean value theorem, 364. Homogeneous func-
tions, 372. Euler’s Theorem, 372. Jacobians, 374. Schwarz’s inequality for
integrals, 378. Approximate values of definite integrals, 380. Simpson’s
Rule, 380.
CHAPTER VIII
THE CONVERGENCE OF INFINITE SERIES AND INFINITE INTEGRALS
SECT. PAGE
165–168. Series of positive terms. Cauchy’s and d’Alembert’s tests of con-
vergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
169. Dirichlet’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
170. Multiplication of series of positive terms . . . . . . . . . . . . . . . . . . . . . . 388
171–174. Further tests of convergence. Abel’s Theorem. Maclaurin’s inte-
gral test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
175. The series


n
−s
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
176. Cauchy’s condensation test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
177–182. Infinite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
183. Series of positive and negative terms . . . . . . . . . . . . . . . . . . . . . . . . . . 416
184–185. Absolutely convergent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
186–187. Conditionally convergent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
188. Alternating series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
189. Abel’s and Dirichlet’s tests of convergence . . . . . . . . . . . . . . . . . . . . 425
190. Series of complex terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
191–194. Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
195. Multiplication of series in general . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
The series

n
k
r
n
and allied series, 385. Transformation of infinite inte-
grals by substitution and integration by parts, 404, 406, 413. The series

a
n
cos nθ,

a
n
sin nθ, 419, 425, 427. Alteration of the sum of a series

by rearrangement, 423. Logarithmic series, 431. Binomial series, 431, 433.
Multiplication of conditionally convergent series, 434, 439. Recurring se-
ries, 437. Difference equations, 438. Definite integrals, 441. Schwarz’s
inequality for infinite integrals, 442.
CONTENTS xiii
CHAPTER IX
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS OF A REAL
VARIABLE
SECT. PAGE
196–197. The logarithmic function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
198. The functional equation satisfied by log x . . . . . . . . . . . . . . . . . . . . . 447
199–201. The behaviour of log x as x tends to infinity or to zero . . . . . . . . 448
202. The logarithmic scale of infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
203. The number e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452
204–206. The exponential function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
207. The general power a
x
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
208. The exponential limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
209. The logarithmic limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
210. Common logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460
211. Logarithmic tests of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466
212. The exponential series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
213. The logarithmic series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
214. The series for arc tan x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
215. The binomial series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480
216. Alternative development of the theory . . . . . . . . . . . . . . . . . . . . . . . . 482
Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484
Integrals containing the exponential function, 460. The hyperbolic func-
tions, 463. Integrals of certain algebraical functions, 464. Euler’s con-

stant, 469, 486. Irrationality of e, 473. Approximation to surds by the bi-
nomial theorem, 480. Irrationality of log
10
n, 483. Definite integrals, 491.
CHAPTER X
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND
CIRCULAR FUNCTIONS
217–218. Functions of a complex variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495
219. Curvilinear integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496
220. Definition of the logarithmic function . . . . . . . . . . . . . . . . . . . . . . . . . 497
221. The values of the logarithmic function . . . . . . . . . . . . . . . . . . . . . . . . 499
CONTENTS xiv
SECT. PAGE
222–224. The exponential function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
225–226. The general power a
z
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
227–230. The trigonometrical and hyperbolic functions . . . . . . . . . . . . . . . . . 512
231. The connection between the logarithmic and inverse trigonomet-
rical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518
232. The exponential series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520
233. The series for cos z and sin z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522
234–235. The logarithmic series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
236. The exponential limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529
237. The binomial series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531
Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534
The functional equation satisfied by Log z, 503. The function e
z
, 509.
Logarithms to any base, 510. The inverse cosine, sine, and tangent of

a complex number, 516. Trigonometrical series, 523, 527, 540. Roots of
transcendental equations, 534. Transformations, 535, 538. Stereographic
projection, 537. Mercator’s projection, 538. Level curves, 539. Definite
integrals, 543.
Appendix I. The proof that every equation has a root . . . . . . . . . . . . . . . 545
Appendix II. A note on double limit problems . . . . . . . . . . . . . . . . . . . . . . . . 553
Appendix III. The circular functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557
Appendix IV. The infinite in analysis and geometry . . . . . . . . . . . . . . . . . . . 560
CHAPTER I
REAL VARIABLES
1. Rational numbers. A fraction r = p/q, where p and q are pos-
itive 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
p/(−q) = (−p)/q, (−p)/(−q) = p/q.
To the rational numbers thus defined we may add the ‘rational number 0’
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. If r and s are rational numbers, then r + s, r − s, rs,
and r/s are rational numbers, unless in the last case s = 0 (when r/s is of course
meaningless).
2. If λ, m, and n are positive rational numbers, and m > n, then
λ(m
2
−n
2
), 2λmn, and λ(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 denomina-
tor 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:
1
1
,
2
1
,
1
2
,
3
1
,
2
2
,
1
3
,
4

1
,
3
2
,
2
3
,
1
4
, . . . .
Show that p/q is the [
1
2
(p + q − 1)(p + q − 2) + q]th term of the series.
[In this series every rational number is repeated indefinitely. Thus 1 occurs
as
1
1
,
2
2
,
3
3
, . . . . We can of course avoid this by omitting every number which has
already occurred in a simpler form, but then the problem of determining the
precise position of p/q becomes more complicated.]
1
[I : 2] REAL VARIABLES 2

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 employed 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 geome-
try; we may be 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 Λ,
produced indefinitely in both directions, and a segment A
0
A
1
of any length.
We call A
0
the origin, or the point 0, and A
1
the point 1, and we regard
these points as representing the numbers 0 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
0
A

r
/A
0
A
1
= r,
A
0
A
r
being a stretch of the line extending in the same direction along the
line as A
0
A
1
, 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 negative rational number r = −s,
A
0
A
1
A
s
A
−1
A
−s
Fig. 1.
it is natural to regard length as a magnitude capable of sign, positive if

the length is measured in one direction (that of A
0
A
1
), and negative if
measured in the other, so that AB = −BA; and to take as the point
representing r the point A
−s
such that
A
0
A
−s
= −A
−s
A
0
= −A
0
A
s
.
[I : 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
A
0
A

r
= r · A
0
A
1
;
and if, as is natural, we take A
0
A
1
as our unit of length, and write
A
0
A
1
= 1, then we have
A
0
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 BC on Λ, we can find as many
rational points as we please on BC.

Suppose, for example, that BC falls within the segment A
1
A
2
. It is
evident that if we choose a positive integer k so that
k · BC > 1,
∗
(1)
and divide A
1
A
2
into k equal parts, then at least one of the points 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 corresponds 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 assumption that this is possible is equivalent to the assumption of what is
known as the Axiom of Archimedes.

[I : 3] REAL VARIABLES 4
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 BC’ 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 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 following 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 n, we can
find another rational number lying on either side of r and differing from r by
less than 1/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 1/n, 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
[I : 3] REAL VARIABLES 5
must be associated a certain entity called its length, which must be a
quantity capable of numerical measurement 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
multiplication. Again, it must be possible to construct a line whose length
is the sum or product of any two given lengths. If the length P Q, along
a given line, is a, and the length QR, along the same straight line, is b,
the length PR must be a + b. 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 + c) = (a + b) + c,
ab = ba, a(bc) = (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 Λ from left to right, we must have AB < AC, and so on. Moreover
it must be possible, on our fundamental line Λ, to find a point P such
that A
0
P is equal to any segment whatever taken along Λ 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
= 2. For instance, we may construct an isosceles right-angled tri-
angle ABC such that AB = AC = 1. Then if BC = x, x
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 require-
ments therefore involve the existence of a length measured by a number x,
[I : 3] REAL VARIABLES 6
A B
C
1
1
x
L M N
P
2 1
x
Fig. 2.
and a point P on Λ such that
A
0
P = x, x
2

= 2.
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
2
/q
2
= m/n,
p having no factor in common with q, and m no factor in common with n.
Then np
2
= mq
2
. Every factor of q
2
must divide np
2
, and as p and q
have no common factor, every factor of q
2
must divide n. Hence n = λq
2
,
where λ is an integer. But this involves m = λp
2
: and as m and n have
no common factor, λ must be unity. Thus m = p

2
, n = q
2
, 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 num-
ber x and a point P , not one of the rational points already constructed,
such that A
0
P = x, x
2
= 2; and (as the reader will remember from ele-
mentary algebra) we write x =
√
2.
The following alternative proof that no rational number can have its square
equal to 2 is interesting.
[I : 4] REAL VARIABLES 7
Suppose, if possible, that p/q is a positive fraction, in its lowest terms, such
that (p/q)
2
= 2 or p
2
= 2q
2
. It is easy to see that this involves (2q − p)
2
=

2(p −q)
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 p/q 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 p/q 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
x
n
+ p
1
x
n−1
+ p
2
x
n−2
+ ···+ p
n
= 0,
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
without a common factor, and b is positive. Writing a/b for x, and multiplying
by b
n−1

, we obtain
−
a
n
b
= p
1
a
n−1
+ p
2
a
n−2
b + ··· + p
n
b
n−1
,
a fraction in its lowest terms equal to an integer, which is absurd. Thus b = 1,
and the root is a. It is evident that a must be a divisor of p
n
.]
4. Show that if p
n
= 1 and neither of
1 + p
1
+ p
2
+ p

3
+ . . . , 1 − p
1
+ p
2
− p
3
+ . . .
is zero, then the equation cannot have a rational root.
5. Find the rational roots (if any) of
x
4
− 4x
3
− 8x
2
+ 13x + 10 = 0.
[The roots can only be integral, and so ±1, ±2, ±5, ±10 are the only possi-
bilities: 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.]
[I : 4] REAL VARIABLES 8
4. Irrational numbers (continued). The result of our geometrical
representation of the rational numbers is therefore to suggest the desirabil-
ity of enlarging our conception of ‘number’ by the introduction of further
numbers of a new kind.
The same conclusion might have been reached without the use of ge-
ometrical language. One of the central problems of algebra is that of the
solution of equations, such as
x
2

= 1, x
2
= 2.
The first equation has the two rational roots 1 and −1. But, if our con-
ception 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 x
3
= 2, x
4
= 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 x
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 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, 1.999 396, 1.999 961 64, . . .
[I : 4] REAL VARIABLES 9
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, 2.002 225, 2.000 244 49, . . .
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 R, differing by as little as we please. For we saw in §3 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 R, 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 δ, δ being any positive rational number as small as we please,
such as .01 or .0001 or .000 001. 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 δ.
We can now prove that an x can be found in L and a y in R such that 2−x
2
and y
2

−2 are as small as we please, say less than δ. Substituting
1
4
δ for δ in the
argument which precedes, we see that we can choose x and y so that y −x <
1
4
δ;
and we may plainly suppose that both x and y are less than 2. Thus
y + x < 4, y
2
− x
2
= (y −x)(y + x) < 4(y −x) < δ;
and since x
2
< 2 and y
2
> 2 it follows a fortiori that 2 −x
2
and y
2
−2 are each
less than δ.
It follows also that there can be no largest member of L or smallest
member of R. For if x is any member of L, then x
2
< 2. Suppose that
x
2

= 2 −δ. Then we can find a member x
1
of L such that x
2
1
differs from 2
by less than δ, and so x
2
1
> x
2
or x
1
> x. Thus there are larger members