zotero-db/storage/C93M2CB7/.zotero-ft-cache

The Project Gutenberg eBook of A Course of Pure Mathematics, by G. H. (Godfrey Harold) Hardy
This eBook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook.
Title: A Course of Pure Mathematics Third Edition
Author: G. H. (Godfrey Harold) Hardy
Release Date: February 5, 2012 [EBook #38769] Most recently updated: August 6, 2021
Language: English
Character set encoding: UTF-8
*** START OF THE PROJECT GUTENBERG EBOOKA COURSE OF PURE MATHEMATICS ***

Produced by Andrew D. Hwang, Brenda Lewis, and the Online Distributed Proofreading Team at http://www.pgdp.net (This file was produced from images generously made available by The Internet Archive/American Libraries.) Revised by Richard Tonsing.
Transcriber’s Note
Minor typographical corrections and presentational changes have been made without comment. Notational modernizations are listed in the transcriber’s note at the end of the book. All changes are detailed in the LATEX source file, which may be downloaded from
www.gutenberg.org/ebooks/38769. This PDF file is optimized for screen viewing, but may easily be recompiled for printing. Please consult the preamble of the LATEX source file for instructions.

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 MACMILLAN AND CO., Ltd. MADRAS  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.

August 1921

G. H. H.

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 definition 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.

October 1914

G. H. H.

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

∂2f ∂x ∂y

and

∂2f ∂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.
1–2. 3–7. 8. 9. 10–11. 12. 13–14. 15. 16. 17. 18. 19.

PAGE
Rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Irrational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Relations of magnitude between real numbers . . . . . . . . . . . . . . . . . 16 Algebraical o√perations with real numbers . . . . . . . . . . . . . . . . . . . . . 18 The number 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Quadratic surds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 The continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 The continuous real variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Sections of the real numbers. Dedekind’s Theorem . . . . . . . . . . . . 30 Points of condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Weierstrass’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Decimals, 1. Gauss’s Theorem, 6. Graphical solution of quadratic equations, 22. Important inequalities, 35. Arithmetical and geometrical means, 35. Schwarz’s Inequality, 36. Cubic and other surds, 38. Algebraical numbers, 41.

20. 21. 22. 23. 24–25. 26–27. 28–29. 30.

CHAPTER II
FUNCTIONS OF REAL VARIABLES
The idea of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 The graphical representation of functions. Coordinates . . . . . . . . 46 Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Algebraical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Transcendental functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Graphical solution of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

CONTENTS

viii

SECT.
31. 32. 33.

PAGE
Functions of two variables and their graphical representation . . 68 Curves in a plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 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 surfaces, 74. Geometrical constructions for irrational numbers, 77. Quadrature of the circle, 79.

34–38. 39–42. 43. 44. 45. 46. 47–49.

CHAPTER III
COMPLEX NUMBERS
Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 The quadratic equation with real coefficients . . . . . . . . . . . . . . . . . . 96 Argand’s diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 De Moivre’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Rational functions of a complex variable . . . . . . . . . . . . . . . . . . . . . . 104 Roots of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Properties of a triangle, 106, 121. Equations with complex coefficients, 107. Coaxal circles, 110. Bilinear and other transformations, 111, 116, 125. Cross ratios, 115. 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.
53–57. 58–61. 62. 63–68. 69–70. 71. 72. 73.
74. 75. 76–77. 78. 79.
80. 81. 82. 83–84. 85–86. 87–88.

PAGE

Properties possessed by a function of n for large values of n . . . 131

Definition of a limit and other definitions . . . . . . . . . . . . . . . . . . . . . 138

Oscillating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

General theorems concerning limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Steadily increasing or decreasing functions . . . . . . . . . . . . . . . . . . . . 157

Alternative proof of Weierstrass’s Theorem . . . . . . . . . . . . . . . . . . . 159

The The

limit limit

of of

xn 1

.. +

.. 1

.

.n.

..... ....

. .

. .

. .

. .

. .

. .

. .

................ ................

. .

. .

. .

. .

. .

. .

. .

..... .....

160 164

n

Some algebraica√l lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 The limit of n( n x − 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Infinite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

The infinite geometrical series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

The representation of functions of a continuous real variable by means of limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

The bounds of a bounded aggregate . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

The bounds of a bounded function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

The limits of indetermination of a bounded function . . . . . . . . . . 180

The general principle of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Limits of complex functions and series of complex terms . . . . . . 185 Applications to zn and the geometrical series . . . . . . . . . . . . . . . . . 188

Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

Oscillation

of

sin nθπ,

144,

146,

181.

Limits

of

nk xn ,

√ n x,

√ n n,

√ n n!,

xn n!

,

m xn, 162, 166. Decimals, 171. Arithmetical series, 175. Harmonical n

series, 176. Equation xn+1 = f (xn), 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 . . . . . . . . . . . . . . . . . . . . 216

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

xm x

− −

am a

,

206.

Orders

of

smallness

and

greatness,

207.

Limit

of

sin x , 209. Infinity of a function, 213. Continuity of cos x and sin x, 213.

x

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 xm, 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 minima of the quotient of two quadratics, 269, 310. Axes of a conic, 273. Lengths and areas in polar coordinates, 307. Differentiation of a determinant, 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. Series 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 integrals, 355, 360. The second mean value theorem, 364. Homogeneous functions, 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 nkrn and allied series, 385. Transformation of infinite integrals by substitution and integration by parts, 404, 406, 413. The series
an cos nθ, an 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 series, 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 ax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 functions, 463. Integrals of certain algebraical functions, 464. Euler’s constant, 469, 486. Irrationality of e, 473. Approximation to surds by the binomial theorem, 480. Irrationality of log10 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 az . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ez, 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 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
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(m2 − n2), 2λmn, and λ(m2 + n2) 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 denominator 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 geometry; 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 A0A1 of any length. We call A0 the origin, or the point 0, and A1 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 Ar such that
A0Ar/A0A1 = r,
A0Ar being a stretch of the line extending in the same direction along the line as A0A1, 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−s

A−1 A0

A1

As

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 A0A1), 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
A0A−s = −A−sA0 = −A0As.

[I : 3]

REAL VARIABLES

3

We thus obtain a point Ar on the line corresponding to every rational value of r, positive or negative, and such that
A0Ar = r · A0A1;
and if, as is natural, we take A0A1 as our unit of length, and write A0A1 = 1, then we have
A0Ar = r. We shall call the points Ar 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 A1A2. It is evident that if we choose a positive integer k so that

k · BC > 1,*

(1)

and divide A1A2 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 A1A2 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 P R 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 A0P 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 x2 = 2. For instance, we may construct an isosceles right-angled triangle ABC such that AB = AC = 1. Then if BC = x, x2 = 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,

[I : 3]

REAL VARIABLES

6

P C

x

1

x

A1B

L

2

Fig. 2.

M1N

and a point P on Λ such that
A0P = x, x2 = 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
p2/q2 = m/n,
p having no factor in common with q, and m no factor in common with n. Then np2 = mq2. Every factor of q2 must divide np2, and as p and q have no common factor, every factor of q2 must divide n. Hence n = λq2, where λ is an integer. But this involves m = λp2: and as m and n have no common factor, λ must be unity. Thus m = p2, n = q2, 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 A0P = x, x2 = 2; an√d (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.

[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 p2 = 2q2. 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

xn + p1xn−1 + p2xn−2 + · · · + pn = 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 bn−1, we obtain

− an b

=

p1an−1 + p2an−2b + · · · + pnbn−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 pn.]
4. Show that if pn = 1 and neither of

1 + p1 + p2 + p3 + . . . , 1 − p1 + p2 − p3 + . . .

is zero, then the equation cannot have a rational root. 5. Find the rational roots (if any) of

x4 − 4x3 − 8x2 + 13x + 10 = 0.

[The roots can only be integral, and so ±1, ±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.]

[I : 4]

REAL VARIABLES

8

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 ‘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
x2 = 1, x2 = 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 x3 = 2, x4 = 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 x2 = 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 − x2

and y2 − 2 argument

are as small as which precedes,

we we

please, say less than δ. see that we can choose

Substituting

1 4

δ

x and y so that

for y−

δ x

in <

the

1 4

δ;

and we may plainly suppose that both x and y are less than 2. Thus

y + x < 4, y2 − x2 = (y − x)(y + x) < 4(y − x) < δ;

and since x2 < 2 and y2 > 2 it follows a fortiori that 2 − x2 and y2 − 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 x2 < 2. Suppose that x2 = 2 − δ. Then we can find a member x1 of L such that x21 differs from 2 by less than δ, and so x21 > x2 or x1 > x. Thus there are larger members

[I : 5]

REAL VARIABLES

10

of L than x; and as x is any 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 Λ such that P divides the points which correspond to members of L from those which correspond to members of R.

L

L L LL RR R R

R

A0

P

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, x2 has a definite meaning. Then x2 cannot be either less than or greater than 2. For suppose, for example, that x2 is less than 2. Then it follows from what precedes that we can find a positive rational number ξ such that ξ2 lies between x2 and 2. That is to say, we can find a member of L greater than x; and this contradicts the supposition that x divides the members of L from those of R. Thus x2 cannot be less than 2, and similarly it cannot be greater than 2. We are therefore driven to the conclu√sion that x2 = 2, and that x is th√e number which in algebra we denote by 2. And of course this number 2 is not rational, for no rational number has its

[I : 5]

REAL VARIABLES

11

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 x2 = 2, almost word for word; for example to x2 = N , where N is any integer which is not a perfect square, or to

x3 = 3, x3 = 7, x4 = 23,

or, as we shall see later on, to x3 = 3x+8. We are thus led to believe in the

existence of irrational numbers x and points P on Λ such tha√t x satisfies equations such as these, even when these lengths cannot (as 2 can) be

constructed by means of elementary geometrical methods.

the

The root

reader will of such an

no doubt equation

remember as xq = n

that in treatises√on is denoted by q n

elementary algebra or n1/q, and that a

meaning is attached to such symbols as

np/q, n−p/q

by means of the equations np/q = (n1/q)p, np/qn−p/q = 1.

And he will remember how, in virtue of these definitions, the ‘laws of indices’

such as

nr × ns = nr+s, (nr)s = nrs

are extended so as to cover the case in which r and s are any rational numbers
whatever.
The reader may now follow one or other of two alternative courses. He
ma√y, if √he pleases, be content to assume that ‘irrational numbers’ such as 2, 3 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 attitude,
he will be well advised to pay careful attention to the sections which follow, in which these questions receive fuller consideration.„

*This is the point of view which was adopted in the first edition of this book. „In these sections I have borrowed freely from Appendix I of Bromwich’s Infinite Series.

[I : 6]

REAL VARIABLES

12

Examples III. 1. Find the differen√ce between 2 and the squares of the
decimals given in § 4 as approximations to 2. 2. Find the differences between 2 and the squares of

1 1

,

3 2

,

7 5

,

17 12

,

41 29

,

99 70

.

√ 3. Show that if m/n is a good approximation to 2, then (m + 2n)/(m + n)

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. 4. If x and y are approximations to 2, by defect and by excess respectively,
and 2 − x2 < δ, y2 − 2 < δ, then y − x < δ.

5. The equation x2 = 4 is satisfied by x = 2. Examine how far the argument

of the preceding sections applies to this equation (writing 4 for 2 throughout).

[If we define the classes L, R as before, they do not include all rational numbers. The rational number 2 is an exception, since 22 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 x2 < 2 for the members of one class and x2 > 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 x3 < 2 and x3 > 2, or x4 < 7 and x4 > 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 ‘x2 < 2’ and Q the property ‘x2 > 2.’ Then 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. 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

[I : 7]

REAL VARIABLES

13

(or Q) were the property of being rational, or of being positive. For the

present, however, we shall 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 importance. We observe first that

it is impossible in any case that L should have a greatest member and

R a least. For if l were the greatest member of L, and r the least of R,

so

that

l

<

r,

then

1 2

(l

+

r)

would

be

a

positive

rational

number

lying

between l 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 l, 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 ‘x2 ≦ 1’ and Q to be ‘x2 > 1’; here l = 1. If P is ‘x2 < 1’ and Q is ‘x2 ≧ 1’, we have an example of the second possibility,
with r = 1. It should be observed that we do not obtain a section at all by taking P to be ‘x2 < 1’ and Q to be ‘x2 > 1’; for the special number 1 escapes

classification (cf. Ex. iii. 5).

7. Irrational numbers (continued ). In the first two cases we say that the section corresponds to a positive rational number a, which is l 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 α.* For we might take P and Q to be the properties expressed by
x ≦ a, x > a
respectively, or by x < a and x ≧ a. In the first case a would be the greatest member of L, and in the second case the least member of R.
*It will be convenient to denote a section, corresponding to a rational number denoted by an English letter, by the corresponding Greek letter.

[I : 8]

REAL VARIABLES

14

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 only sections in which the lower class L has no greatest number.
There being this correspondence between the positive rational numbers and the sections defined 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, α > α′ would mean the same as a > a′, if α and α′ 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 generali-
sation 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

[I : 8]

REAL VARIABLES

15

the word positive occasionally.

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

1 2

>

1 3

,

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

1 2

>

1 3

and

way affected by any doubt as

or real numbers. Sometimes,

1 3
to

>wh14etwheerca12n,

13in, faenr d12

of course, the context

i41>nawr14eh; iatchrhiet(hiemn.gfee.)triec‘n12ac’leofricasccuitnriosnnioss

sufficient

to

fix

its

interpretation.

When

we

say

(see

§ 9)

that

1 2

<

1 3

,

we

must

mean

by

‘

1 2

’

the

real

number

1 2

.

The reader should observe, moreover, that no particular logical importance

is to be attached to the precise form of definition of a ‘real number’ 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 math-

ematics 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 mathematics

[I : 9]

REAL VARIABLES

16

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 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 + 1 < 0 and Q to be x + 1 ≧ 0. This section plainly corresponds to the negative rational number −1. If we took P to be x3 < −2 and Q to be x3 > −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
*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 inconvenient 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.

[I : 9]

REAL VARIABLES

17

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 α, β, γ, . . . ; the rational numbers of its lower and upper classes by the corresponding English letters a, A; b, B; c, C; . . . . The classes themselves we denote by (a), (A), . . . .
If α and β are two real numbers, there are three possibilities: (i) every a is a b and every A a B; in this case (a) is identical with (b) and (A) with (B);
(ii) every a is a b, but not all A’s are B’s; in this case (a) is a proper part of (b),* 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 α = β, in case (ii) α < β, and in case (iii) α > β. It is clear that, when α and β are both rational, these definitions agree

β

α

(i)

β

α

(ii)

β

α

(iii)

Fig. 4.

with the ideas of equality and inequality between rational numbers which we began by taking for granted; and that any positive number is greater than any negative number.
It will be convenient to define at this stage the negative −α of a positive number α. If (a), (A) are the classes which constitute α, 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 −α. Similarly we can define −α
*I.e. is included in but not identical with (b).

[I : 10]

REAL VARIABLES

18

when α is negative or zero; if α is negative, −α is positive. It is plain also that −(−α) = α. Of the two numbers α and −α one is always positive (unless α = 0). The one which is positive we denote by |α| and call the modulus of α.
Examples IV. 1. Prove that 0 = −0.
2. Prove that β = α, β < α, or β > α according as α = β, α > β, or α < β. 3. If α = β and β = γ, then α = γ. 4. If α ≦ β, β < γ, or α < β, β ≦ γ, then α < γ. 5. Prove that −β = −α, −β < −α, or −β > −α, according as α = β, α < β, or α > β. 6. Prove that α > 0 if α is positive, and α < 0 if α is negative. 7. Prove that α ≦ √|α|. √ 8. Prove that 1 < 2 < 3 < 2. 9. Prove that, if α and β are two different real numbers, we can always find an infinity of rational numbers lying between α and β. [All these results are immediate consequences of our definitions.]
10. Algebraical operations with real numbers. We now proceed to define the meaning of the elementary algebraical operations such as addition, as applied to real numbers in general.
(i) Addition. In order to define the sum of two numbers α and β, 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 C; and so C − c cannot be less than s − r. But
C − c = (A − a) + (B − b);
and we can choose a, b, A, B so that both A − a and B − b are as small as we like; and this plainly contradicts our hypothesis.

[I : 10]

REAL VARIABLES

19

If every rational number belongs to (c) or to (C), the classes (c), (C) form a section of the rational numbers, that is to say, a number γ. If there is one which does not, we add it to (C). We have now a section or real number γ, which must clearly be rational, since it corresponds to the least member of (C). In any case we call γ the sum of α and β, and write
γ = α + β.
If both α and β are rational, they are the least members of the upper classes (A) and (B). In this case it is clear that α + β is the least member of (C), so that our definition agrees with our previous ideas of addition.
(ii) Subtraction. We define α − β by the equation
α − β = α + (−β).
The idea of subtraction accordingly presents no fresh difficulties.
Examples V. 1. Prove that α + (−α) = 0.
2. Prove that α + 0 = 0 + α = α. 3. Prove that α + β = β + α. [This follows at once from the fact that the classes (a + b) and (b + a), or (A + B) and (B + A), are the same, since, e.g., a + b = b + a when a and b are rational.] 4. Prove that α + (β + γ) = (α + β) + γ. 5. Prove that α − α = 0. 6. Prove that α − β = −(β − α). 7. From the definition of subtraction, and Exs. 4, 1, and 2 above, it follows that
(α − β) + β = {α + (−β)} + β = α + {(−β) + β} = α + 0 = α.
We might therefore define the difference α − β = γ by the equation γ + β = α. 8. Prove that α − (β − γ) = α − β + γ. 9. Give a definition of subtraction which does not depend upon a previous
definition of addition. [To define γ = α − β, form the classes (c), (C) for which

[I : 11]

REAL VARIABLES

20

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 |α| − |β| ≦ |α ± β| ≦ |α| + |β|.

11. Algebraical operations with real numbers (continued ). (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 practically 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 (C). This depends, as in the case of addition, on showing that we can choose a, A, b, and B so that C − c is as small as we please. Here we use the identity
C − c = AB − ab = (A − a)B + a(B − b).
Finally we include negative numbers within the scope of our definition by agreeing that, if α and β are positive, then
(−α)β = −αβ, α(−β) = −αβ, (−α)(−β) = αβ.
(iv) Division. In order to define division, we begin by defining the reciprocal 1/α of a number α (other than zero). Confining ourselves in the first instance to positive numbers and sections of positive rational numbers, we define the reciprocal of a positive number α by means of the lower class (1/A) and the upper class (1/a). We then define the reciprocal of a negative number −α by the equation 1/(−α) = −(1/α). Finally we define α/β by the equation
α/β = α × (1/β).

[I : 13]

REAL VARIABLES

21

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:

1. α × 0 = 0 × α = 0. 2. α × 1 = 1 × α = α.

5. α(βγ) = (αβ)γ. 6. α(β + γ) = αβ + αγ.

3. α × (1/α) = 1.

7. (α + β)γ = αγ + βγ.

4. αβ = βα.

8. |αβ| = |α| |β|.

√ 12. The number 2. Let us now return for a moment to the partic-
ular irrational number which we discussed in §§ 4–5. We there constructed

a section by means of the inequalities x2 < 2, x2 > 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 rationa√l 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) (AA′), 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

√

√√

( 2)2 = 2 2 = 2.

Again

√

√ √ √√ √

(− 2)2 = (− 2)(− 2) = 2 2 = ( 2)2 = 2.

√

√

Thus the equation x2 = 2 has the two roots 2 and − 2. Similarly we

could discuss the e√quatio√ns x√2 = 3, x3 = 7, . . . and the corresponding irrational numbers 3, − 3, 3 7, . . . .

[I : 13]

REAL VARIABLES

22

13. Quadratic surds. A number of the form ±√a, where a is a

positive rational number which is not the square of another rati√onal number, is called a pu√re 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.

√

The two numbers a ± b are the roots of the quadratic equation

x2 − 2ax + a2 − b = 0.

Conversely, the equation x2 + 2px + q = 0, where p and q are rational, and p2 − q > 0, has as its roots the two quadratic surds −p ± p2 − 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

√

√

√

2 + 2 + 2 + 2 + 2 + 2.

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 constructed 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, 2 + 2, 2 + 2 + 2.

2. The quadratic equation ax2+2bx+c = 0 has two real roots„ if b2−ac > 0.
*See Ch. II, Misc. Exs. 22. „I.e. there are two values of x for which ax2 + 2bx + c = 0. If b2 − 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 b2 = ac 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.

[I : 14]

REAL VARIABLES

23

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 ± b2 √− ac}/a. It is easy to construct these lengths geometrically, first constructing b2 − ac. A much more elegant, though less straightforward, construction is the following.
Draw a circle of unit radius, a diameter P Q, and the tangents at the ends of the diameters.
P P′
M

N

Q′

Y

Q

X

Fig. 5.

Take P P ′ = −2a/b and QQ′ = −c/2b, having regard to sign.* Join P ′Q′, cutting the circle in M and N . Draw P M and P N , cutting QQ′ in X and Y .
Then QX and QY are the roots of the equation with their proper signs.„

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 BC in X and Y . Then BX and BY are the roots.

3. If ac is positive P P ′ and QQ′ will be drawn in the same direction. Verify that P ′Q′ will not meet the circle if b2 < ac, while if b2 = ac it will be a tangent. Verify also that if b2 = ac the circle in the second construction will touch BC.

4. Prove that

√ pq

=

√ p

×

√ q,

p2q

=

√ p q.

*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.
„I have taken this construction from Klein’s Le¸cons sur certaines questions de g´eom´etrie ´el´ementaire (French translation by J. Griess, Paris, 1896).

[I : 14]

REAL VARIABLES

24

14. Some theorems concerning quadratic surds. Two pure quadratic surds are said to be similar if they can be expressed as rational multiples of the same surd, and otherwise to be dissimilar. Thus

√√ 8 = 2 2,

√

25 2

=

5 2

2,

√ and so 8,

25 2

are similar surds.

On the other hand, if M

and N

are

integers√which ha√ve no common factor, and neither of which is a perfect

square, M and N are dissimilar surds. For suppose, if possible,

√ M

=

p q

t u

,

√ N

=

r s

t u

,

where all t√he letters denote integers. Then M N is evidently rational, and therefore (Ex. ii. 3) integral.
Thus M N = P 2, where P is an integer. Let a, b, c, . . . be the prime

factors of P , so that

M N = a2αb2βc2γ . . . ,

where α, β, γ, . . . are positive integers. Then M N is divisible by a2α, and therefore either (1) M is divisible by a2α, or (2) N is divisible by a2α, 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 a2α, b2β, c2γ, . . . , so that M must be divisible by some
of these factors and N by the remainder. Thus

M = P12, N = P22,

where P12 denotes the product of some of the factors a2α, b2β, c2γ, . . . and P22 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

√

√

A + B = C + D,

then either (i) A = C, B = D or (ii) B and D are both squares of rational numbers.

[I : 14]

REAL VARIABLES

25

For B − D is rational, and so is √√ B − D = C − A.

If B is not equal to D (in which case it is obvious that A is also equal

to C), it follows that

√√

√√

B + D = (B − D)/( B − D)

√

√

is also rational. Hence √B and D√are rational. √

√

√ Coro√llary. If A + B = C + D, then A − B = C − D (unless

B and D are both rational).

√

√

Examples VIII. 1. Prove ab initio that 2 and 3 are not similar

surds. 2.

Prove

that

√ a

and

1/a, where a is rational, are similar surds (unless

both are rational).

√√

√

√

are

3. If a and b are rational, then√ rational. The same is true of a

a+√ b − b,

cannot unless

be rational a = b.

unless

a and

b

4. If

√√ √√

A + B = C + D,

√√

t√hen e√ither (a) A = C and B = D, or (b) A = D and B = C, or (c) A, B,

C, D are all rational or all similar surds. [Square the given equation and

apply the theorem ab√ove.]

√

√

5. Neither (a + b)3 nor (a − b)3 can be rational unless b is rational.

m

6. Prove that if x = p + √q, where p and is any integer, can be expressed in the form

q P

a+reQr√atqio, nwahl,ertehePn

xm, and

where Q are

rational. For example,

(p + √q)2 = p2 + q + 2p√q, (p + √q)3 = p3 + 3pq + (3p2 + q)√q.

Deduce that any polynomial in x with rational coefficients (i.e. any expression

of the form

a0xn + a1xn−1 + · · · + an,

where a0, . . . , an are rational numbers) can be expressed in the form P + Q√q.

[I : 15]

REAL VARIABLES

26

√ 7. If a + b, where b is not a perfect squ√are, is the root of an algebraical equation with rational coefficients, then a − b is another root of the same

equation. 8. Express 1/(p
and denominator by

+ p

−√q√) qin.]

the

form

prescribed

in

Ex.

6.

[Multiply

numerator

9. Deduce from Exs. 6 and 8 that any expression of the form G(x)/H(x),

where G(x) expressed in

and the

H(x) are form P +

Qp√olqy,nwomheiarelsPinanxdwQitharreartaiotnioanlacl.oefficients,

can

be

√ x

1+0.√yIf,

p, q, and where

p2

−q

are

positive,

we

can

express

p + √q in the form

x

=

1 2

{p

+

p2 − q},

y

=

1 2

{p

−

p2 − q}.

11. Determine where p and q are

the conditions rational, in the

tfhoramt it√mx a+y√bye,pwohsseirbelextaonedxpyraerses

p + √q, rational.

12. If a2 − b is positive, the necessary and sufficient conditions that

√

√

a+ b+ a− b

should

be

rational

are

that

a2

−b

and

1 2

{a

+

√ a2 − b}

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 Λ of § 2 is composed of
points corresponding to all the numbers of the arithmetical continuum, and of no others.* The points of the 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.
*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.

[I : 15]

REAL VARIABLES

27

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

z=

3

√ 4 + 15 +

3

√ 4 − 15.

Our argument for supposing that the expression for z has a mea√ning might be

as y2

follows. We first show, = 15, and we can then,

as as

in in

§ 12, § 10,

that there is a number√y = define the numbers 4 + 15,

15√such 4 − 15.

that Now

consider the equation in z1,

z13

=

4

+

√ 15.

The right-hand side of this equation is not rational: but exactly the same rea-

soning which leads us to suppose that there is a real number x such that x3 = 2

(or any other rational
number z1 such that z13 we can define z2 = 3 4

numb√er) also leads = √4+ 15. We thus

us to define

tzh1e=co3nc4lu+si√on15t,haantdthsiemreilaisrlya

− 15; and then, as in § 10, we define z = z1 + z2.

Now it is easy to verify that

z3 = 3z + 8.

And we might have given a direct proof of the existence of a unique number z such that z3 = 3z + 8. It is easy to see that there cannot be two such numbers. For if z13 = 3z1 + 8 and z23 = 3z2 + 8, we find on subtracting and dividing by z1 − z2 that z12 + z1z2 + z22 = 3. But if z1 and z2 are positive z13 > 8, z23 > 8 and therefore z1 > 2, z2 > 2, z12 + z1z2 + z22 > 12, and so the equation just found is impossible. And it is easy to see that neither z1 nor z2 can be negative. For if z1 is negative and equal to −ζ, ζ is positive and ζ3 − 3ζ + 8 = 0, or 3 − ζ2 = 8/ζ. Hence 3 − ζ2 > 0, and so ζ < 2. But then 8/ζ > 4, and so 8/ζ cannot be equal to 3 − ζ2, which is less than 3.
Hence there is at most one z such that z3 = 3z + 8. And it cannot be
rational. For any rational root of this equation must be integral and a factor
of 8 (Ex. ii. 3), and it is easy to verify that no one of 1, 2, 4, 8 is a root. Thus z3 = 3z + 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

[I : 15]

REAL VARIABLES

28

classes L, R according as x3 < 3x + 8 or x3 > 3x + 8. It is easy to see that if x3 > 3x + 8 and y is any number greater than x, then also y3 > 3y + 8. For suppose if possible y3 ≦ 3y + 8. Then since x3 > 3x + 8 we obtain on subtracting y3 − x3 < 3(y − x), or y2 + xy + x2 < 3, which is impossible; for y is positive and x > 2 (since x3 > 8). Similarly we can show that if x3 < 3x + 8 and y < x then also y3 < 3y + 8.
Finally, it is evident that the classes L and R both exist; and they form a section of the positive rational numbers or positive real number z which satisfies the equation z3 = 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 x3 = 3x + 8 could be applied (though the application would be a little more difficult) to the equation
x5 = x + 16,
and would lead us to the conclusion that a unique positive real number exists which satisfies this equation. In this case, however, 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 x5 = x+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 A0A1, 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
*A proof will be found in Ch. VII.

[I : 16]

REAL VARIABLES

29

is usually denoted by π. And it has been shown* (though the proof is unfortunately long and difficult) that this number π is not the root of any algebraical equation with integral coefficients, such, for example, as
π2 = n, π3 = n, π5 = π + n,
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 π 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 re-

garded from two points of view. We may think of them as an aggregate,

the ‘arithmetical continuum’ defined in the preceding section, or individ-

puaalrltyi.cuAlanrdspwehciefniedwenuthminbkero(fsuthchemasin1d,i−vid12 ,ua√ll2y,,

we may or π) or

think either of a 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’, ‘x is the mea-

sure of a length’, ‘x 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 ‘x’ 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

*See Hobson’s Trigonometry (3rd edition), pp. 305 et seq., or the same writer’s Squaring the Circle (Cambridge, 1913).

[I : 17]

REAL VARIABLES

30

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 x 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 think of other examples of variables with different fields of variation. 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 considered ‘sec-

tions’ of the rational numbers, i.e. modes of division of the rational num-

bers (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 mutu-

ally 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 ≦ 2 and Q be x > 2. It is important to observe

that a pair of properties which suffice to define a section of the rational numbers

*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 Stetigkeit 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 notion of an irrational number contained in §§ 6–12.

[I : 17]

REAL VARIABLES

31

may not suffic√e to define on√e of the real numbers. This is so, for example, with the pair ‘x < 2’ and ‘x > 2’ or (if we confine ourselves to positive numbers)

with ‘x2 < 2’ and ‘x2 > 2’. Every rational number possesses on√e or other of the properties, but not every real number, since in either case 2 escapes

classification.

There are now two possibilities.* Either L has a greatest member l, or

R has a least member r. Both of these events cannot occur. For if L had

a

greatest

member

l,

and

R

a

least

member

r,

the

number

1 2

(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 L1 and R1 denote the classes formed from L and R by taking only the rational members of L and R. Then the classes L1 and R1 form a section of the rational numbers. There are now two cases to distinguish.

It may happen that L1 has a greatest member α. In this case α must be also the greatest member of L. For if not, we could find a greater, say β.

There are rational numbers lying between α and β, and these, being less

than β, belong to L, and therefore to L1; and this is plainly a contradiction. Hence α is the greatest member of L.

On the other hand it may happen that L1 has no greatest member. In this case the section of the rational numbers formed by L1 and R1 is a real number α. This number α must belong to L or to R. If it belongs to L

we can show, 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

*There were three in § 6. „This was not the case in § 6.

[I : 18]

REAL VARIABLES

32

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.
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 α, 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 α 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 [β, γ], that is to say of all numbers x such that β ≦ x ≦ γ. 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 β and to R all numbers greater than γ. It is clear that the conclusion stated in Dedekind’s Theorem still holds if we substitute ‘the real numbers of the interval [β, γ]’ for ‘the real numbers’, and that the number α in this case satisfies the inequalities β ≦ α ≦ γ.
18. Points of accumulation. A system of real numbers, or of the points on a straight line corresponding to them, defined 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
*The reader will hardly require to be reminded that this course is adopted solely for reasons of linguistic convenience.

[I : 18]

REAL VARIABLES

33

then that we are given a set of points, which we will denote by S. Take any point ξ, which may or may not belong to S. Then there are two possibilities. Either (i) it is possible to choose a positive number δ so that the interval [ξ −δ, ξ +δ] does not contain any point of S, other than ξ itself,* or (ii) this is not possible.

Suppose, for example, that S consists of the points corresponding to all the

positive integers. If ξ is itself a positive integer, we can take δ to be any number

less than 1, and (i) will be true; or, if ξ is halfway between two positive integers,

we

can

take

δ

to

be

any

number

less

than

1 2

.

On

the

other

hand,

if

S

consists

of

all the rational points, then, whatever the value of ξ, (ii) is true; for any interval

whatever contains an infinity of rational points.

Let us suppose that (ii) is true. Then any interval [ξ − δ, ξ + δ], however small its length, contains at least one point ξ1 which belongs to S and does not coincide with ξ; and this whether ξ itself be a member of S or not. In this case we shall say that ξ is a point of accumulation of S. It is easy to see that the interval [ξ − δ, ξ + δ] must contain, not merely one, but infinitely many points of S. For, when we have determined ξ1, we can take an interval [ξ − δ1, ξ + δ1] surrounding ξ but not reaching as far as ξ1. But this interval also must contain a point, say ξ2, which is a member of S and does not coincide with ξ. Obviously we may repeat this argument, with ξ2 in the place of ξ1; and so on indefinitely. In this way we can determine as many points
ξ1, ξ2, ξ3, . . .

as we please, all belonging to S, and all lying inside the interval [ξ −δ, ξ +δ]. A point of accumulation of S 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 posi-
tive 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.
*This clause is of course unnecessary if ξ does not itself belong to S.

[I : 19]

REAL VARIABLES

34

3.

If

S

consists

of

the

points

1,

1 2

,

1 3

,

.

.

.

,

there

is

one

point

of

accumulation,

viz. the origin.

4. If S consists of all the positive rational points, the points of accumulation

are the origin and all positive points of the line.

19. Weierstrass’s Theorem. The general theory of sets 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 fundamental 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 [α, β], then at least one point of the interval is a point of accumulation of S.

We divide the points of the line Λ into two classes in the following manner. The point P belongs to L if there are an 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 α belongs to L and β to R, condition (ii) is satisfied also.
Hence there is a point ξ such that, however small be δ, ξ − δ belongs to L and ξ + δ to R, so that the interval [ξ − δ, ξ + δ] contains an infinity of points of S. Hence ξ is a point of accumulation of S.

This point may of course coincide with α or β, as for instance when α = 0,

β

=

1,

and

S

consists

of

the

points

1,

1 2

,

1 3

,

.

.

.

.

In

this

case

0

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 x,

y, z; (2) for all values of x, y, z subject to αx + βy + γz = 0; (3) for all values

of x, y, z subject to both αx + βy + γz = 0 and Ax + By + Cz = 0?

2. Any positive rational number can be expressed in one and only one way

in the form

a1

+

a2 1·2

+

1

a3 ·2·

3

+

·

·

·

+

1

·

2

ak ·3

.

.

.

k

,

[I : 19]

REAL VARIABLES

35

where a1, a2, . . . , ak are integers, and

0 ≦ a1, 0 ≦ a2 < 2, 0 ≦ a3 < 3, . . . 0 < ak < k.

3. Any positive rational number can be expressed in one and one way only as a simple continued fraction

1

a1 +

, 1

a2 +

1

a3 +

1

···+

an

where a1, a2, . . . 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. 45–49.]
4. Find the rational roots (if any) of 9x3 − 6x2 + 15x − 10 = 0.

5. A line AB is divided at C in aurea sectione (Euc. ii. 11)—i.e. so that AB · AC = BC2. Show that the ratio AC/AB is irrational.

[A direct geometrical proof will be found in Bromwich’s Infinite Series, § 143,

p. 363.]

6. A is irrational. rational, be rational?

In

what

circumstances

can

aA cA

+ +

b d

,

where

a,

b,

c,

d

are

7. Some elementary inequalities. In what follows a1, a2, . . . denote

positive numbers and aq1 − aq2 have

(including zero) and p, the same sign, we have

q, . (ap1

.. −

positive ap2)(aq1 −

integers. Since aq2) ≧ 0, or

ap1

−

ap2

ap1+q + ap2+q ≧ ap1aq2 + aq1ap2,

(1)

an inequality which may also be written in the form

ap1+q + ap2+q 2

≧

ap1 + ap2 2

aq1 + aq2 2

.

(2)

By repeated application of this formula we obtain

a1p+q+r+...

+ a2p+q+r+... 2

≧

ap1 + ap2 2

aq1 + aq2 2

ar1 + ar2 2

...,

(3)

[I : 19]

REAL VARIABLES

36

and in particular

ap1 + ap2 2

≧

a1 + a2 2

p
.

(4)

When p = q = 1 in (1), or p = 2 in (4), the inequalities are merely different forms

of the inequality a21 + a22 ≧ 2a1a2, 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

1 2

n(n

−

1)

inequalities of the type (1) which can be formed with n numbers a1, a2, . . . , an,

and add the results, we obtain the inequality

n ap+q ≧ ap aq,

(5)

or

ap+q /n ≧ {( ap) /n} {( aq) /n} .

(6)

Hence we can deduce an obvious extension of (3) which the reader may formulate for himself, and in particular the inequality

( ap) /n ≧ {( a) /n}p .

(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 a1, a2, . . . , an is not less than their geometric mean. Suppose that ar and as 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 G be their geometric mean. We may suppose G > 0, as the truth of the proposition is obvious when G = 0. If now we replace ar and as by
a′r = G, a′s = aras/G,
we do not alter the value of the geometric mean; and, since
a′r + a′s − ar − as = (ar − G)(as − G)/G ≦ 0,
we certainly do not increase the arithmetic mean. It is clear that we may repeat this argument until we have replaced each of
a1, a2, . . . , an 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.

[I : 19]

REAL VARIABLES

37

10. Schwarz’s inequality. Suppose that a1, a2, . . . , an and b1, b2, . . . , bn are any two sets of numbers positive or negative. It is easy to verify the identity

( arbr)2 = a2r a2s − (arbs − asbr)2,

where r and s assume the values 1, 2, . . . , n. It follows that

( arbr)2 ≦ a2r b2r,

an inequality usually known as Schwarz’s (though due originally to Cauchy). 11. If a1, a2, . . . , an are all positive, and sn = a1 + a2 + · · · + an, then

(1

+

a1)(1

+

a2) .

.

.

(1

+

an)

≦

1

+

sn

+

s2n 2!

+

·

·

·

+

snn n!

.

(Math. Trip. 1909.)
12. If a1, a2, . . . , an and b1, b2, . . . , bn are two sets of positive numbers, arranged in descending order of magnitude, then

(a1 + a2 + · · · + an)(b1 + b2 + · · · + bn) ≦ n(a1b1 + a2b2 + · · · + anbn).

13. If a, b, c, . . . k and A, B, C, . . . K are two sets of numbers, and all of the first set are positive, then

aA + bB + · · · + kK a+b+···+k

lies between the algebraically least and greatest of A, B, . . . , K. 14. If √p, √q are dissimilar surds, and a + b√p + c√q + d√pq = 0, where

a,

b, c, d are [Express

√raptiionntahl,etfhoernmaM=+0,Nb√=q,0,wch=ere0,Md

= 0. and N

are

rational,

and

apply

the theorem of § 14.] √ √ √ 15. Show that if a 2 + b 3 + c 5 = 0, where a, b, c are rational numbers,

then a = 0, b = 0, c = 0.

a

16. finite

Any polynomial number of terms

in of

√p the

afnordm√Aq,(√wipt)hmr(a√tiqo)nna,lwchoeerffeicmienatnsd(in.ea. raenyinstuegmerosf,

and A rational), can be expressed in the form

a

+

√ bp

+

√ cq

+

√ d pq,

[I : 19]

REAL VARIABLES

38

where a, b, c, d 17. Express

aare+rabt√iopn+al.c√q d + e√p + f √q

,

where

a,

b,

etc.

are

rational,

in

the

form

A

+

√ Bp

+

√ Cq

+

√ D pq,

where A, B, C, D are rational.

[Evidently

a + b√p + c√q (a + b√p + c√q)(d + e√p − f √q)

d + e√p + f √q =

(d + e√p)2 − f 2q

α + β√p + γ√q + δ√pq

=

ϵ + ζ√p

,

where α, β, etc. are rational numbers which can easily be found. The required

reduction may now denominator by ϵ −

ζb√ep.eaFsoilry ecxoammpplleet,epdrobvye

multiplication that

of

numerator

and

1

+

√1 2

+

√ 3

=

1 2

+

1 4

√ 2

−

1 4

√ 6.]

18. If a, b, x, y are rational numbers such that (ay − bx)2 + 4(a − x)(b − y) = 0,

then either (i) x = a, y = b or (ii) 1 − ab and 1 − xy are squares of rational

numbers.

(Math. Trip. 1903.)

19. If all the values of x and y given by

ax2 + 2hxy + by2 = 1, a′x2 + 2h′xy + b′y2 = 1

(where a, h, b, a′, h′, b′ are rational) are rational, then

(h − h′)2 − (a − a′)(b − b′), (ab′ − a′b)2 + 4(ah′ − a′h)(bh′ − b′h)

are both squares of √rational √numbers.

√ (M√ath. Trip. 1899.)

20. Show that √2 and√ 3 are cubic functions of 2 + 3, with√ratio√nal

coefficients, and that 2 − 6 + 3 is the ratio of two linear functions of 2 + 3.

(Math. Trip. 1905.)

[I : 19]

REAL VARIABLES

39

21. The expression

a + 2m a − m2 + a − 2m a − m2

is

equal

to

2m

if

2m2

>

a

>

m2,

and

√ t√o 2 a

−

m2

if

a

>

2m2.

22. Show that any polynomial in 3 2, with rational coefficients, can be ex-

pressed in the form

a + b√3 2 + c√3 4,

where a, b, c are rational. More generally, if p is any rational number, any polynomial in

√ mp

with

rational coefficients can be expressed in the form

a0 + a1α + a2α2 + · · · + am−1αm−1,
where a0, a1, . . . are rational and α = m √p. For any such polynomial is of the form
b0 + b1α + b2α2 + · · · + bkαk,

where the b’s are rational. If k ≦ m − 1, this is already of the form required. If k > m−1, let αr be any power of α higher than the (m−1)th. Then r = λm+s,

where λ is an integer and 0 ≦ s ≦ m − 1; and αr = αλm+s = pλαs. Hence we

can get 23.

Erixdporfesasll(p√3o2w−ers1)o5f

α hig√her and ( 3 2

than t√he − 1)/( 3 2

(m − + 1)

1)th. in the

form

a

+

√ b32

+

√ c 3 4,

where a, b, c a√re rati√onal. [Multiply numerator and denominator of the second expression by 3 4 − 3 2 + 1.]

24. If

√√ a + b 3 2 + c 3 4 = 0,

where a, b, c√are [Let y = 3 2.

rational, Then y3

then a = = 2 and

0,

b

=

0,

c

=

0.

cy2 + by + a = 0.

Hence 2cy2 + 2by + ay3 = 0 or

ay2 + 2cy + 2b = 0.

Multiplying these two quadratic equations by a and c and subtracting, we obtain (ab−2c2)y+a2−2bc = 0, or y = −(a2−2bc)/(ab−2c2), a rational number, which is impossible. The only alternative is that ab − 2c2 = 0, a2 − 2bc = 0.

[I : 19]

REAL VARIABLES

40

Hence ab = 2c2, a4 = 4b2c2. If neither a nor b is zero, we can divide the s√econd equation by the first, which gives a3 = 2b3: and this is impossible, since 3 2 cannot be equal to the rational number a/b. Hence ab = 0, c = 0, and it follows from the original e√quatio√n that a, b,√and c √are all zero.
As a corollary, if a + b 3 2 + c 3 4 = d + e 3 2 + f 3 4, then a = d, b = e, c = f .
It may be proved, more generally, that if

a0 + a1p1/m + · · · + am−1p(m−1)/m = 0,

p not being a perfect mth power, then a0 = a1 = · · · = am−1 = 0; but the proof

is less simple.] √

√

25. If A + 3 B = C + 3 D, then either A = C, B = D, or B and D are both

cubes of ra√tional n√umber√s. 26. If 3 A + 3 B + 3 C = 0, t√hen e√ither√one of A, B, C is zero, and the
other two e√qual and opposite, or 3 A, 3 B, 3 C are rational multiples of the same surd 3 X.

27. Find rational numbers α, β such that

3

√ 7+5 2

=

√ α + β 2.

28. If (a − b3)b > 0, then

3

a+

9b3 + a 3b

a − b3 3b

+

3

a−

9b3 + a 3b

a − b3 3b

is rational. [Each of the numbers under a cube root is of the form

α+β

a − b3 3 3b

where α and β are rational.] 29. If α = √n p, any polynomial in α is the root of an equation of degree n,
with rational coefficients. [We can express the polynomial (x say) in the form
x = l1 + m1α + · · · + r1α(n−1),

[I : 19]

REAL VARIABLES

41

where l1, m1, . . . are rational, as in Ex. 22. Similarly

x2 = l2 + m2a + . . . + r2a(n−1), .............................. xn = ln + mna + . . . + rna(n−1).

Hence

L1x + L2x2 + · · · + Lnxn = ∆,

where ∆ is the determinant

l1 m1 . . . r1 l2 m2 . . . r2 ................
ln mn . . . rn

and L1, L2, . . . the minors 30. Apply this process

of to

lx1,=l2p, .+. .√.] q,

and

deduce

the

theorem

of

§

14.

31. Show that y = a + bp1/3 + cp2/3 satisfies the equation

y3 − 3ay2 + 3y(a2 − bcp) − a3 − b3p − c3p2 + 3abcp = 0.

32. A√lgebraical numbers. We have seen that some irrational numbers (such as 2) are roots of equations of the type
a0xn + a1xn−1 + · · · + an = 0,
where a0, a1, . . . , an are integers. Such irrational numbers are called algebraical numbers: all other irrational numbers, such as π (§ 15), are called transcendental numbers. Show that if x is an algebraical number, then so are kx, where k is any rational number, and xm/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
a0xm + a1xm−1 + . . . + am = 0, b0yn + b1yn−1 + . . . + bn = 0,

[I : 19]

REAL VARIABLES

42

where the a’s and b’s are integers. Write x + y = z, y = z − x in the second, and eliminate x. We thus get an equation of similar form

c0zp + c1zp−1 + · · · + cp = 0,

satisfied by z. Similarly for the other cases.]

34. If

a0xn + a1xn−1 + · · · + an = 0,

where a0, a1, . . . , an are any algebraical numbers, then x is an algebraical number. [We have n + 1 equations of the type

a0,ramr r + a1,ramr r−1 + · · · + amr,r = 0 (r = 0, 1, . . . , n),

in which the coefficients a0,r, a1,r, . . . are integers. Eliminate a0, a1, . . . , an

between these 35. Apply

and this

the original equation for x.] process to the equation x2 −

√ 2x 2

+

√ 3

=

0.

[The result is x8 − 16x6 + 58x4 − 48x2 + 9 = 0.]

36. Find equations, with rational coefficients, satisfied by

√√ 1 + 2 + 3,

√√ √3 + √2 ,

3− 2

√√

√√ √√

3 + 2 + 3 − 2, 3 2 + 3 3.

37. If x3 = x + 1, then x3n = anx + bn + cn/x, where

an+1 = an + bn, bn+1 = an + bn + cn, cn+1 = an + cn.

38. If x6 + x5 − 2x4 − x3 + x2 + 1 = 0 and y = x4 − x2 + x − 1, then y satisfies

a quadratic equation with rational coefficients.

(Math. Trip. 1903.)

[It will be found that y2 + y + 1 = 0.]

CHAPTER II

FUNCTIONS OF REAL VARIABLES

20. The idea of a function. Suppose that x and y are two contin-

uous real variables, which we may suppose to be represented geometrically

by distances A0P = x, B0Q = y measured from fixed points A0, B0 along two straight lines Λ, 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

1 2

x,

or

x2 + 1.

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 x, 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 mathematics. 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 of y;

(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.

It is indeed the case that these particular characteristics are possessed

by many of the most important functions. But the consideration 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

43

[II : 20]

FUNCTIONS OF REAL VARIABLES

44

values of y.

Examples X.

1.

Let

y

=

x

or

2x

or

1 2

x

or

x2 + 1.

Nothing

further

need

be said at present about cases such as these.

2. Let y = 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 of x.

the functional relation makes the same value The same would be true were y equal to 1 or

of −

y
1 2

co√rrespond to all values or 2 instead of 0. Such

a function of x is called a constant.

3. Let y2 corresponding

= to

x. Then if each value

xofisx,povsizit.iv±e√thxi.s

equation defines two values of y If x = 0, y = 0. 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 p0 per unit of area on the piston, which balances the weight W , so that

W = Ap0.

Let v0 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 pro-
vided the gas is not compressed too much. When v is decreased and p increased
*I borrow this instructive example from Prof. H. S. Carslaw’s Introduction to the Calculus.

[II : 20]

FUNCTIONS OF REAL VARIABLES

45

beyond a certain point, the relation between them is no longer expressed with tolerable exactness by the equation (i). It is known that a 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

p

+

α v2

(v − β) = γ,

(ii)

where α, β, γ 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 form 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 contemplating 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 defining p as a function of v. It is an example of a function which for some values of v is defined by one formula and for other values of v is defined 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 defined as a function of v for negative values of v; a ‘negative volume’ means nothing, and so negative values of v do not present themselves for consideration at all.

5. Suppose that a perfectly elastic ball is dropped (without rotation) from

a

height

1 2

gτ

2

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

=

1 2

gt2

if

0

≦

t

≦

τ,

h

=

1 2

g(2τ

−

t)2

if

τ

≦

t

≦

3τ ,

and generally

h

=

1 2

g(2nτ

−

t)2

if (2n − 1)τ ≦ t ≦ (2n + 1)τ , h being the depth of the ball, at time t, below its

original position. Obviously h is a function of t which is only defined for positive

values of t.

6. Suppose that y is defined as being the largest prime factor of x. This

is of

an instance of x, viz. integral

avadlueefisn.it‘iTohnewlahrigchestonplryimaeppfalicetsortoofa13p1 aorrtiocfu√lar2

class or of

of π’

values means

nothing, and so our defining relation fails to define for such values of x as these.

[II : 21]

FUNCTIONS OF REAL VARIABLES

46

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. Th√us y = 7 if x = −11/7: but y is not defined for x = 2, ‘the denominator of 2’ being a meaningless form of words.
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. Suppose 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 = f (x)
(or F (x), ϕ(x), ψ(x), . . . , as the case may be). The nature of particular functions may, in very many cases, be illus-
trated and made easily intelligible as follows. Draw two lines OX, OY at right angles to one another and produced indefinitely in both directions. We can represent values of x and y by distances measured from O along the lines OX, OY respectively, regard being paid, of course, to sign, and the positive directions of measurement 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 OAP B. 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 ′′.

[II : 21]

FUNCTIONS OF REAL VARIABLES

47

Y

B′

P′

B

P

b

O

a

AX

B′′

P ′′

Fig. 6.

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 OY the

axis of x and the axis of y, or together the axes of coordinates, and O 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 corresponding point P (or points P , P ′, 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 = 0,

(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

*If B = 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.

[II : 22]

FUNCTIONS OF REAL VARIABLES

48

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 locus, 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 +C = 0 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 represents 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 geometrical loci defined by equations. An equation of the form
(x − α)2 + (y − β)2 = ρ2,
or x2 + y2 + 2Gx + 2F y + C = 0,
where G2 + F 2 − C > 0, represents a circle. The equation
Ax2 + 2Hxy + By2 + 2Gx + 2F y + C = 0
(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, M P = y. If OP = r and M OP = θ, θ being an angle between 0 and 2π (measured in the positive direction), it is evident that
x = r cos θ, y = r sin θ,
r = x2 + y2, cos θ : sin θ : 1 :: x : y : r,

[II : 23]

FUNCTIONS OF REAL VARIABLES

49

and that the position of P is equally well determined by a knowledge of
r and θ. We call r and θ the polar coordinates of P . The former, it should be observed, is essentially positive.*

N

P

r

y

θ

O

x

M

Fig. 7.

If P moves on a locus there will be some relation between r and θ, say r = f (θ) or θ = 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
r cos(θ − α) = p,
where p and α are constants. The equation r = 2a cos θ represents a circle passing through the origin; and the general equation of a circle is of the form
r2 + c2 − 2rc cos(θ − α) = A2,
where A, c, and α 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, π)—correspond to the same point. The distinction between the two systems may be illustrated by means of the equation l/r = 1 − e cos θ, where l > 0, e > 1. According to our definitions r must be positive and therefore cos θ < 1/e: the equation represents one branch only of a hyperbola, the other having the equation −l/r = 1 − e cos θ. With the system of coordinates which admits negative values of r, the equation represents the whole hyperbola.

[II : 23]

FUNCTIONS OF REAL VARIABLES

50

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

a0xm + a1xm−1 + · · · + am,

where a0, a1, . . . , am are constants. The simplest polynomials are the simple powers y = x, x2, x3, . . . , xm, . . . . The graph of the function xm 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

x

=

1 2

,

2,

3,

−

1 2

,

−2,

3

give

y

=

1 4

,

4,

9,

1 4

,

4, 9.

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 corresponding 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, a curve 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 = x2 is in fact such a curve. This cannot be proved by merely constructing any number of isolated points on the curve, although the more such points we construct the more probable it will appear.

[II : 23]

FUNCTIONS OF REAL VARIABLES

51

P1 y = x2

(−1, 1)

(1, 1)

P0

N

(0, 0)

Fig. 8.

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 = x2 is everywhere convex to the axis of x. Let P0, P1 (Fig. 8) be the points (x0, x20), (x1, x21). Then the coordinates of a point on the chord P0P1 are x = λx0 + µx1, y = λx20 + µx21, where λ and µ are positive numbers whose sum is 1. And
y − x2 = (λ + µ)(λx20 + µx21) − (λx0 + µx1)2 = λµ(x1 − x0)2 ≧ 0,
so that the chord lies entirely above the curve.
The curve y = x4 is similar to y = x2 in general appearance, but flatter near O, and steeper beyond the points A, A′ (Fig. 9), and y = xm, 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 indistinguishable from the thick line in the figure.
The reader should next consider the curves given by y = xm, when m is odd. The fundamental difference between the two cases is that whereas when m is even (−x)m = xm, so that the curve is symmetrical about OY , when m is odd (−x)m = −xm, so that y is negative when x is negative.

[II : 23]

FUNCTIONS OF REAL VARIABLES
y = x4 y = x2

52
y = x3 y=x
A

A′

A

M

O

N

Fig. 9.

O A′
Fig. 10.

Fig. 10 shows the curves y = x, y = x3, and the form to which y = xm 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 = xm we can at once derive that of Cxm, 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 (x), 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 y = 7x4, y = 3x5, y = x10.
[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,
*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.

[II : 24]

FUNCTIONS OF REAL VARIABLES

53

as x gets larger and larger, and will see that, in such a polynomial as
x10 + 3x5 + 7x4
(or even x10 + 30x5 + 700x4), it is the first term which is of really preponderant importance when x is fairly large. Thus even when x = 4, x10 > 1,000,000, while 30x5 < 35,000 and 700x4 < 180,000; while if x = 10 the preponderance of the first term is still more marked.]
2. Compare the relative magnitudes of x12, 1,000,000x6, 1,000,000,000,000x when x = 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 ax2 + 2bx + c. [Here y − {(ac − b2)/a} = a{x + (b/a)}2. If we take new axes parallel to the old and passing through the point x = −b/a, y = (ac − b2)/a, the new equation is y′ = ax′2. The curve is a parabola.] 4. Trace the curves y = x3 − 3x + 1, y = x2(x − 1), y = x(x − 1)2.

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

R(x)

=

P (x) Q(x)

.

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 xp + axp−1 + bxp−2 + · · · + 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 x/x, which is a rational function. On removing the common factor x we obtain 1/1 = 1. But the

[II : 24]

FUNCTIONS OF REAL VARIABLES

54

original function is not always equal to 1: it is equal to 1 only so long as x ̸= 0. If x = 0 it takes the form 0/0, which is meaningless. Thus the function x/x is equal to 1 if x ̸= 0 and is undefined when x = 0. It therefore differs from the function 1, which is always equal to 1.
(3) Such a function as

x

1 +

1

+

x

1 −

1

1 x

+

x

1 −

2

may be reduced, by the ordinary rules of algebra, to the form

x2(x − 2) (x − 1)2(x + 1) ,

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 = 0 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 denominator vanishes. Thus (x2 − 7)/(x2 − 3x + 2) is not defined when x = 1 or 2.
(5) Generally we agree, in dealing with expressions such as those considered 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)/Q(y), where P and Q are polynomials, we substitute y = P1(x)/Q1(x), we obtain on simplification an equation of the form z = P2(x)/Q2(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

[II : 25]

FUNCTIONS OF REAL VARIABLES

55

rational has reference solely to the way in which the variable x appears in the

function. Thus

x2

+√x

+

√ 3

x32−π

is a rational function. The use of the word rational arises as follows. The rational function
P (x)/Q(x) may be generated from x by a finite number of operations upon x, including only multiplication of x by itself or a constant, addition 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 procedure is very much like that by which all rational numbers can be obtained from unity, a procedure exemplified in the equation

5 3

=

1

+

1 1

+ +

1 1

+ +

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

x x2 +

1

+

x2

+

2x + 7 √ 11x − 3 2
9x + 1

17

+

2 x3

,

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 of y = 1/x, y = 1/x2, y = 1/x3, . . . .
[The figures show the graphs of the first two curves. It should be observed that since 1/0, 1/02, . . . are meaningless expressions, these functions are not
defined for x = 0.]

[II : 26]

FUNCTIONS OF REAL VARIABLES

56

(1, 1)

(−1, −1)

y = 1/x

y = 1/x2

Fig. 11.

Fig. 12.

2. Trace y = x + (1/x), x − (1/x), x2 + (1/x2), x2 − (1/x2) and ax + (b/x) taking various values, positive and negative, for a and b.

3. Trace

y

=

x x

+ −

1 1

,

x+1 2

1

x2 + 1

x − 1 , (x − 1)2 , x2 − 1 .

4. Trace y = 1/(x − a)(x − b), 1/(x − a)(x − b)(x − c), where a < b < c. 5. Sketch the general form assumed by the curves y = 1/xm as m becomes larger and larger, considering separately the cases in which m is odd or even.

26. C. Explicit Algebraical Functions. The next important class of functions is that of explicit algebraical functions. These are functions which can be generated from x by a finite number of operations such as those used in generating rational functions, together with a finite number of operations of root extraction. Thus

√

√

√1 + x − √3 1 − x ,

√ x+

√ x + x,

1+x+ 31−x

√ x2 +√x + 3

2 3

x32−π

√ are explicit algebraical functions, and so is xm/n (i.e. n xm), where m and n

are any integers.

[II : 27]

FUNCTIONS OF REAL VARIABLES

57

It should be noticed that√there is an ambiguity of notation involved

in suc√h an equation as y = x. We have, up to the present, regarded

(e.g.) 2 as de√noting the positive square root of 2, and it would be natural

to denote by x, where x √is any positive number, the positive square

rhooowteovferxo, fitnenwmhiocrheccaosnevyen=ientxtowroeugladrdbe√ax

one-valued function of x. It is as standing for the two-valued

function whose two values are the positive and negative square roots of x.

T√he reader will observe that, when this course is adopted, the function x differs fundamentally from rational functions in two respects. In

the first place a rational function is always def√ined for all values of x with a certain number of isolated exceptions. But x 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 3 x, on the other hand, is one-valued and defined for all

values of x.

Examples XIII. 1. (x − a)(b − x), 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

(x − a)(x − b)(x − c) (a < b < c),

x(x2 − a2), 3 (x − a)2(b − x) (a < b),

√

√

√1 + x − √1 − x ,

√ x + x.

1+x+ 1−x

3. Trace the curves y2 = x, y3 = x, y2 = x3. 4. Draw the graphs of the functions

y = a2 − x2, y = b 1 − (x2/a2).

27. D. Implicit Algebraical Functions. It is easy to verify that

if

√

√

y = √1 + x − √3 1 − x ,

1+x+ 31−x

[II : 27]

FUNCTIONS OF REAL VARIABLES

58

then

1+y 1−y

6

=

(1 (1

+ −

x)3 x)2

;

or if

√

√

y = x + x + x,

then y4 − (4y2 + 4y + 1)x = 0.

Each of these equations may be expressed in the form

ym + R1ym−1 + · · · + Rm = 0,

(1)

where R1, R2, . . . , Rm 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. 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

y

=

x x

+ −

√x √
x

+ +

x

+

√x √

+

√ 31 √

+

x

.

x+ x− 31+x

Then we have the equations

y

=

x x

+ −

u u

+ +

v v

+ −

w w

,

u2 = x, v2 = x + u, w3 = 1 + x,

and we have only to eliminate u, 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 y = f (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 mth 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.

[II : 27]

FUNCTIONS OF REAL VARIABLES

59

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 = 1, y is a rational function.
2. If m = 2, the equation is y2 + R1y + R2 = 0, so that

y

=

1 2

{−R1

±

R12 − 4R2}.

This function is defined for all values of x for which R12 ≧ 4R2. It has two values if R12 > 4R2 and one if R12 = 4R2.
If m = 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.
3. Consider the functions defined by the equations

y2 − 2y − x2 = 0, y2 − 2y + x2 = 0, y4 − 2y2 + x2 = 0,

in each case obtaining y as an explicit function of x, and stating for what values of x it is defined.
4. Find algebraical equations, with coefficients rational in x, satisfied by each of the functions

√ x+

1/x,

√ 3x+

3

1/x,

√ x + x,

√ x + x + x.

5. Consider the equation y4 = x2. [Here y2 = ±x. If x is positive, y =

√x:

if negative,

y

=

√ −x.

Thus

the

function has two values for all values of x save x = 0.]

6. An algebraical function of an algebraical function of x is itself an alge-

braical function of x.

[II : 28]

FUNCTIONS OF REAL VARIABLES

60

[For we have

ym + R1(z)ym−1 + . . . + Rm(z) = 0,

where

zn + S1(x)zn−1 + . . . + Sn(x) = 0.

Eliminating z we find an equation of the form

yp + T1(x)yp−1 + . . . + Tp(x) = 0.
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
y5 − y − x = 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 whole 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 presuppose 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.

[II : 28]

FUNCTIONS OF REAL VARIABLES

61

Examples XV. 1. Draw the graphs of cos x√, sin x, and a cos x + b sin x.
[Since a cos x + b sin x = β√cos(x − α), whe√re β = a2 + b2, and α is an angle whose cosine and sine are a/ a2 + b2 and b/ a2 + b2, the graphs of these three functions are similar in character.]
2. Draw the graphs of cos2 x, sin2 x, a cos2 x + b sin2 x. 3. Suppose the graphs of f (x) and F (x) drawn. Then the graph of
f (x) cos2 x + F (x) sin2 x

is a wavy curve which oscillates between the curves y = f (x), y = F (x). Draw the graph when f (x) = x, F (x) = x2.

4. and −2

cSohso21w(pth+aqt)txh,etogurachpihnogfecaocshpixn+tucrons.qxSklieetschbetthweegernapthhowsehoefn2(pco−sq12)(/p(−p+q)qx)

is small.

(Math. Trip. 1908.)

5. Draw the graphs of x + sin x, (1/x) + sin x, x sin x, (sin x)/x.

6. Draw the graph of sin(1/x).

[If y = sin(1/x), then y = 0 when x = 1/mπ, where m is any integer.

Similarly

y

=

1

when

x

=

1/(2m

+

1 2

)π

and

y

=

−1

when

x

=

1/(2m

−

1 2

)π.

The curve is entirely comprised between the lines y = −1 and y = 1 (Fig. 13).

It oscillates up and down, the rapidity of the oscillations becoming greater and

greater as x approaches 0. For x = 0 the function is undefined. When x is large

y is small.* The negative half of the curve is similar in character to the positive

half.]

7. Draw the graph of x sin(1/x). [This curve is comprised between the lines y = −x and y = x just as the last curve is comprised between the lines y = −1 and y = 1 (Fig. 14).]

8. Draw the graphs of x2 sin(1/x), (1/x) sin(1/x), sin2(1/x), {x sin(1/x)}2, a cos2(1/x) + b sin2(1/x), sin x + sin(1/x), sin x sin(1/x).

9. Draw the graphs of cos x2, sin x2, a cos x2 + b sin x2.

10. Draw the graphs of arc cos x and arc sin x. [If y = arc cos x, x = cos y. This enables us to draw the graph of x, considered as a function of y, and the same curve shows y as a function of x. It is clear that y is only defined for −1 ≦ x ≦ 1, and is infinitely many-valued for these values of x. As the reader no doubt remembers, there is, when −1 < x < 1, a

*See Chs. IV and V for explanations as to the precise meaning of this phrase.

[II : 28]

FUNCTIONS OF REAL VARIABLES

62

Fig. 13.

Fig. 14.

value of y between 0 and π, say α, and the other values of y are given by the formula 2nπ ± α, where n is any integer, positive or negative.]
11. Draw the graphs of
tan x, cot x, sec x, cosec x, tan2 x, cot2 x, sec2 x, cosec2 x.
12. Draw the graphs of arc tan x, arc cot x, arc sec x, arc cosec x. Give formulae (as in Ex. 10) expressing all the values of each of these functions in terms of any particular value.
13. Draw the graphs of tan(1/x), cot(1/x), sec(1/x), cosec(1/x). 14. Show that cos x and sin x are not rational functions of x. [A function is said to be periodic, with period a, if f (x) = f (x + a) for all values of x for which f (x) is defined. Thus cos x and sin x have the period 2π. It is easy to see that no periodic function can be a rational function, unless it is a constant. For suppose that
f (x) = P (x)/Q(x),
where P and Q are polynomials, and that f (x) = f (x+a), each of these equations holding for all values of x. Let f (0) = k. Then the equation P (x) − kQ(x) = 0 is satisfied by an infinite number of values of x, viz. x = 0, a, 2a, etc., and therefore for all values of x. Thus f (x) = k for all values of x, i.e. f (x) is a constant.]
15. Show, more generally, that no function with a period can be an algebraical function of x.

[II : 29]

FUNCTIONS OF REAL VARIABLES

63

[Let the equation which defines the algebraical function be

ym + R1ym−1 + · · · + Rm = 0

(1)

where R1, . . . are rational functions of x. This may be put in the form P0ym + P1ym−1 + · · · + Pm = 0,

where P0, P1, . . . are polynomials in x. Arguing as above, we see that P0km + P1km−1 + · · · + Pm = 0

for all values of x. Hence y = k satisfies the equation (1) for all values of x, and one set of values of our algebraical function reduces to a constant.
Now divide (1) by y − k and repeat the argument. Our final conclusion is that our algebraical function has, for any value of x, the same set of values k, k′, . . . ; i.e. it is composed of a certain number of constants.]
16. The inverse sine and inverse cosine are not rational or algebraical functions. [This follows from the fact that, for any value of x between −1 and +1, arc sin x and arc cos x have infinitely many values.]

29. F. Other classes of transcendental functions. Next in importance to the trigonometrical functions come the exponential and logarithmic functions, which will be discussed in Chs. IX and X. But these functions are beyond our range at present. And most of the other classes of transcendental functions whose properties have been studied, such as the elliptic functions, Bessel’s and Legendre’s functions, Gamma-functions, and so forth, lie altogether beyond the scope of this book. There are however some elementary types of functions which, though of much less importance theoretically than the rational, algebraical, or trigonometrical functions, are particularly instructive as illustrations of the possible varieties of the functional relation.
Examples XVI. 1. Let y = [x], where [x] denotes the greatest integer
not greater than x. The graph is shown in Fig. 15a. The left-hand end points of the thick lines, but not the right-hand ones, belong to the graph.
2. y = x − [x]. (Fig. 15b.)

[II : 29]

FUNCTIONS OF REAL VARIABLES

64

01 2

01 2

Fig. 15a.

Fig. 15b.

3. y = x − [x]. (Fig. 15c.)

4. y = [x] + x − [x]. (Fig. 15d.)

5. y = (x − [x])2, [x] + (x − [x])2.

6.

y

=

√ [ x],

[x2],

√ x

−

√ [ x],

x2

−

[x2],

[1

−

x2].

01 2

01 2

Fig. 15c.

Fig. 15d.

7. Let y be defined as the largest prime factor of x (cf. Exs. x. 6). Then

[II : 29]

FUNCTIONS OF REAL VARIABLES

65

y is defined only for integral values of x. If
x = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, . . . ,
then
y = 1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, . . . .
The graph consists of a number of isolated points. 8. Let y be the denominator of x (Exs. x. 7). In this case y is defined only
for rational values of x. We can mark off as many points on the graph as we please, but the result is not in any ordinary sense of the word a curve, and there are no points corresponding to any irrational values of x.
Draw the straight line joining the points (N − 1, N ), (N, N ), where N is a positive integer. Show that the number of points of the locus which lie on this line is equal to the number of positive integers less than and prime to N .
9. Let y = 0 when x is an integer, y = x when x is not an integer. The graph is derived from the straight line y = x by taking out the points
. . . (−1, −1), (0, 0), (1, 1), (2, 2), . . .
and adding the points (−1, 0), (0, 0), (1, 0), . . . on the axis of x. The reader may possibly regard this as an unreasonable function. Why, he
may ask, if y is equal to x for all values of x save integral values, should it not be equal to x for integral values too? The answer is simply, why should it? The function y does in point of fact answer to the definition of a function: there is a relation between x and y such that when x is known y is known. We are perfectly at liberty to take this relation to be what we please, however arbitrary and apparently futile. This function y is, of course, a quite different function from that one which is always equal to x, whatever value, integral or otherwise, x may have.
10. Let y = 1 when x is rational, but y = 0 when x is irrational. The graph consists of two series of points arranged upon the lines y = 1 and y = 0. To the eye it is not distinguishable from two continuous straight lines, but in reality an infinite number of points are missing from each line.
11. Let y = x when x is irrational and y = (1 + p2)/(1 + q2) when x is a rational fraction p/q.

[II : 29]

FUNCTIONS OF REAL VARIABLES

66

Fig. 16.
The irrational values of x contribute to the graph a curve in reality discontinuous, but apparently not to be distinguished from the straight line y = x.
Now consider the rational values of x. First let x be positive. Then (1 + p2)/(1 + q2) cannot be equal to p/q unless p = q, i.e. x = 1. Thus all the points which correspond to rational values of x lie off the line, except the one point (1, 1). Again, if p < q, (1 + p2)/(1 + q2) > p/q; if p > q, (1 + p2)/(1 + q2) < p/q. Thus the points lie above the line y = x if 0 < x < 1, below if x > 1. If p and q are large, (1 + p2)/(1 + q2) is nearly equal to p/q. Near any value of x we can find any number of rational fractions with large numerators and denominators. Hence the graph contains a large number of points which crowd round the line y = x. Its general appearance (for positive values of x) is that of a line surrounded by a swarm of isolated points which gets denser and denser as the points approach the line.
The part of the graph which corresponds to negative values of x consists of the rest of the discontinuous line together with the reflections of all these

[II : 30]

FUNCTIONS OF REAL VARIABLES

67

isolated points in the axis of y. Thus to the left of the axis of y the swarm of points is not round y = x but round y = −x, which is not itself part of the graph. See Fig. 16.

30. Graphical solution of equations containing a single unknown number. Many equations can be expressed in the form

f (x) = ϕ(x),

(1)

where f (x) and ϕ(x) are functions whose graphs are easy to draw. And if the curves
y = f (x), y = ϕ(x)
intersect in a point P whose abscissa is ξ, then ξ is a root of the equation (1).

Examples XVII. 1. The quadratic equation ax2 + 2bx + c = 0.
This may be solved graphically in a variety of ways. For instance we may draw the graphs of
y = ax + 2b, y = −c/x,
whose intersections, if any, give the roots. Or we may take
y = x2, y = −(2bx + c)/a.
But the most elementary method is probably to draw the circle
a(x2 + y2) + 2bx + c = 0, √
whose centre is (−b/a, 0) and radius { b2 − ac}/a. The abscissae of its intersections with the axis of x are the roots of the equation.
2. Solve by any of these methods
x2 + 2x − 3 = 0, x2 − 7x + 4 = 0, 3x2 + 2x − 2 = 0.
3. The equation xm + ax + b = 0. This may be solved by constructing the curves y = xm, y = −ax − b. Verify the following table for the number of

[II : 31]

FUNCTIONS OF REAL VARIABLES

68

roots of xm + ax + b = 0 :
b positive, two or none, (a) m even
b negative, two;

a positive, one, (b) m odd
a negative, three or one.

Construct numerical examples to illustrate all possible cases.
4. Show that the equation tan x = ax + b has always an infinite number of roots.
5. Determine the number of roots of

sin x = x,

sin x

=

1 3

x,

sin x

=

1 8

x,

sin x

=

1 120

x.

6. Show that if a is small and positive (e.g. a = .01), the equation

x

−

a

=

1 2

π

sin2

x

has three roots. Consider also the case in which a is small and negative. Explain how the number of roots varies as a varies.

31. Functions of two variables and their graphical represen-
tation. In § 20 we considered two variables connected by a relation. We
may similarly consider three variables (x, y, and z) connected by a relation such that when the values of x and y are both given, the value or values of z are known. In this case we call z a function of the two variables x and y; x and y the independent variables, z the dependent variable; and we express this dependence of z upon x and y by writing
z = f (x, y).
The remarks of § 20 may all be applied, mutatis mutandis, to this more
complicated case. The method of representing such functions of two variables graphically
is exactly the same in principle as in the case of functions of a single variable. We must take three axes, OX, OY , OZ in space of three dimensions,

[II : 32]

FUNCTIONS OF REAL VARIABLES

69

each axis being perpendicular to the other two. The point (a, b, c) is the point whose distances from the planes Y OZ, ZOX, XOY , measured parallel to OX, OY , OZ, are a, b, and c. Regard must of course be paid to sign, lengths measured in the directions OX, OY , OZ being regarded as positive. The definitions of coordinates, axes, origin are the same as before.
Now let z = f (x, y).
As x and y vary, the point (x, y, z) will move in space. The aggregate of all the positions it assumes is called the locus of the point (x, y, z) or the graph of the function z = f (x, y). When the relation between x, y, and z which defines z can be expressed in an analytical formula, this formula is called the equation of the locus. It is easy to show, for example, that the equation
Ax + By + Cz + D = 0
(the general equation of the first degree) represents a plane, and that the equation of any plane is of this form. The equation
(x − α)2 + (y − β)2 + (z − γ)2 = ρ2,
or x2 + y2 + z2 + 2F x + 2Gy + 2Hz + C = 0,
where F 2 + G2 + H2 − C > 0, represents a sphere; and so on. For proofs of these propositions we must again refer to text-books of Analytical Geometry.

32. Curves in a plane. We have hitherto used the notation

y = f (x)

(1)

to express functional dependence of y upon x. It is evident that this notation is most appropriate in the case in which y is expressed explicitly in terms of x by means of a formula, as when for example

y = x2, sin x, a cos2 x + b sin2 x.

[II : 32]

FUNCTIONS OF REAL VARIABLES

70

We have however very often to deal with functional relations which it is impossible or inconvenient to express in this form. If, for example, y5 − y − x = 0 or x5 + y5 − ay = 0, it is known to be impossible to express y explicitly as an algebraical function of x. If

x2 + y2 + 2Gx + 2F y + C = 0,

y can indeed be so expressed, viz. by the formula √
y = −F + F 2 − x2 − 2Gx − C;

but the functional dependence of y upon x is better and more simply expressed by the original equation.
It will be observed that in these two cases the functional relation is fully expressed by equating a function of the two variables x and y to zero, i.e. by means of an equation

f (x, y) = 0.

(2)

We shall adopt this equation as the standard method of expressing the functional relation. It includes the equation (1) as a special case, since y − f (x) is a special form of a function of x and y. We can then speak of the locus of the point (x, y) subject to f (x, y) = 0, the graph of the function y defined by f (x, y) = 0, the curve or locus f (x, y) = 0, and the equation of this curve or locus.
There is another method of representing curves which is often useful. Suppose that x and y are both functions of a third variable t, which is to be regarded as essentially auxiliary and devoid of any particular geometrical significance. We may write

x = f (t), y = F (t).

(3)

If a particular value is assigned to t, the corresponding values of x and of y are known. Each pair of such values defines a point (x, y). If we construct all the points which correspond in this way to different values

[II : 33]

FUNCTIONS OF REAL VARIABLES

71

of t, we obtain the graph of the locus defined by the equations (3). Suppose for example
x = a cos t, y = a sin t.
Let t vary from 0 to 2π. Then it is easy to see that the point (x, y) describes the circle whose centre is the origin and whose radius is a. If t varies beyond these limits, (x, y) describes the circle over and over again. We can in this case at once obtain a direct relation between x and y by squaring and adding: we find that x2 + y2 = a2, t being now eliminated.
Examples XVIII. 1. The points of intersection of the two curves whose
equations are f (x, y) = 0, ϕ(x, y) = 0, where f and ϕ are polynomials, can be determined if these equations can be solved as a pair of simultaneous equations in x and y. The solution generally consists of a finite number of pairs of values of x and y. The two equations therefore generally represent a finite number of isolated points.
2. Trace the curves (x + y)2 = 1, xy = 1, x2 − y2 = 1. 3. The curve f (x, y) + λϕ(x, y) = 0 represents a curve passing through the points of intersection of f = 0 and ϕ = 0. 4. What loci are represented by
(α) x = at + b, y = ct + d, (β) x/a = 2t/(1 + t2), y/a = (1 − t2)/(1 + t2),
when t varies through all real values?
33. Loci in space. In space of three dimensions there are two fundamentally different kinds of loci, of which the simplest examples are the plane and the straight line.
A particle which moves along a straight line has only one degree of freedom. Its direction of motion is fixed; its position can be completely fixed by one measurement of position, e.g. by its distance from a fixed point on the line. If we take the line as our fundamental line Λ of Chap. I, the position of any of its points is determined by a single coordinate x. A particle which moves in a plane, on the other hand, has two degrees of freedom; its position can only be fixed by the determination of two coordinates.

[II : 33]

FUNCTIONS OF REAL VARIABLES

72

A locus represented by a single equation
z = f (x, y)
plainly belongs to the second of these two classes of loci, and is called a surface. It may or may not (in the obvious simple cases it will) satisfy our common-sense notion of what a surface should be.
The considerations of § 31 may evidently be generalised so as to give
definitions of a function f (x, y, z) of three variables (or of functions of any
number of variables). And as in § 32 we agreed to adopt f (x, y) = 0 as the
standard form of the equation of a plane curve, so now we shall agree to adopt
f (x, y, z) = 0
as the standard form of equation of a surface. The locus represented by two equations of the form z = f (x, y) or
f (x, y, z) = 0 belongs to the first class of loci, and is called a curve. Thus a straight line may be represented by two equations of the type Ax + By + Cz + D = 0. A circle in space may be regarded as the intersection of a sphere and a plane; it may therefore be represented by two equations of the forms
(x − α)2 + (y − β)2 + (z − γ)2 = ρ2, Ax + By + Cz + D = 0.
Examples XIX. 1. What is represented by three equations of the type
f (x, y, z) = 0? 2. Three linear equations in general represent a single point. What are the
exceptional cases? 3. What are the equations of a plane curve f (x, y) = 0 in the plane XOY ,
when regarded as a curve in space? [f (x, y) = 0, z = 0.] 4. Cylinders. What is the meaning of a single equation f (x, y) = 0,
considered as a locus in space of three dimensions? [All points on the surface satisfy f (x, y) = 0, whatever be the value of z. The
curve f (x, y) = 0, z = 0 is the curve in which the locus cuts the plane XOY . The locus is the surface formed by drawing lines parallel to OZ through all points of this curve. Such a surface is called a cylinder.]

[II : 33]

FUNCTIONS OF REAL VARIABLES

73

5. Graphical representation of a surface on a plane. Contour Maps. It might seem to be impossible to represent a surface adequately by a drawing on a plane; and so indeed it is: but a very fair notion of the nature of the surface may often be obtained as follows. Let the equation of the surface be z = f (x, y).
If we give z a particular value a, we have an equation f (x, y) = a, which we may regard as determining a plane curve on the paper. We trace this curve and mark it (a). Actually the curve (a) is the projection on the plane XOY of the section of the surface by the plane z = a. We do this for all values of a (practically, of course, for a selection of values of a). We obtain some such figure as is shown in Fig. 17. It will at once suggest a contoured Ordnance Survey map: and in fact this is the principle on which such maps are constructed. The contour line 1000 is the projection, on the plane of the sea level, of the section of the surface of the land by the plane parallel to the plane of the sea level and 1000 ft. above it.*

5000 4000 3000 5000
2000
1000
Fig. 17.
6. Draw a series of contour lines to illustrate the form of the surface 2z = 3xy.
7. Right circular cones. Take the origin of coordinates at the vertex of *We assume that the effects of the earth’s curvature may be neglected.

[II : 33]

FUNCTIONS OF REAL VARIABLES

74

the cone and the axis of z along the axis of the cone; and let α be the semivertical angle of the cone. The equation of the cone (which must be regarded as extending both ways from its vertex) is x2 + y2 − z2 tan2 α = 0.
8. Surfaces of revolution in general. The cone of Ex. 7 cuts ZOX in two lines whose equations may be combined in the equation x2 = z2 tan2 α. That is to say, the equation of the surface generated by the revolution of the curve y = 0, x2 = z2 tan2 α round the axis of z is derived from the second of these equations by changing x2 into x2 + y2. Show generally that the equation of the surface generated by the revolution of the curve y = 0, x = f (z), round the axis of z, is
x2 + y2 = f (z).

9. Cones in general. A surface formed by straight lines passing through

a fixed point is called a cone: the point is called the vertex. A particular case

is given by the right circular cone of Ex. 7. Show that the equation of a cone

whose vertex is O is of the form f (z/x, z/y) = 0, and that any equation of this

form represents a cone. [If (x, y, z) lies on the cone, so must (λx, λy, λz), for any

value of λ.]

10. Ruled surfaces. Cylinders and cones are special cases of surfaces

composed of straight lines. Such surfaces are called ruled surfaces.

The two equations

x = az + b, y = cz + d,

(1)

represent the intersection of two planes, i.e. a straight line. Now suppose that a, b, c, d instead of being fixed are functions of an auxiliary variable t. For any particular value of t the equations (1) give a line. As t varies, this line moves and generates a surface, whose equation may be found by eliminating t between the two equations (1). For instance, in Ex. 7 the equations of the line which generates the cone are

x = z tan α cos t, y = z tan α sin t,

where t is the angle between the plane XOZ and a plane through the line and the axis of z.
Another simple example of a ruled surface may be constructed as follows. Take two sections of a right circular cylinder perpendicular to the axis and at a distance l apart (Fig. 18a). We can imagine the surface of the cylinder to be

[II : 33]

FUNCTIONS OF REAL VARIABLES

75

made up of a number of thin parallel rigid rods of length l, such as P Q, the ends
of the rods being fastened to two circular rods of radius a.
Now let us take a third circular rod of the same radius and place it round
the surface of the cylinder at a distance h from one of the first two rods (see
Fig. 18a, where P q = h). Unfasten the end Q of the rod P Q and turn P Q about P until Q can be fastened to the third circular rod in the position Q′. The angle qOQ′ = α in the figure is evidently given by

l2 − h2 = qQ′2 =

2a

sin

1 2

α

2.

Let all the other rods of which the cylinder was composed be treated in the same way. We obtain a ruled surface whose form is indicated in Fig. 18b. It is entirely built up of straight lines; but the surface is curved everywhere, and is in general shape not unlike certain forms of table-napkin rings (Fig. 18c).

P

O

q

Q′

Q
Fig. 18a.

Fig. 18b.

Fig. 18c.

MISCELLANEOUS EXAMPLES ON CHAPTER II.

1. Show that if y = f (x) = (ax + b)/(cx − a) then x = f (y).

2. If f (x) = f (−x) for all values of x, f (x) is called an even function. If f (x) = −f (−x), it is called an odd function. Show that any function of x,

defined for all values of x, is the sum of an even and an odd function of x.

[Use

the

identity

f (x)

=

1 2

{f

(x)

+

f (−x)}

+

1 2

{f

(x)

−

f (−x)}.]

[II : 33]

FUNCTIONS OF REAL VARIABLES

76

3. Draw the graphs of the functions

3 sin x + 4 cos x, sin √π sin x . 2

4. Draw the graphs of the functions

(Math. Trip. 1896.)

sin x(a cos2 x + b sin2 x), sin x (a cos2 x + b sin2 x), x

sin x 2 .
x

5. Draw the graphs of the functions x[1/x], [x]/x. 6. Draw the graphs of the functions

(i) arc cos(2x2 − 1) − 2 arc cos x,

(ii)

arc

tan

a+x 1 − ax

−

arc

tan

a

−

arc

tan

x,

where the symbols arc cos a, arc tan a denote, for any value of a, the least positive (or zero) angle, whose cosine or tangent is a.
7. Verify the following method of constructing the graph of f {ϕ(x)} by means of the line y = x and the graphs of f (x) and ϕ(x): take OA = x along OX, draw AB parallel to OY to meet y = ϕ(x) in B, BC parallel to OX to meet y = x in C, CD parallel to OY to meet y = f (x) in D, and DP parallel to OX to meet AB in P ; then P is a point on the graph required.
8. Show that the roots of x3 + px + q = 0 are the abscissae of the points of intersection (other than the origin) of the parabola y = x2 and the circle

x2 + y2 + (p − 1)y + qx = 0.

9. The roots of x4 + nx3 + px2 + qx + r = 0 are the abscissae of the points

of

intersection

of

the

parabola

x2

=

y

−

1 2

nx

and

the

circle

x2

+

y2

+

(

1 8

n2

−

1 2

pn

+

1 2

n

+

q)x

+

(p

−

1

−

1 4

n2)y

+

r

=

0.

10. Discuss the graphical solution of the equation xm + ax2 + bx + c = 0

[II : 33]

FUNCTIONS OF REAL VARIABLES

77

by means of the curves y = xm, y = −ax2 − bx − c. Draw up a table of the

various possible numbers of roots.

√

11. Solve the equation sec θ + cosec θ = 2 2; and show that the equation

sec θ + cosec θ = c has two roots between 0 and 2π if c2 < 8 and four if c2 > 8.

12. Show that the equation

2x = (2n + 1)π(1 − cos x),

where n is a positive integer, has 2n + 3 roots and no more, indicating their

localities roughly.

(Math. Trip. 1896.)

13.

Show

that the equation

2 3

x

sin

x

=

1

has

four roots between −π

and π.

14. Discuss the number and values of the roots of the equations

(1)

cot x

+

x

−

3 2

π

=

0,

(2) x2 + sin2 x = 1,

(3) tan x = 2x/(1 + x2),

(4)

sin

x

−

x

+

1 6

x3

=

0,

(5) (1 − cos x) tan α − x + sin x = 0.

15. The polynomial of the second degree which assumes, when x = a, b, c

the values α, β, γ is

(x α (a

− −

b)(x b)(a

− −

c) c)

+

β

(x (b

− −

c)(x − a) c)(b − a)

+

γ

(x (c

− −

a)(x − b) a)(c − b) .

Give a similar formula for the polynomial of the (n − 1)th degree which assumes,

when x = a1, a2, . . . an, the values α1, α2, . . . αn. 16. Find a polynomial in x of the second degree which for the values 0, 1, 2

of x takes the values 1/c, 1/(c + 1), 1/(c + 2); and show that when x = c + 2 its

value is 1/(c + 1).

(Math. Trip. 1911.)

17. Show that if x is a rational function of y, and y is a rational function

of x, then Axy + Bx + Cy + D = 0.

18. If y is an algebraical function of x, then x is an algebraical function of y.

19. Verify that the equation

cos

1 2

πx

=

1

−

x2

x + (x − 1)

2−x 3

[II : 33]

FUNCTIONS OF REAL VARIABLES

78

is approximately true for all values of x between 0 and 1. [Take

2 3

,

5 6

,

1,

and

use

tables.

For

which

of

these

values

is

the

formula

x

=

0,

1 6

,

exact?]

1 3

,

1 2

,

20. What is the form of the graph of the functions

z = [x] + [y], z = x + y − [x] − [y]?

21. What is the form of the graph of the functions z = sin x + sin y, z = sin x sin y, z = sin xy, z = sin(x2 + y2)?
22. Geometrical constructions for irrational numbers. In Chapter I we√indicated one or two simple geometrical constructions for a length equal to 2, starting from a given unit length. We also showed how to construct the roots of any quadratic equation ax2 + 2bx + c = 0, it being supposed that we can construct lines whose lengths are equal to any of the ratios of the coefficients a, b, c, as is certainly the case if a, b, c are rational. All these constructions were what may be called Euclidean constructions; they depended on the ruler and compasses only.
It is fairly obvious that we can construct by these methods the length measured by any irrational number which is defined by any combination of square roots, however complicated. Thus

√

√

4 17 + 3√11 − 17 − 3√11

17 − 3 11 17 + 3 11

is a case in point. This expression contains a fourth root, but this√is of course the square root of a square root. We sh√ould begin by c√onstructing 11, e.g. as the mean between 1 and 11: then 17 + 3 11 and 17 − 3 11, and so on. Or these two mixed surds might be constructed directly as the roots of x2 −34x+190 = 0.
Conversely, only irrationals of this kind can be constructed by Euclidean methods. Starting from a unit length we can construct any rational length. And hence we can construct the line Ax + By + C = 0, provided that the ratios of A, B, C are rational, and the circle

(x − α)2 + (y − β)2 = ρ2

(or x2 + y2 + 2gx + 2f y + c = 0), provided that α, β, ρ are rational, a condition which implies that g, f , c are rational.

[II : 33]

FUNCTIONS OF REAL VARIABLES

79

Now in any Euclidean construction each new point introduced into the figure is determined as the intersection of two lines or circles, or a line and a circle. But if the coefficients are rational, such a pair of equations as

Ax + By + C = 0, x2 + y2 + 2gx + 2f y + c = 0

give, on solution, values of x and y of the form m + n√p, where m, n, p are rational: for if we substitute for x in terms of y in the second equation we obtain a quadratic in y with rational coefficients. Hence the coordinates of all points obtained by means of lines and circles with rational coefficients are expressible by rational numbers and quadratic surds. And so the same is true of the distance
(x1 − x2)2 + (y1 − y2)2 between any two points so obtained. With the irrational distances thus constructed we may proceed to construct
a number of lines and circles whose coefficients may now themselves involve quadratic surds. It is evident, however, that all the lengths which we can construct by the use of such lines and circles are still expressible by square roots only, though our surd expressions may now be of a more complicated form. And this remains true however often our constructions are repeated. Hence Euclidean

methods will construct any surd expression involving square roots only, and no others.

One of the famous problems of antiquity was that of the duplication of the cube, tha√t is to say of the construc√tion by Euclidean methods of a length measured by 3 2. It can be shown that 3 2 cannot be expressed by means of any finite combination of rational numbers and square roots, and so that the problem is an impossible one. See Hobson, S√quaring the Circle, pp. 47 et seq.; the first stage of the proof, viz. the proof that 3 2 cannot be a root of a quadratic equation ax2 + 2bx + c = 0 with rational coefficients, was given in Ch. I (Misc. Exs. 24).

23. Approximate quadrature of the circle. Let O be the centre of a

circle of radius R.

On the tangent at A take AP

=

11 5

R

and

AQ

=

13 5

R,

in

the same direction. On AO take AN = OP and draw N M parallel to OQ and

cutting AP in M . Show that

AM/R

=

13 25

√ 146,

and that to take AM as being equal to the circumference of the circle would lead to a value of π correct to five places of decimals. If R is the earth’s radius, the error in supposing AM to be its circumference is less than 11 yards.

[II : 33]

FUNCTIONS OF REAL VARIABLES

80

24. Show that the only lengths which can be constructed with the ruler only,

starting from a given unit len√gth, are rational lengths. 25. Constructions for 3 2. O is the vertex and S the focus of the parabola
y2 = 4x, and P is one of its points of intersection with the parabola x2 = 2y.

Show that O√P meets the latus rectum of the first parabola in a point Q such that SQ = 3 2.

26. Take a circle of unit diameter, a diameter OA and the tangent at A.

Draw a chord OBC cutting the circle at B and the tangent at C. On this line

take OM = BC. Taking O as origin and OA as axis of x, show that the locus

of M is the curve

(x2 + y2)x − y2 = 0

(the Cissoid of Diocles). Sketch the curve. Take along the axis of y a length OD = 2. Let AD cut th√e curve in P and OP cut the tangent to the circle at A in Q. Show that AQ = 3 2.

CHAPTER III
COMPLEX NUMBERS
34. Displacements along a line and in a plane. The ‘real number’ x, with which we have been concerned in the two preceding chapters, may be regarded from many different points of view. It may be regarded as a pure number, destitute of geometrical significance, or a geometrical significance may be attached to it in at least three different ways. It may be regarded as the measure of a length, viz. the length A0P along the line Λ of Chap. I. It may be regarded as the mark of a point, viz. the point P whose distance from A0 is x. Or it may be regarded as the measure of a displacement or change of position on the line Λ. It is on this last point of view that we shall now concentrate our attention.
Imagine a small particle placed at P on the line Λ and then displaced to Q. We shall call the displacement or change of position which is needed to transfer the particle from P to Q the displacement P Q. To specify a displacement completely three things are needed, its magnitude, its sense forwards or backwards along the line, and what may be called its point of application, i.e. the original position P of the particle. But, when we are thinking merely of the change of position produced by the displacement, it is natural to disregard the point of application and to consider all displacements as equivalent whose lengths and senses are the same. Then the displacement is completely specified by the length P Q = x, the sense of the displacement being fixed by the sign of x. We may therefore, without ambiguity, speak of the displacement [x],* and we may write P Q = [x].
We use the square bracket to distinguish the displacement [x] from the length or number x.„ If the coordinate of P is a, that of Q will be a + x;
*It is hardly necessary to caution the reader against confusing this use of the symbol [x] and that of Chap. II (Exs. xvi. and Misc. Exs.).
„Strictly speaking we ought, by some similar difference of notation, to distinguish the actual length x from the number x which measures it. The reader will perhaps be inclined to consider such distinctions futile and pedantic. But increasing experience of mathematics will reveal to him the great importance of distinguishing clearly between things which, however intimately connected, are not the same. If cricket were a math-
81

[III : 34]

COMPLEX NUMBERS

82

the displacement [x] therefore transfers a particle from the point a to the point a + x.
We come now to consider displacements in a plane. We may define the displacement P Q as before. But now more data are required in order to specify it completely. We require to know: (i) the magnitude of the displacement, i.e. the length of the straight line P Q; (ii) the direction of the displacement, which is determined by the angle which P Q makes with some fixed line in the plane; (iii) the sense of the displacement; and (iv) its point of application. Of these requirements we may disregard the fourth, if we consider two displacements as equivalent if they are the same

Y Q S

P

R

A

O

X

B

Fig. 19.

in magnitude, direction, and sense. In other words, if P Q and RS are equal and parallel, and the sense of motion from P to Q is the same as that of motion from R to S, we regard the displacements P Q and RS as equivalent, and write
P Q = RS.
Now let us take any pair of coordinate axes in the plane (such as OX, OY in Fig. 19). Draw a line OA equal and parallel to P Q, the sense of motion from O to A being the same as that from P to Q. Then P Q and OA are equivalent displacements. Let x and y be the coordinates
ematical science, it would be very important to distinguish between the motion of the batsman between the wickets, the run which he scores, and the mark which is put down in the score-book.

[III : 35]

COMPLEX NUMBERS

83

of A. Then it is evident that OA is completely specified if x and y are given. We call OA the displacement [x, y] and write
OA = P Q = RS = [x, y].

35. Equivalence of displacements. Multiplication of displacements by numbers. If ξ and η are the coordinates of P , and ξ′ and η′ those of Q, it is evident that
x = ξ′ − ξ, y = η′ − η.

The displacement from (ξ, η) to (ξ′, η′) is therefore

[ξ′ − ξ, η′ − η].

It is clear that two displacements [x, y], [x′, y′] are equivalent if, and only if, x = x′, y = y′. Thus [x, y] = [x′, y′] if and only if

x = x′, y = y′.

(1)

The reverse displacement QP would be [ξ − ξ′, η − η′], and it is natural to agree that

[ξ − ξ′, η − η′] = −[ξ′ − ξ, η′ − η], QP = −P Q,

these equations being really definitions of the meaning of the symbols −[ξ′ − ξ, η′ − η], −P Q. Having thus agreed that

−[x, y] = [−x, −y],

it is natural to agree further that

α[x, y] = [αx, αy],

(2)

[III : 36]

COMPLEX NUMBERS

84

where α is any real number, positive or negative. Thus (Fig. 19) if

OB

=

−

1 2

OA

then

OB

=

−

1 2

OA

=

−

1 2

[x,

y]

=

[−

1 2

x,

−

1 2

y].

The equations (1) and (2) define the first two important ideas connected with displacements, viz. equivalence of displacements, and multiplication of displacements by numbers.

36. Addition of displacements. We have not yet given any definition which enables us to attach any meaning to the expressions
P Q + P ′Q′, [x, y] + [x′, y′].

Common sense at once suggests that we should define the sum of two
displacements as the displacement which is the result of the successive
application of the two given displacements. In other words, it suggests that if QQ1 be drawn equal and parallel to P ′Q′, so that the result of successive displacements P Q, P ′Q′ on a particle at P is to transfer it first to Q and then to Q1 then we should define the sum of P Q and P ′Q′ as being P Q1. If then we draw OA equal and parallel to P Q, and OB equal and parallel to P ′Q′, and complete the parallelogram OACB, we have

P Q + P ′Q′ = P Q1 = OA + OB = OC.

Let us consider the consequences of adopting this definition. If the

coordinates of B are x′, y′, then those of the middle point of AB are

1 2

(x

+

x′

),

1 2

(y

+ y′),

and

those

of

C

are

x + x′,

y + y′.

Hence

[x, y] + [x′, y′] = [x + x′, y + y′],

(3)

which may be regarded as the symbolic definition of addition of displacements. We observe that
[x′, y′] + [x, y] = [x′ + x, y′ + y] = [x + x′, y + y′] = [x, y] + [x′, y′]