A COURSE

OF

PURE MATHEMATICS

CAMBRIDGE UNIVERSITY PRESS

C. F. CLAY, Manager

LONDON: FETTER LANE, E.C. 4

NEW YORK : THE MACMILLAN CO.

BOMBAY

CALCUTTA

MADRAS

MACMILLAN AND CO., Ltd.

TORONTO : THE MACMILLAN CO. OF

CANADA, Ltd.

TOKYO : MARUZEN-KABUSHIKI-KAISHA

ALL RIGHTS RESERVED

A COURSE

OF

PURE MATHEMATICS

BY

G. H. HARDY, M.A., F.R.S.

FELLOW OF NEW COLLEGE

SAVILIAN PROFESSOR OF GEOMETRY IN THE UNIVERSITY

OF OXFORD

LATE FELLOW OF TRINITY COLLEGE, CAMBRIDGE

THIRD EDITION

Cambridge

at the University Press

1921

First Edition 1908

Second Edition 1914

Third Edition 1921

PREFACE TO THE THIRD EDITION

No extensive changes have been made in this edition. The most important are in §§ 80–82, which I have rewritten in accordance with suggestions

made by Mr S. Pollard.

The earlier editions contained no satisfactory account of the genesis of

the circular functions. I have made some attempt to meet this objection

in § 158 and Appendix III. Appendix IV is also an addition.

It is curious to note how the character of the criticisms I have had to

meet has changed. I was too meticulous and pedantic for my pupils of

fifteen years ago: I am altogether too popular for the Trinity scholar of

to-day. I need hardly say that I find such criticisms very gratifying, as the

best evidence that the book has to some extent fulfilled the purpose with

which it was written.

G. H. H.

August 1921

EXTRACT FROM THE PREFACE TO THE

SECOND EDITION

The principal changes made in this edition are as follows. I have 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.

G. H. H.

October 1914

EXTRACT FROM THE PREFACE TO THE FIRST

EDITION

This book has been designed primarily for the use of first year students

at the Universities whose abilities reach or approach something like what is

usually described as ‘scholarship standard’. I hope that it may be useful to

other classes of readers, but it is this class whose wants I have considered

first. It is in any case a book for mathematicians: I have nowhere made

any attempt to meet the needs of students of engineering or indeed any

class of students whose interests are not primarily mathematical.

I regard the book as being really elementary. There are plenty of hard

examples (mainly at the ends of the chapters): to these I have added,

wherever space permitted, an outline of the solution. But I have done my

best to avoid the inclusion of anything that involves really difficult ideas.

For instance, I make no use of the ‘principle of convergence’: uniform

convergence, double series, infinite products, are never alluded to: and

I prove no general theorems whatever concerning the inversion of limitoperations—I never even define ∂

2

f

∂x ∂y and ∂

2

f

∂y ∂x. In the last two chapters I

have occasion once or twice to integrate a power-series, but I have confined

myself to the very simplest cases and given a special discussion in each

instance. Anyone who has read this book will be in a position to read with

profit Dr Bromwich’s Infinite Series, where a full and adequate discussion

of all these points will be found.

September 1908

CONTENTS

CHAPTER I

REAL VARIABLES

SECT. PAGE

1–2. Rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

3–7. Irrational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

8. Real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

9. Relations of magnitude between real numbers . . . . . . . . . . . . . . . . . 16

10–11. Algebraical operations with real numbers . . . . . . . . . . . . . . . . . . . . . 18

12. The number √

2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

13–14. Quadratic surds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

15. The continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

16. The continuous real variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

17. Sections of the real numbers. Dedekind’s Theorem . . . . . . . . . . . . 30

18. Points of condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

19. Weierstrass’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Decimals, 1. Gauss’s Theorem, 6. Graphical solution of quadratic

equations, 22. Important inequalities, 35. Arithmetical and geometrical means, 35. Schwarz’s Inequality, 36. Cubic and other surds, 38.

Algebraical numbers, 41.

CHAPTER II

FUNCTIONS OF REAL VARIABLES

20. The idea of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

21. The graphical representation of functions. Coordinates . . . . . . . . 46

22. Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

23. Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

24–25. Rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

26–27. Algebraical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

28–29. Transcendental functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

30. Graphical solution of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

CONTENTS viii

SECT. PAGE

31. Functions of two variables and their graphical representation . . 68

32. Curves in a plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

33. Loci in space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Trigonometrical functions, 60. Arithmetical functions, 63. Cylinders, 72.

Contour maps, 72. Cones, 73. Surfaces of revolution, 73. Ruled surfaces, 74. Geometrical constructions for irrational numbers, 77. Quadrature of the circle, 79.

CHAPTER III

COMPLEX NUMBERS

34–38. Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

39–42. Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

43. The quadratic equation with real coefficients . . . . . . . . . . . . . . . . . . 96

44. Argand’s diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

45. De Moivre’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

46. Rational functions of a complex variable . . . . . . . . . . . . . . . . . . . . . . 104

47–49. Roots of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Properties of a triangle, 106, 121. Equations with complex coefficients, 107. Coaxal circles, 110. Bilinear and other transformations, 111, 116, 125. Cross ratios, 114. Condition that four points

should be concyclic, 116. Complex functions of a real variable, 116.

Construction of regular polygons by Euclidean methods, 120. Imaginary

points and lines, 124.

CHAPTER IV

LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE

50. Functions of a positive integral variable . . . . . . . . . . . . . . . . . . . . . . . 128

51. Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

52. Finite and infinite classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

CONTENTS ix

SECT. PAGE

53–57. Properties possessed by a function of n for large values of n . . . 131

58–61. Definition of a limit and other definitions . . . . . . . . . . . . . . . . . . . . . 138

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

63–68. General theorems concerning limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

69–70. Steadily increasing or decreasing functions . . . . . . . . . . . . . . . . . . . . 157

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

72. The limit of x

n

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

73. The limit of

1 +

1

n

n

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

74. Some algebraical lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

75. The limit of n(

√n x − 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

76–77. Infinite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

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

79. The representation of functions of a continuous real variable by

means of limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

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

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

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

83–84. The general principle of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 183

85–86. Limits of complex functions and series of complex terms . . . . . . 185

87–88. Applications to z

n and the geometrical series . . . . . . . . . . . . . . . . . 188

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

Oscillation of sin nθπ, 144, 146, 181. Limits of n

kx

n,

√n x,

√n n,

√n

n!, x

n

n!

,

m

n

x

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

series, 176. Equation xn+1 = f(xn), 190. Expansions of rational functions, 191. Limit of a mean value, 193.

CHAPTER V

LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND

DISCONTINUOUS FUNCTIONS

89–92. Limits as x → ∞ or x → −∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

CONTENTS x

SECT. PAGE

93–97. Limits as x → a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

98–99. Continuous functions of a real variable . . . . . . . . . . . . . . . . . . . . . . . . 210

100–104. Properties of continuous functions. Bounded functions. The

oscillation of a function in an interval . . . . . . . . . . . . . . . . . . . . 215

105–106. Sets of intervals on a line. The Heine-Borel Theorem . . . . . . . . . . 223

107. Continuous functions of several variables . . . . . . . . . . . . . . . . . . . . . . 228

108–109. Implicit and inverse functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

Limits and continuity of polynomials and rational functions, 204, 212.

Limit of x

m − a

m

x − a

, 206. Orders of smallness and greatness, 207. Limit of

sin x

x

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

Classification of discontinuities, 214.

CHAPTER VI

DERIVATIVES AND INTEGRALS

110–112. Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

113. General rules for differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

114. Derivatives of complex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

115. The notation of the differential calculus . . . . . . . . . . . . . . . . . . . . . . . 246

116. Differentiation of polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

117. Differentiation of rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

118. Differentiation of algebraical functions . . . . . . . . . . . . . . . . . . . . . . . . 253

119. Differentiation of transcendental functions . . . . . . . . . . . . . . . . . . . . 255

120. Repeated differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

121. General theorems concerning derivatives. Rolle’s Theorem . . . . 262

122–124. Maxima and minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

125–126. The Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

127–128. Integration. The logarithmic function . . . . . . . . . . . . . . . . . . . . . . . . . 277

129. Integration of polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

130–131. Integration of rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

CONTENTS xi

SECT. PAGE

132–139. Integration of algebraical functions. Integration by rationalisation. Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

140–144. Integration of transcendental functions . . . . . . . . . . . . . . . . . . . . . . . . 298

145. Areas of plane curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

146. Lengths of plane curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

Derivative of x

m, 241. Derivatives of cos x and sin x, 241. Tangent and

normal to a curve, 241, 257. Multiple roots of equations, 249, 309. Rolle’s

Theorem for polynomials, 251. Leibniz’ Theorem, 259. Maxima and 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. Se-

ries for cosx and sinx, 325. Binomial series, 325. Tangent to a curve,

331, 346, 374. Points of inflexion, 331. Curvature, 333, 372. Osculating

CONTENTS xii

conics, 334, 372. Differentiation of implicit functions, 346. Fourier’s inte-

grals, 355, 360. The second mean value theorem, 364. Homogeneous func-

tions, 372. Euler’s Theorem, 372. Jacobians, 374. Schwarz’s inequality for

integrals, 378. Approximate values of definite integrals, 380. Simpson’s

Rule, 380.

CHAPTER VIII

THE CONVERGENCE OF INFINITE SERIES AND INFINITE INTEGRALS

SECT. PAGE

165–168. Series of positive terms. Cauchy’s and d’Alembert’s tests of con-

vergence …………………………………………. 382

169. Dirichlet’s Theorem ……………………………………. 388

170. Multiplication of series of positive terms …………………. 388

171–174. Further tests of convergence. Abel’s Theorem. Maclaurin’s inte-

gral test …………………………………………. 390

175. The series

P

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

P

n k r n and allied series, 385. Transformation of infinite inte-

grals by substitution and integration by parts, 404, 406, 413. The series

P

a n cosnθ,

P

a n sinnθ, 419, 425, 427. Alteration of the sum of a series

by rearrangement, 423. Logarithmic series, 431. Binomial series, 431, 433.

Multiplication of conditionally convergent series, 434, 439. Recurring se-

ries, 437. Difference equations, 438. Definite integrals, 441. Schwarz’s

inequality for infinite integrals, 442.

CONTENTS xiii

CHAPTER IX

THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS OF A REAL

VARIABLE

SECT. PAGE

196–197. The logarithmic function ……………………………….. 444

198. The functional equation satisfied by logx ………………… 447

199–201. The behaviour of logx as x tends to infinity or to zero …….. 448

202. The logarithmic scale of infinity …………………………. 450

203. The number e ………………………………………… 452

204–206. The exponential function ………………………………. 453

207. The general power a x ………………………………….. 456

208. The exponential limit ………………………………….. 457

209. The logarithmic limit ………………………………….. 459

210. Common logarithms …………………………………… 460

211. Logarithmic tests of convergence ………………………… 466

212. The exponential series …………………………………. 471

213. The logarithmic series …………………………………. 475

214. The series for arctanx …………………………………. 476

215. The binomial series ……………………………………. 480

216. Alternative development of the theory …………………… 482

Miscellaneous Examples ……………………………….. 484

Integrals containing the exponential function, 460. The hyperbolic func-

tions, 463. Integrals of certain algebraical functions, 464. Euler’s con-

stant, 469, 486. Irrationality of e, 473. Approximation to surds by the bi-

nomial theorem, 480. Irrationality of log 10 n, 483. Definite integrals, 491.

CHAPTER X

THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND

CIRCULAR FUNCTIONS

217–218. Functions of a complex variable …………………………. 495

219. Curvilinear integrals …………………………………… 496

220. Definition of the logarithmic function ……………………. 497

221. The values of the logarithmic function …………………… 499

CONTENTS xiv

SECT. PAGE

222–224. The exponential function ………………………………. 505

225–226. The general power a z ………………………………….. 507

227–230. The trigonometrical and hyperbolic functions …………….. 512

231. The connection between the logarithmic and inverse trigonomet-

rical functions ……………………………………. 518

232. The exponential series …………………………………. 520

233. The series for cosz and sinz ……………………………. 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 Logz, 503. The function e z , 509.

Logarithms to any base, 510. The inverse cosine, sine, and tangent of

a complex number, 516. Trigonometrical series, 523, 527, 540. Roots of

transcendental equations, 534. Transformations, 535, 538. Stereographic

projection, 537. Mercator’s projection, 538. Level curves, 539. Definite

integrals, 543.

Appendix I. The proof that every equation has a root …………… 545

Appendix II. A note on double limit problems …………………… 553

Appendix III. The circular functions ……………………………. 557

Appendix IV. The infinite in analysis and geometry ………………. 560

CHAPTER I

REAL VARIABLES

1. Rational numbers. A fraction r = p/q, where p and q are pos-

itive or negative integers, is called a rational number. We can suppose

(i) that p and q have no common factor, as if they have a common factor

we can divide each of them by it, and (ii) that q is positive, since

p/(−q) = (−p)/q, (−p)/(−q) = p/q.

To the rational numbers thus defined we may add the ‘rational number 0’

obtained by taking p = 0.

We assume that the reader is familiar with the ordinary arithmetical

rules for the manipulation of rational numbers. The examples which follow

demand no knowledge beyond this.

Examples I. 1. If r and s are rational numbers, then r + s, r − s, rs,

and r/s are rational numbers, unless in the last case s = 0 (when r/s is of course

meaningless).

2. If λ, m, and n are positive rational numbers, and m > n, then

λ(m 2 − n 2 ), 2λmn, and λ(m 2 + n 2 ) are positive rational numbers. Hence show

how to determine any number of right-angled triangles the lengths of all of

whose sides are rational.

3. Any terminated decimal represents a rational number whose denomina-

tor contains no factors other than 2 or 5. Conversely, any such rational number

can be expressed, and in one way only, as a terminated decimal.

[The general theory of decimals will be considered in Ch. IV.]

4. The positive rational numbers may be arranged in the form of a simple

series as follows:

1

1 ,

2

1 ,

1

2 ,

3

1 ,

2

2 ,

1

3 ,

4

1 ,

3

2 ,

2

3 ,

1

4 , ….

Show that p/q is the [ 1

2 (p + q − 1)(p + q − 2) + q]th term of the series.

[In this series every rational number is repeated indefinitely. Thus 1 occurs

as

1

1 ,

2

2 ,

3

3 ,…. We can of course avoid this by omitting every number which has

already occurred in a simpler form, but then the problem of determining the

precise position of p/q becomes more complicated.]

1

[I:2] REAL VARIABLES 2

2. The representation of rational numbers by points on a line.

It is convenient, in many branches of mathematical analysis, to make a

good deal of use of geometrical illustrations.

The use of geometrical illustrations in this way does not, of course,

imply that analysis has any sort of dependence upon geometry: they are

illustrations and nothing more, and are employed merely for the sake of

clearness of exposition. This being so, it is not necessary that we should

attempt any logical analysis of the ordinary notions of elementary geome-

try; we may be content to suppose, however far it may be from the truth,

that we know what they mean.

Assuming, then, that we know what is meant by a straight line, a

segment of a line, and the length of a segment, let us take a straight line Λ,

produced indefinitely in both directions, and a segment A 0 A 1 of any length.

We call A 0 the origin, or the point 0, and A 1 the point 1, and we regard

these points as representing the numbers 0 and 1.

In order to obtain a point which shall represent a positive rational

number r = p/q, we choose the point A r such that

A 0 A r /A 0 A 1 = r,

A 0 A r being a stretch of the line extending in the same direction along the

line as A 0 A 1 , a direction which we shall suppose to be from left to right

when, as in Fig. 1, the line is drawn horizontally across the paper. In

order to obtain a point to represent a negative rational number r = −s,

A 0 A 1 A s A −1 A −s

Fig. 1.

it is natural to regard length as a magnitude capable of sign, positive if

the length is measured in one direction (that of A 0 A 1 ), and negative if

measured in the other, so that AB = −BA; and to take as the point

representing r the point A −s such that

A 0 A −s = −A −s A 0 = −A 0 A s .

[I:3] REAL VARIABLES 3

We thus obtain a point A r on the line corresponding to every rational

value of r, positive or negative, and such that

A 0 A r = r · A 0 A 1 ;

and if, as is natural, we take A 0 A 1 as our unit of length, and write

A 0 A 1 = 1, then we have

A 0 A r = r.

We shall call the points A r the rational points of the line.

3. Irrational numbers. If the reader will mark off on the line all

the points corresponding to the rational numbers whose denominators are

1, 2, 3,… in succession, he will readily convince himself that he can cover

the line with rational points as closely as he likes. We can state this more

precisely as follows: if we take any segment BC on Λ, we can find as many

rational points as we please on BC.

Suppose, for example, that BC falls within the segment A 1 A 2 . It is

evident that if we choose a positive integer k so that

k · BC > 1, ∗ (1)

and divide A 1 A 2 into k equal parts, then at least one of the points of

division (say P) must fall inside BC, without coinciding with either B or C.

For if this were not so, BC would be entirely included in one of the k parts

into which A 1 A 2 has been divided, which contradicts the supposition (1).

But P obviously corresponds to a rational number whose denominator is k.

Thus at least one rational point P lies between B and C. But then we can

find another such point Q between B and P, another between B and Q,

and so on indefinitely; i.e., as we asserted above, we can find as many as

we please. We may express this by saying that BC includes infinitely many

rational points.

∗ The assumption that this is possible is equivalent to the assumption of what is

known as the Axiom of Archimedes.

[I:3] REAL VARIABLES 4

The meaning of such phrases as ‘infinitely many’ or ‘an infinity of ’, in such

sentences as ‘BC includes infinitely many rational points’ or ‘there are an infinity

of rational points on BC’ or ‘there are an infinity of positive integers’, will be

considered more closely in Ch. IV. The assertion ‘there are an infinity of positive

integers’ means ‘given any positive integer n, however large, we can find more

than n positive integers’. This is plainly true whatever n may be, e.g. for

n = 100,000 or 100,000,000. The assertion means exactly the same as ‘we can

find as many positive integers as we please’.

The reader will easily convince himself of the truth of the following assertion,

which is substantially equivalent to what was proved in the second paragraph

of this section: given any rational number r, and any positive integer n, we can

find another rational number lying on either side of r and differing from r by

less than 1/n. It is merely to express this differently to say that we can find

a rational number lying on either side of r and differing from r by as little as

we please. Again, given any two rational numbers r and s, we can interpolate

between them a chain of rational numbers in which any two consecutive terms

differ by as little as we please, that is to say by less than 1/n, where n is any

positive integer assigned beforehand.

From these considerations the reader might be tempted to infer that an

adequate view of the nature of the line could be obtained by imagining it to

be formed simply by the rational points which lie on it. And it is certainly

the case that if we imagine the line to be made up solely of the rational

points, and all other points (if there are any such) to be eliminated, the

figure which remained would possess most of the properties which common

sense attributes to the straight line, and would, to put the matter roughly,

look and behave very much like a line.

A little further consideration, however, shows that this view would

involve us in serious difficulties.

Let us look at the matter for a moment with the eye of common sense,

and consider some of the properties which we may reasonably expect a

straight line to possess if it is to satisfy the idea which we have formed of

it in elementary geometry.

The straight line must be composed of points, and any segment of it by

all the points which lie between its end points. With any such segment

[I:3] REAL VARIABLES 5

must be associated a certain entity called its length, which must be a

quantity capable of numerical measurement in terms of any standard or

unit length, and these lengths must be capable of combination with one

another, according to the ordinary rules of algebra, by means of addition or

multiplication. Again, it must be possible to construct a line whose length

is the sum or product of any two given lengths. If the length PQ, along

a given line, is a, and the length QR, along the same straight line, is b,

the length PR must be a+b. Moreover, if the lengths OP, OQ, along one

straight line, are 1 and a, and the length OR along another straight line is b,

and if we determine the length OS by Euclid’s construction (Euc. vi. 12)

for a fourth proportional to the lines OP, OQ, OR, this length must be ab,

the algebraical fourth proportional to 1, a, b. And it is hardly necessary to

remark that the sums and products thus defined must obey the ordinary

‘laws of algebra’; viz.

a + b = b + a, a + (b + c) = (a + b) + c,

ab = ba, a(bc) = (ab)c, a(b + c) = ab + ac.

The lengths of our lines must also obey a number of obvious laws concerning

inequalities as well as equalities: thus if A, B, C are three points lying

along Λ from left to right, we must have AB < AC, and so on. Moreover

it must be possible, on our fundamental line Λ, to find a point P such

that A 0 P is equal to any segment whatever taken along Λ or along any

other straight line. All these properties of a line, and more, are involved

in the presuppositions of our elementary geometry.

Now it is very easy to see that the idea of a straight line as composed of

a series of points, each corresponding to a rational number, cannot possibly

satisfy all these requirements. There are various elementary geometrical

constructions, for example, which purport to construct a length x such

that x 2 = 2. For instance, we may construct an isosceles right-angled tri-

angle ABC such that AB = AC = 1. Then if BC = x, x 2 = 2. Or we may

determine the length x by means of Euclid’s construction (Euc. vi. 13) for

a mean proportional to 1 and 2, as indicated in the figure. Our require-

ments therefore involve the existence of a length measured by a number x,

[I:3] REAL VARIABLES 6

A B

C

1

1

x

L M N

P

2 1

x

Fig. 2.

and a point P on Λ such that

A 0 P = x, x 2 = 2.

But it is easy to see that there is no rational number such that its square

is 2. In fact we may go further and say that there is no rational number

whose square is m/n, where m/n is any positive fraction in its lowest terms,

unless m and n are both perfect squares.

For suppose, if possible, that

p 2 /q 2 = m/n,

p having no factor in common with q, and m no factor in common with n.

Then np 2 = mq 2 . Every factor of q 2 must divide np 2 , and as p and q

have no common factor, every factor of q 2 must divide n. Hence n = λq 2 ,

where λ is an integer. But this involves m = λp 2 : and as m and n have

no common factor, λ must be unity. Thus m = p 2 , n = q 2 , as was to be

proved. In particular it follows, by taking n = 1, that an integer cannot

be the square of a rational number, unless that rational number is itself

integral.

It appears then that our requirements involve the existence of a num-

ber x and a point P, not one of the rational points already constructed,

such that A 0 P = x, x 2 = 2; and (as the reader will remember from ele-

mentary algebra) we write x =

√ 2.

The following alternative proof that no rational number can have its square

equal to 2 is interesting.

[I:4] REAL VARIABLES 7

Suppose, if possible, that p/q is a positive fraction, in its lowest terms, such

that (p/q) 2 = 2 or p 2 = 2q 2 . It is easy to see that this involves (2q − p) 2 =

2(p − q) 2 ; and so (2q − p)/(p − q) is another fraction having the same property.

But clearly q < p < 2q, and so p − q < q. Hence there is another fraction equal

to p/q and having a smaller denominator, which contradicts the assumption that

p/q is in its lowest terms.

Examples II. 1. Show that no rational number can have its cube equal

to 2.

2. Prove generally that a rational fraction p/q in its lowest terms cannot

be the cube of a rational number unless p and q are both perfect cubes.

3. A more general proposition, which is due to Gauss and includes those

which precede as particular cases, is the following: an algebraical equation

x n + p 1 x n−1 + p 2 x n−2 + ··· + p n = 0,

with integral coefficients, cannot have a rational but non-integral root.

[For suppose that the equation has a root a/b, where a and b are integers

without a common factor, and b is positive. Writing a/b for x, and multiplying

by b n−1 , we obtain

− a

n

b

= p 1 a n−1 + p 2 a n−2 b + ··· + p n b n−1 ,

a fraction in its lowest terms equal to an integer, which is absurd. Thus b = 1,

and the root is a. It is evident that a must be a divisor of p n .]

4. Show that if p n = 1 and neither of

1 + p 1 + p 2 + p 3 + …, 1 − p 1 + p 2 − p 3 + …

is zero, then the equation cannot have a rational root.

5. Find the rational roots (if any) of

x 4 − 4x 3 − 8x 2 + 13x + 10 = 0.

[The roots can only be integral, and so ±1, ±2, ±5, ±10 are the only possi-

bilities: whether these are roots can be determined by trial. It is clear that we

can in this way determine the rational roots of any such equation.]

[I:4] REAL VARIABLES 8

4. Irrational numbers (continued). The result of our geometrical

representation of the rational numbers is therefore to suggest the desirabil-

ity of enlarging our conception of ‘number’ by the introduction of further

numbers of a new kind.

The same conclusion might have been reached without the use of ge-

ometrical language. One of the central problems of algebra is that of the

solution of equations, such as

x 2 = 1, x 2 = 2.

The first equation has the two rational roots 1 and −1. But, if our con-

ception of number is to be limited to the rational numbers, we can only

say that the second equation has no roots; and the same is the case with

such equations as x 3 = 2, x 4 = 7. These facts are plainly sufficient to make

some generalisation of our idea of number desirable, if it should prove to

be possible.

Let us consider more closely the equation x 2 = 2.

We have already seen that there is no rational number x which satisfies

this equation. The square of any rational number is either less than or

greater than 2. We can therefore divide the rational numbers into two

classes, one containing the numbers whose squares are less than 2, and

the other those whose squares are greater than 2. We shall confine our

attention to the positive rational numbers, and we shall call these two

classes the class L, or the lower class, or the left-hand class, and the class R,

or the upper class, or the right-hand class. It is obvious that every member

of R is greater than all the members of L. Moreover it is easy to convince

ourselves that we can find a member of the class L whose square, though

less than 2, differs from 2 by as little as we please, and a member of R

whose square, though greater than 2, also differs from 2 by as little as we

please. In fact, if we carry out the ordinary arithmetical process for the

extraction of the square root of 2, we obtain a series of rational numbers,

viz.

1, 1.4, 1.41, 1.414, 1.4142, …

whose squares

1, 1.96, 1.9881, 1.999396, 1.99996164, …

[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, 2.002225, 2.00024449, …

are all greater than 2 but approximate to 2 as closely as we please.

The reasoning which precedes, although it will probably convince the reader,

is hardly of the precise character required by modern mathematics. We can

supply a formal proof as follows. In the first place, we can find a member of L

and a member of 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 .000001. In this chain there must be a last which belongs

to L and a first which belongs to R, and these two numbers differ by less than δ.

We can now prove that an x can be found in L and a y in R such that 2−x 2

and y 2 −2 are as small as we please, say less than δ. Substituting

1

4 δ for δ in the

argument which precedes, we see that we can choose x and y so that y−x <

1

4 δ;

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

y + x < 4, y 2 − x 2 = (y − x)(y + x) < 4(y − x) < δ;

and since x 2 < 2 and y 2 > 2 it follows a fortiori that 2−x 2 and y 2 −2 are each

less than δ.

It follows also that there can be no largest member of L or smallest

member of R. For if x is any member of L, then x 2 < 2. Suppose that

x 2 = 2−δ. Then we can find a member x 1 of L such that x 2

1

differs from 2

by less than δ, and so x 2

1

> x 2 or x 1 > x. Thus there are larger members

[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 posi-

tive 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 at-

tributes 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 cor-

responding point P on Λ such that P divides the points which correspond

to members of L from those which correspond to members of R.

A 0 P

R L R L R L R L R L

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, x 2 has a definite meaning. Then x 2 cannot be either less than or

greater than 2. For suppose, for example, that x 2 is less than 2. Then it

follows from what precedes that we can find a positive rational number ξ

such that ξ 2 lies between x 2 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 x 2 cannot be less than 2, and similarly

it cannot be greater than 2. We are therefore driven to the conclusion that

x 2 = 2, and that x is the number which in algebra we denote by

√ 2. And

of course this number

√ 2 is not rational, for no rational number has its

[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

x 2 = 2, almost word for word; for example to x 2 = N, where N is any

integer which is not a perfect square, or to

x 3 = 3, x 3 = 7, x 4 = 23,

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

existence of irrational numbers x and points P on Λ such that x satisfies

equations such as these, even when these lengths cannot (as

√ 2 can) be

constructed by means of elementary geometrical methods.

The reader will no doubt remember that in treatises on elementary algebra

the root of such an equation as x q = n is denoted by

q

√ n or n 1/q , and that a

meaning is attached to such symbols as

n p/q , n −p/q

by means of the equations

n p/q = (n 1/q ) p , n p/q n −p/q = 1.

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

such as

n r × n s = n r+s , (n r ) s = n rs

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

may, 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 difference 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 − x 2 < δ, y 2 − 2 < δ, then y − x < δ.

5. The equation x 2 = 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 2 2 is neither less than or greater

than 4.]

6. Irrational numbers (continued). In § 4 we discussed a special

mode of division of the positive rational numbers x into two classes, such

that x 2 < 2 for the members of one class and x 2 > 2 for those of the others.

Such a mode of division is called a section of the numbers in question. It

is plain that we could equally well construct a section in which the numbers

of the two classes were characterised by the inequalities x 3 < 2 and x 3 > 2,

or x 4 < 7 and x 4 > 7. Let us now attempt to state the principles of the

construction of such ‘sections’ of the positive rational numbers in quite

general terms.

Suppose that P and Q stand for two properties which are mutually

exclusive and one of which must be possessed by every positive rational

number. Further, suppose that every such number which possesses P is less

than any such number which possesses Q. Thus P might be the property

‘x 2 < 2’ and Q the property ‘x 2 > 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 ‘x 2 5 1’ and Q to be ‘x 2 > 1’; here l = 1.

If P is ‘x 2 < 1’ and Q is ‘x 2 = 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 ‘x 2 < 1’ and Q to be ‘x 2 > 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 5 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 de-

noted 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 0 would mean the

same as a > a 0 , if α and α 0 are the sections which correspond to a and a 0 .

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 sec-

tions 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 how-

ever 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 sec-

tions 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 ra-

tional 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

1

3

>

1

4

we can infer

1

2

>

1

4 ; the inference is in no

way affected by any doubt as to whether

1

2 ,

1

3 , and

1

4

are arithmetical fractions

or real numbers. Sometimes, of course, the context in which (e.g.) ‘ 1

2 ’ occurs is

sufficient to fix its interpretation. When we say (see § 9) that

1

2

<

q

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 neg-

ative rational numbers belong to the lower class and zero and all positive

rational numbers to the upper. We describe this section as the real num-

ber 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 x 3 < −2

and Q to be x 3 > −2, we should obtain a negative real number which is not

rational.

9. Relations of magnitude between real numbers. It is plain

that, now that we have extended our conception of number, we are bound

to make corresponding extensions of our conceptions of equality, inequality,

addition, multiplication, and so on. We have to show that these ideas can

be applied to the new numbers, and that, when this extension of them

is made, all the ordinary laws of algebra retain their validity, so that we

can operate with real numbers in general in exactly the same way as with

the rational numbers of § 1. To do all this systematically would occupy a

∗ 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 En-

glish 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

α

β

α

β

α

β

(i)

(ii)

(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 5 β, β < γ, or α < β, β 5 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 5 |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| ? 5 |α ± β| 5 |α| + |β|.

11. Algebraical operations with real numbers (continued).

(iii) Multiplication. When we come to multiplication, it is most con-

venient 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. Nat-

urally 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.

3. α × (1/α) = 1.

4. αβ = βα.

5. α(βγ) = (αβ)γ.

6. α(β + γ) = αβ + αγ.

7. (α + β)γ = αγ + βγ.

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 x 2 < 2, x 2 > 2. This was a section

of the positive rational numbers only; but we replace it (as was explained

in § 8) by a section of all the rational numbers. We denote the section or

number thus defined by the symbol

√ 2.

The classes by means of which the product of

√ 2 by itself is defined

are (i) (aa 0 ), where a and a 0 are positive rational numbers whose squares

are less than 2, (ii) (AA 0 ), where A and A 0 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 x 2 = 2 has the two roots

√ 2 and − √ 2. Similarly we

could discuss the equations x 2 = 3, x 3 = 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 rational num-

ber, is called a pure quadratic surd. A number of the form a ±

√ b, where

a is rational, and

√ b is a pure quadratic surd, is sometimes called a mixed

quadratic surd.

The two numbers a ±

√ b are the roots of the quadratic equation

x 2 − 2ax + a 2 − b = 0.

Conversely, the equation x 2 + 2px + q = 0, where p and q are rational, and

p 2 − q > 0, has as its roots the two quadratic surds −p ±

p p 2

− q.

The only kind of irrational numbers whose existence was suggested by

the geometrical considerations of § 3 are these quadratic surds, pure and

mixed, and the more complicated irrationals which may be expressed in a

form involving the repeated extraction of square roots, such as

√ 2 +

q

2 +

√ 2 +

r

2 +

q

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,

q

2 +

√ 2,

r

2 +

q

2 +

√ 2.

2. The quadratic equation ax 2 +2bx+c = 0 has two real roots † if b 2 −ac > 0.

∗ See Ch. II, Misc. Exs. 22.

† I.e. there are two values of x for which ax 2

+ 2bx + c = 0. If b 2 − ac < 0 there

are no such values of x. The reader will remember that in books on elementary algebra

the equation is said to have two ‘complex’ roots. The meaning to be attached to this

statement will be explained in Ch. III.

When b 2 = 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 ±

√ b 2

− ac}/a. It is easy

to construct these lengths geometrically, first constructing

√ b 2

− ac. A much

more elegant, though less straightforward, construction is the following.

Draw a circle of unit radius, a diameter PQ, and the tangents at the ends

of the diameters.

Q ′ Q X Y

P

P ′

N

M

Fig. 5.

Take PP 0 = −2a/b and QQ 0 = −c/2b, having regard to sign. ∗ Join P 0 Q 0 ,

cutting the circle in M and N. Draw PM and PN, cutting QQ 0 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 PP 0 and QQ 0 will be drawn in the same direction. Verify

that P 0 Q 0 will not meet the circle if b 2 < ac, while if b 2 = ac it will be a tangent.

Verify also that if b 2 = ac the circle in the second construction will touch BC.

4. Prove that

√ pq = √ p × √ q,

p

p 2 q = 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,

q

25

2

=

5

2

√ 2,

and so

√ 8,

q

25

2

are similar surds. On the other hand, if M and N are

integers which have no common factor, and neither of which is a perfect

square,

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

√ M =

p

q

r

t

u ,

√ N =

r

s

r

t

u ,

where all the letters denote integers.

Then

√ MN is evidently rational, and therefore (Ex. ii. 3) integral.

Thus MN = P 2 , where P is an integer. Let a, b, c,… be the prime

factors of P, so that

MN = a 2α b 2β c 2γ …,

where α, β, γ,… are positive integers. Then MN is divisible by a 2α , and

therefore either (1) M is divisible by a 2α , or (2) N is divisible by a 2α , or

(3) M and N are both divisible by a. The last case may be ruled out,

since M and N have no common factor. This argument may be applied to

each of the factors a 2α , b 2β , c 2γ ,…, so that M must be divisible by some

of these factors and N by the remainder. Thus

M = P

2

1 ,

N = P

2

2 ,

where P 2

1

denotes the product of some of the factors a 2α , b 2β , c 2γ ,… and

P 2

2

the product of the rest. Hence M and N are both perfect squares,

which is contrary to our hypothesis.

Theorem. If A, B, C, D are rational and

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.

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

both are rational).

3. If a and b are rational, then

√ a+ √ b cannot be rational unless √ a and √ b

are rational. The same is true of

√ a − √ b, unless a = b.

4. If

√ A + √ B = √ C + √ D,

then either (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 above.]

5. Neither (a +

√ b) 3

nor (a −

√ b) 3

can be rational unless

√ b is rational.

6. Prove that if x = p +

√ q, where p and q are rational, then x m , where

m is any integer, can be expressed in the form P + Q √ q, where P and Q are

rational. For example,

(p +

√ q) 2

= p 2 + q + 2p √ q, (p +

√ q) 3

= p 3 + 3pq + (3p 2 + q) √ q.

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

of the form

a 0 x n + a 1 x n−1 + ··· + a n ,

where a 0 , …, a n 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 square, is the root of an algebraical

equation with rational coefficients, then a −

√ b is another root of the same

equation.

8. Express 1/(p+ √ q) in the form prescribed in Ex. 6. [Multiply numerator

and denominator by p −

√ q.]

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

where G(x) and H(x) are polynomials in x with rational coefficients, can be

expressed in the form P + Q √ q, where P and Q are rational.

10. If p, q, and p 2 − q are positive, we can express

p p + √ q in the form

√ x + √ y, where

x =

1

2 {p +

p

p 2 − q}, y =

1

2 {p −

p

p 2 − q}.

11. Determine the conditions that it may be possible to express

p p + √ q,

where p and q are rational, in the form

√ x + √ y, where x and y are rational.

12. If a 2 − b is positive, the necessary and sufficient conditions that

q

a +

√

b +

q

a −

√

b

should be rational are that a 2 − b and

1

2 {a +

√ a 2

− 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

q

4 +

√ 15 +

3

q

4 −

√ 15.

Our argument for supposing that the expression for z has a meaning might be

as follows. We first show, as in § 12, that there is a number y =

√ 15 such that

y 2 = 15, and we can then, as in § 10, define the numbers 4+

√ 15, 4− √ 15. Now

consider the equation in z 1 ,

z 3

1

= 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 x 3 = 2

(or any other rational number) also leads us to the conclusion that there is a

number z 1 such that z 3

1

= 4+ √ 15. We thus define z 1 =

3

p

4 +

√ 15, and similarly

we can define z 2 =

3

p

4 −

√ 15; and then, as in § 10, we define z = z

1 + z 2 .

Now it is easy to verify that

z 3 = 3z + 8.

And we might have given a direct proof of the existence of a unique number z

such that z 3 = 3z + 8. It is easy to see that there cannot be two such numbers.

For if z 3

1

= 3z 1 + 8 and z 3

2

= 3z 2 + 8, we find on subtracting and dividing by

z 1 −z 2 that z 2

1 +z 1 z 2 +z 2 2

= 3. But if z 1 and z 2 are positive z 3

1

> 8, z 3

2

> 8 and

therefore z 1 > 2, z 2 > 2, z 2

1

+ z 1 z 2 + z 2

2

> 12, and so the equation just found is

impossible. And it is easy to see that neither z 1 nor z 2 can be negative. For if

z 1 is negative and equal to −ζ, ζ is positive and ζ 3 −3ζ +8 = 0, or 3−ζ 2 = 8/ζ.

Hence 3 − ζ 2 > 0, and so ζ < 2. But then 8/ζ > 4, and so 8/ζ cannot be equal

to 3 − ζ 2 , which is less than 3.

Hence there is at most one z such that z 3 = 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 z 3 = 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 x 3 < 3x + 8 or x 3 > 3x + 8. It is easy to see that if

x 3 > 3x + 8 and y is any number greater than x, then also y 3 > 3y + 8. For

suppose if possible y 3 5 3y+8. Then since x 3 > 3x+8 we obtain on subtracting

y 3 − x 3 < 3(y − x), or y 2 + xy + x 2 < 3, which is impossible; for y is positive

and x > 2 (since x 3 > 8). Similarly we can show that if x 3 < 3x + 8 and y < x

then also y 3 < 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 z 3 = 3z + 8. The reader who knows how to solve cubic equations

by Cardan’s method will be able to obtain the explicit expression of z directly

from the equation.

(ii) The direct argument applied above to the equation x 3 = 3x + 8

could be applied (though the application would be a little more difficult)

to the equation

x 5 = 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 x 5 = 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 A 0 A 1 , 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

π 2 = n, π 3 = n, π 5 = 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 ex-

ceptional 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-

ually. And when we think of them individually, we may think either of a

particular specified number (such as 1, − 1

2 ,

√ 2, or π) or we may think of

any number, an unspecified number, the number x. This last is our point

of view when we make such assertions as ‘x is a number’, ‘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 5

√ 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 suffice to define one 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 ‘x 2 < 2’ and ‘x 2 > 2’. Every rational number possesses one or other

of the properties, but not every real number, since in either case

√ 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 L 1 and R 1 denote the classes formed from L and R by taking

only the rational members of L and R. Then the classes L 1 and R 1 form

a section of the rational numbers. There are now two cases to distinguish.

It may happen that L 1 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 L 1 ; and this is plainly a contradiction.

Hence α is the greatest member of L.

On the other hand it may happen that L 1 has no greatest member. In

this case the section of the rational numbers formed by L 1 and R 1 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 5 x 5 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 5 α 5 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 what-

ever, 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 ξ 1 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 ξ 1 , we can take

an interval [ξ −δ 1 ,ξ +δ 1 ] surrounding ξ but not reaching as far as ξ 1 . But

this interval also must contain a point, say ξ 2 , which is a member of S and

does not coincide with ξ. Obviously we may repeat this argument, with

ξ 2 in the place of ξ 1 ; and so on indefinitely. In this way we can determine

as many points

ξ 1 , ξ 2 , ξ 3 , …

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 man-

ner. 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

a 1 +

a 2

1 · 2

+

a 3

1 · 2 · 3

+ ··· +

a k

1 · 2 · 3…k

,

[I:19] REAL VARIABLES 35

where a 1 , a 2 , …, a k are integers, and

0 5 a 1 , 0 5 a 2 < 2, 0 5 a 3 < 3, … 0 < a k < k.

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

as a simple continued fraction

a 1 +

1

a 2 +

1

a 3 +

1

··· +

1

a n

,

where a 1 , a 2 ,… are positive integers, of which the first only may be zero.

[Accounts of the theory of such continued fractions will be found in text-

books of algebra. For further information as to modes of representation of

rational and irrational numbers, see Hobson, Theory of Functions of a Real

Variable, pp. 45–49.]

4. Find the rational roots (if any) of 9x 3 − 6x 2 + 15x − 10 = 0.

5. A line AB is divided at C in aurea sectione (Euc. ii. 11)—i.e. so that

AB · AC = BC 2 . 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. In what circumstances can

aA + b

cA + d , where a, b, c, d are

rational, be rational?

7. Some elementary inequalities. In what follows a 1 , a 2 ,… denote

positive numbers (including zero) and p, q,… positive integers. Since a p

1

− a p

2

and a q

1 − a

q

2

have the same sign, we have (a p

1 − a

p

2 )(a

q

1 − a

q

2 ) = 0, or

a p+q

1

+ a p+q

2

= a p

1 a

q

2 + a

q

1 a

p

2 ,

(1)

an inequality which may also be written in the form

a p+q

1

+ a p+q

2

2

=

? a p

1 + a

p

2

2

?? a q

1 + a

q

2

2

?

. (2)

By repeated application of this formula we obtain

a p+q+r+…

1

+ a p+q+r+…

2

2

=

? a p

1 + a

p

2

2

?? a q

1 + a

q

2

2

?? a r

1 + a r 2

2

?

…, (3)

[I:19] REAL VARIABLES 36

and in particular

a p

1 + a

p

2

2

=

? a

1 + a 2

2

? p

. (4)

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

of the inequality a 2

1 + a 2 2

= 2a 1 a 2 , 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 a 1 , a 2 , …, a n ,

and add the results, we obtain the inequality

n P a p+q =

P a p P a q ,

(5)

or

?P

a p+q ? /n = {( P a p )/n}{( P a q )/n}. (6)

Hence we can deduce an obvious extension of (3) which the reader may formulate

for himself, and in particular the inequality

( P a p )/n = {( P 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 a 1 , a 2 , …, a n is not less than their

geometric mean. Suppose that a r and a s 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 a r and a s by

a 0 r = G, a 0 s = a r a s /G,

we do not alter the value of the geometric mean; and, since

a 0 r + a 0 s − a r − a s = (a r − G)(a s − G)/G 5 0,

we certainly do not increase the arithmetic mean.

It is clear that we may repeat this argument until we have replaced each of

a 1 , a 2 , …, a n by G; at most n repetitions will be necessary. As the final value

of the arithmetic mean is G, the initial value cannot have been less.

[I:19] REAL VARIABLES 37

10. Schwarz’s inequality. Suppose that a 1 , a 2 , …, a n and b 1 , b 2 , …, b n

are any two sets of numbers positive or negative. It is easy to verify the identity

( P a r b r ) 2 =

P a 2

r

P a 2

s −

P (a

r b s − a s b r ) 2 ,

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

( P a r b r ) 2 5

P a 2

r

P b 2

r ,

an inequality usually known as Schwarz’s (though due originally to Cauchy).

11. If a 1 , a 2 , …, a n are all positive, and s n = a 1 + a 2 + ··· + a n , then

(1 + a 1 )(1 + a 2 )…(1 + a n ) 5 1 + s n +

s 2

n

2!

+ ··· +

s n

n

n!

.

(Math. Trip. 1909.)

12. If a 1 , a 2 , …, a n and b 1 , b 2 , …, b n are two sets of positive numbers,

arranged in descending order of magnitude, then

(a 1 + a 2 + ··· + a n )(b 1 + b 2 + ··· + b n ) 5 n(a 1 b 1 + a 2 b 2 + ··· + a n b n ).

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

[Express

√ p in the form M +N √ q, where M 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.

16. Any polynomial in

√ p and √ q, with rational coefficients (i.e. any sum of

a finite number of terms of the form A( √ p) m ( √ q) n , where m and n are integers,

and A rational), can be expressed in the form

a + b √ p + c √ q + d √ pq,

[I:19] REAL VARIABLES 38

where a, b, c, d are rational.

17. Express

a + b √ p + c √ q

d + e √ p + f √ q

, where a, b, etc. are rational, in the form

A + B √ p + C √ q + D √ pq,

where A, B, C, D are rational.

[Evidently

a + b √ p + c √ q

d + e √ p + f √ q

=

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

(d + e √ p) 2 − f 2 q

=

α + β √ p + γ √ q + δ √ pq

? + ζ √ p

,

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

reduction may now be easily completed by multiplication of numerator and

denominator by ? − ζ √ p. For example, prove that

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

ax 2 + 2hxy + by 2 = 1, a 0 x 2 + 2h 0 xy + b 0 y 2 = 1

(where a, h, b, a 0 , h 0 , b 0 are rational) are rational, then

(h − h 0 ) 2 − (a − a 0 )(b − b 0 ), (ab 0 − a 0 b) 2 + 4(ah 0 − a 0 h)(bh 0 − b 0 h)

are both squares of rational numbers. (Math. Trip. 1899.)

20. Show that

√ 2 and √ 3 are cubic functions of √ 2 + √ 3, with rational

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

q

a + 2m

p

a − m 2 +

q

a − 2m

p

a − m 2

is equal to 2m if 2m 2 > a > m 2 , and to 2 √ a − m 2 if a > 2m 2 .

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

m

√ p with

rational coefficients can be expressed in the form

a 0 + a 1 α + a 2 α 2 + ··· + a m−1 α m−1 ,

where a 0 , a 1 ,… are rational and α =

m

√ p. For any such polynomial is of the

form

b 0 + b 1 α + b 2 α 2 + ··· + b k α k ,

where the b’s are rational. If k 5 m − 1, this is already of the form required. If

k > m−1, let α r be any power of α higher than the (m−1)th. Then r = λm+s,

where λ is an integer and 0 5 s 5 m − 1; and α r = α λm+s = p λ α s . Hence we

can get rid of all powers of α higher than the (m − 1)th.

23. Express (

3

√ 2 − 1) 5

and (

3

√ 2 − 1)/(

3

√ 2 + 1) in the form a + b

3

√ 2 + c

3

√ 4,

where a, b, c are rational. [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 rational, then a = 0, b = 0, c = 0.

[Let y =

3

√ 2. Then y 3

= 2 and

cy 2 + by + a = 0.

Hence 2cy 2 + 2by + ay 3 = 0 or

ay 2 + 2cy + 2b = 0.

Multiplying these two quadratic equations by a and c and subtracting, we

obtain (ab−2c 2 )y+a 2 −2bc = 0, or y = −(a 2 −2bc)/(ab−2c 2 ), a rational number,

which is impossible. The only alternative is that ab − 2c 2 = 0, a 2 − 2bc = 0.

[I:19] REAL VARIABLES 40

Hence ab = 2c 2 , a 4 = 4b 2 c 2 . If neither a nor b is zero, we can divide the

second equation by the first, which gives a 3 = 2b 3 : 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 equation 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

a 0 + a 1 p 1/m + ··· + a m−1 p (m−1)/m = 0,

p not being a perfect mth power, then a 0 = a 1 = ··· = a m−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 rational numbers.

26. If

3

√ A +

3

√ B +

3

√ C = 0, then either one of A, B, C is zero, and the

other two equal 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

q

7 + 5 √ 2 = α + β √ 2.

28. If (a − b 3 )b > 0, then

3

s

a +

9b 3 + a

3b

r

a − b 3

3b

+

3

s

a −

9b 3 + a

3b

r

a − b 3

3b

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

(

α + β

r

a − b 3

3b

) 3

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 = l 1 + m 1 α + ··· + r 1 α (n−1) ,

[I:19] REAL VARIABLES 41

where l 1 , m 1 ,… are rational, as in Ex. 22.

Similarly

x 2 = l 2 + m 2 a + … + r 2 a (n−1) ,

…………………………

x n = l n + m n a + … + r n a (n−1) .

Hence

L 1 x + L 2 x 2 + ··· + L n x n = ∆,

where ∆ is the determinant

?

l 1 m 1 … r 1

l 2 m 2 … r 2

…………….

l n m n … r n

?

and L 1 , L 2 ,… the minors of l 1 , l 2 ,….]

30. Apply this process to x = p +

√ q, and deduce the theorem of § 14.

31. Show that y = a + bp 1/3 + cp 2/3 satisfies the equation

y 3 − 3ay 2 + 3y(a 2 − bcp) − a 3 − b 3 p − c 3 p 2 + 3abcp = 0.

32. Algebraical numbers. We have seen that some irrational numbers

(such as

√ 2) are roots of equations of the type

a 0 x n + a 1 x n−1 + ··· + a n = 0,

where a 0 , a 1 , …, a n 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 x m/n , where m and n are any integers.

33. If x and y are algebraical numbers, then so are x+y, x−y, xy and x/y.

[We have equations

a 0 x m + a 1 x m−1 + … + a m = 0,

b 0 y n + b 1 y n−1 + … + b n = 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

c 0 z p + c 1 z p−1 + ··· + c p = 0,

satisfied by z. Similarly for the other cases.]

34. If

a 0 x n + a 1 x n−1 + ··· + a n = 0,

where a 0 , a 1 , …, a n are any algebraical numbers, then x is an algebraical

number. [We have n + 1 equations of the type

a 0,r a m r

r

+ a 1,r a m r −1

r

+ ··· + a m r ,r = 0 (r = 0, 1, …, n),

in which the coefficients a 0,r , a 1,r ,… are integers. Eliminate a 0 , a 1 , …, a n

between these and the original equation for x.]

35. Apply this process to the equation x 2 − 2x √ 2 +

√ 3 = 0.

[The result is x 8 − 16x 6 + 58x 4 − 48x 2 + 9 = 0.]

36. Find equations, with rational coefficients, satisfied by

1 +

√ 2 + √ 3,

√ 3 + √ 2

√ 3 − √ 2 ,

q √

3 +

√ 2 +

q √

3 −

√ 2,

3

√ 2 +

3

√ 3.

37. If x 3 = x + 1, then x 3n = a n x + b n + c n /x, where

a n+1 = a n + b n , b n+1 = a n + b n + c n , c n+1 = a n + c n .

38. If x 6 +x 5 −2x 4 −x 3 +x 2 +1 = 0 and y = x 4 −x 2 +x−1, then y satisfies

a quadratic equation with rational coefficients. (Math. Trip. 1903.)

[It will be found that y 2 + 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 A 0 P = x, B 0 Q = y measured from fixed points A 0 , B 0 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 x

2

+ 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 x

2 +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 the functional relation makes the same value of y correspond to all values

of x. The same would be true were y equal to 1 or − 1

2

or

√ 2 instead of 0. Such

a function of x is called a constant.

3. Let y 2 = x. Then if x is positive this equation defines two values of y

corresponding to each value of x, viz. ± √ x. 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 p 0 per

unit of area on the piston, which balances the weight W, so that

W = Ap 0 .

Let v 0 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 approx-

imation 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

v 2

?

(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 gt

2

if 0 5 t 5 τ, h =

1

2 g(2τ−t)

2

if τ 5 t 5 3τ,

and generally

h =

1

2 g(2nτ − t)

2

if (2n − 1)τ 5 t 5 (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 an instance of a definition which only applies to a particular class of values

of x, viz. integral values. ‘The largest prime factor of

11

3

or of

√ 2 or of π’ 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. Thus 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 OAPB.

Imagine the point P marked on the diagram. This marking of the point P

may be regarded as showing that the value of y for x = a is b.

If to the value a of x correspond several values of y (say b, b 0 , b 00 ), we

have, instead of the single point P, a number of points P, P 0 , P 00 .

[II:21] FUNCTIONS OF REAL VARIABLES 47

O A X

Y

a

b

B P

B ′ P ′

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 0 , b 00 ,…) of y, and the corresponding point P (or

points P, P 0 , P 00 ,…), 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 = ρ 2 ,

or

x 2 + y 2 + 2Gx + 2Fy + C = 0,

where G 2 + F 2 − C > 0, represents a circle. The equation

Ax 2 + 2Hxy + By 2 + 2Gx + 2Fy + 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, MP = y. If

OP = r and MOP = θ, θ being an angle between 0 and 2π (measured in

the positive direction), it is evident that

x = rcosθ, y = rsinθ,

r =

p

x 2 + y 2 , cosθ : sinθ : 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. ∗

O M

x

r

y

θ

N P

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

rcos(θ − α) = p,

where p and α are constants. The equation r = 2acosθ represents a circle

passing through the origin; and the general equation of a circle is of the

form

r 2 + c 2 − 2rccos(θ − α) = A 2 ,

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 − ecosθ, where l > 0, e > 1. According to our definitions r must be positive

and therefore cosθ < 1/e: the equation represents one branch only of a hyperbola, the

other having the equation −l/r = 1 − ecosθ. 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 rep-

resentation. The examples which follow will give the reader a better

notion of the infinite variety of possible types of functions.

A. Polynomials. A polynomial in x is a function of the form

a 0 x m + a 1 x m−1 + ··· + a m ,

where a 0 , a 1 , …, a m are constants. The simplest polynomials are the

simple powers y = x, x 2 , x 3 , …, x m ,…. The graph of the function x m is

of two distinct types, according as m is even or odd.

First let m = 2. Then three points on the graph are (0,0), (1,1),

(−1,1). Any number of additional points on the graph may be found by

assigning other special values to x: thus the values

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 = x 2 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

(−1,1)

(1,1)

(0,0)

P 0

P 1

N

y = x 2

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 = x 2 is everywhere convex to the axis of x.

Let P 0 , P 1 (Fig. 8) be the points (x 0 ,x 2

0 ), (x 1 ,x 2 1 ). Then the coordinates of a

point on the chord P 0 P 1 are x = λx 0 + µx 1 , y = λx 2

0 + µx 2 1 , where λ and µ are

positive numbers whose sum is 1. And

y − x 2 = (λ + µ)(λx 2

0 + µx

2

1 ) − (λx 0 + µx 1 )

2

= λµ(x 1 − x 0 ) 2 = 0,

so that the chord lies entirely above the curve.

The curve y = x 4 is similar to y = x 2 in general appearance, but flatter

near O, and steeper beyond the points A, A 0 (Fig. 9), and y = x m , where

m is even and greater than 4, is still more so. As m gets larger and larger

the flatness and steepness grow more and more pronounced, until the curve

is practically indistinguishable from the thick line in the figure.

The reader should next consider the curves given by y = x m , when m is

odd. The fundamental difference between the two cases is that whereas

when m is even (−x) m = x m , so that the curve is symmetrical about OY ,

when m is odd (−x) m = −x m , so that y is negative when x is negative.

[II:23] FUNCTIONS OF REAL VARIABLES 52

O M N

A

A ′

y = x 2

y = x 4

Fig. 9.

O

A

A ′

y = x

y = x 3

Fig. 10.

Fig. 10 shows the curves y = x, y = x 3 , and the form to which y = x m

approximates for larger odd values of m.

It is now easy to see how (theoretically at any rate) the graph of any

polynomial may be constructed. In the first place, from the graph of y = x m

we can at once derive that of Cx m , where C is a constant, by multiplying

the ordinate of every point of the curve by C. And if we know the graphs

of f(x) and F(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 = 7x 4 , y = 3x 5 , y = x 10 .

[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

x 10 + 3x 5 + 7x 4

(or even x 10 +30x 5 +700x 4 ), it is the first term which is of really preponderant

importance when x is fairly large. Thus even when x = 4, x 10 > 1,000,000,

while 30x 5 < 35,000 and 700x 4 < 180,000; while if x = 10 the preponderance of

the first term is still more marked.]

2. Compare the relative magnitudes of x 12 , 1,000,000x 6 , 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 ax 2 + 2bx + c.

[Here y −{(ac−b 2 )/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−b 2 )/a, the new equation

is y 0 = ax 02 . The curve is a parabola.]

4. Trace the curves y = x 3 − 3x + 1, y = x 2 (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 x p + ax p−1 + bx p−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 6= 0.

If x = 0 it takes the form 0/0, which is meaningless. Thus the function x/x is

equal to 1 if x 6= 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

?

1

x + 1

+

1

x − 1

???

1

x

+

1

x − 2

?

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

x 2 (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 (x 2 − 7)/(x 2 − 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 = P 1 (x)/Q 1 (x),

we obtain on simplification an equation of the form z = P 2 (x)/Q 2 (x).

(6) It is in no way presupposed in the definition of a rational function that

the constants which occur as coefficients should be rational numbers. The word

[II:25] FUNCTIONS OF REAL VARIABLES 55

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

function. Thus

x 2 + x +

√ 3

x

3

√ 2 − 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 oper-

ations 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 stan-

dard type of rational function. The most general kind of function which can be

obtained in this way is sufficiently illustrated by the example

x

x 2 + 1

+

2x + 7

x 2 +

11x − 3 √ 2

9x + 1

!, ?

17 +

2

x 3

?

,

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 depend-

ing 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/x 2 , y = 1/x 3 , ….

[The figures show the graphs of the first two curves. It should be observed

that since 1/0, 1/0 2 , … are meaningless expressions, these functions are not

defined for x = 0.]

[II:26] FUNCTIONS OF REAL VARIABLES 56

y = 1/x

(−1,−1)

(1,1)

Fig. 11.

y = 1/x 2

Fig. 12.

2. Trace y = x+(1/x), x−(1/x), x 2 +(1/x 2 ), x 2 −(1/x 2 ) and ax+(b/x)

taking various values, positive and negative, for a and b.

3. Trace

y =

x + 1

x − 1 ,

? x + 1

x − 1

? 2

,

1

(x − 1) 2

,

x 2 + 1

x 2 − 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/x m 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 func-

tions which can be generated from x by a finite number of operations such

as those used in generating rational functions, together with a finite num-

ber of operations of root extraction. Thus

√ 1 + x −

3

√ 1 − x

√ 1 + x +

3

√ 1 − x ,

√ x +

q

x +

√ x,

x 2 + x +

√ 3

x

3

√ 2 − π

! 2

3

are explicit algebraical functions, and so is x m/n (i.e.

n

√ x m ), 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 such an equation as y =

√ x. We have, up to the present, regarded

(e.g.)

√ 2 as denoting the positive square root of 2, and it would be natural

to denote by

√ x, where x is any positive number, the positive square

root of x, in which case y =

√ x would be a one-valued function of x. It is

however often more convenient to regard

√ x as standing for the two-valued

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

The reader will observe that, when this course is adopted, the func-

tion

√ x differs fundamentally from rational functions in two respects. In

the first place a rational function is always defined for all values of x with

a certain number of isolated exceptions. But

√ 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.

p (x − a)(b − x), where a < b, is defined only for

a 5 x 5 b. If a < x < b it has two values: if x = a or b only one, viz. 0.

2. Consider similarly

p

(x − a)(x − b)(x − c) (a < b < c),

p

x(x 2 − a 2 ),

3

p

(x − a) 2 (b − x) (a < b),

√ 1 + x − √ 1 − x

√ 1 + x + √ 1 − x ,

q

x +

√ x.

3. Trace the curves y 2 = x, y 3 = x, y 2 = x 3 .

4. Draw the graphs of the functions

y =

p

a 2 − x 2 , y = b

p

1 − (x 2 /a 2 ).

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

if

y =

√ 1 + x −

3

√ 1 − x

√ 1 + x +

3

√ 1 − x ,

[II:27] FUNCTIONS OF REAL VARIABLES 58

then

?

1 + y

1 − y

? 6

=

(1 + x) 3

(1 − x) 2

;

or if

y =

√ x +

q

x +

√ x,

then

y 4 − (4y 2 + 4y + 1)x = 0.

Each of these equations may be expressed in the form

y m + R 1 y m−1 + ··· + R m = 0, (1)

where R 1 , R 2 , …, R m are rational functions of x: and the reader will

easily verify that, if y is any one of the functions considered in the last set

of examples, y satisfies an equation of this form. 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 + p x + √ x +

3

√ 1 + x

x −

√ x + p x + √ x −

3

√ 1 + x .

Then we have the equations

y =

x + u + v + w

x − u + v − w

,

u 2 = x, v 2 = x + u, w 3 = 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 m th degree in y, whose

coefficients are rational functions of x. There is plainly no loss of generality

in supposing the first coefficient to be unity.

[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 y 2 + R 1 y + R 2 = 0, so that

y =

1

2 {−R 1

±

q

R 2

1 − 4R 2 }.

This function is defined for all values of x for which R 2

1

= 4R 2 . It has two values

if R 2

1

> 4R 2 and one if R 2

1

= 4R 2 .

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

y 2 − 2y − x 2 = 0, y 2 − 2y + x 2 = 0, y 4 − 2y 2 + x 2 = 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 + p 1/x,

3

√ x +

3

p

1/x,

q

x +

√ x,

r

x +

q

x +

√ x.

5. Consider the equation y 4 = x 2 .

[Here y 2 = ±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

y m + R 1 (z)y m−1 + … + R m (z) = 0,

where

z n + S 1 (x)z n−1 + … + S n (x) = 0.

Eliminating z we find an equation of the form

y p + T 1 (x)y p−1 + … + T p (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

y 5 − 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 in-

cludes 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 func-

tions. 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 proper-

ties. ∗

∗ The definitions of the circular functions given in elementary trigonometry presup-

pose 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 cosx, sinx, and acosx+bsinx.

[Since acosx+bsinx = β cos(x−α), where β =

√ a 2

+ b 2 , and α is an angle

whose cosine and sine are a/ √ a 2 + b 2 and b/ √ a 2 + b 2 , the graphs of these three

functions are similar in character.]

2. Draw the graphs of cos 2 x, sin 2 x, acos 2 x + bsin 2 x.

3. Suppose the graphs of f(x) and F(x) drawn. Then the graph of

f(x)cos 2 x + F(x)sin 2 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) = x 2 .

4. Show that the graph of cospx+cosqx lies between those of 2cos

1

2 (p−q)x

and −2cos

1

2 (p+q)x, touching each in turn. Sketch the graph when (p−q)/(p+q)

is small. (Math. Trip. 1908.)

5. Draw the graphs of x + sinx, (1/x) + sinx, xsinx, (sinx)/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 xsin(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 x 2 sin(1/x), (1/x)sin(1/x), sin 2 (1/x), {xsin(1/x)} 2 ,

acos 2 (1/x) + bsin 2 (1/x), sinx + sin(1/x), sinxsin(1/x).

9. Draw the graphs of cosx 2 , sinx 2 , acosx 2 + bsinx 2 .

10. Draw the graphs of arccosx and arcsinx.

[If y = arccosx, x = cosy. 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 5 x 5 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

tanx, cotx, secx, cosecx, tan 2 x, cot 2 x, sec 2 x, cosec 2 x.

12. Draw the graphs of arctanx, arccotx, arcsecx, arccosecx. Give for-

mulae (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 cosx and sinx 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 cosx and sinx 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 alge-

braical function of x.

[II:29] FUNCTIONS OF REAL VARIABLES 63

[Let the equation which defines the algebraical function be

y m + R 1 y m−1 + ··· + R m = 0 (1)

where R 1 , … are rational functions of x. This may be put in the form

P 0 y m + P 1 y m−1 + ··· + P m = 0,

where P 0 , P 1 , … are polynomials in x. Arguing as above, we see that

P 0 k m + P 1 k m−1 + ··· + P m = 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 0 , …; i.e. it is composed of a certain number of constants.]

16. The inverse sine and inverse cosine are not rational or algebraical func-

tions. [This follows from the fact that, for any value of x between −1 and +1,

arcsinx and arccosx have infinitely many values.]

29. F. Other classes of transcendental functions. Next in

importance to the trigonometrical functions come the exponential and log-

arithmic 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 el-

liptic 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

0 1 2

Fig. 15a.

0 1 2

Fig. 15b.

3. y =

p x − [x]. (Fig. 15c.)

4. y = [x] +

p x − [x]. (Fig. 15d.)

5. y = (x − [x]) 2 , [x] + (x − [x]) 2 .

6. y = [ √ x], [x 2 ],

√ x − [ √ x], x 2

− [x 2 ], [1 − x 2 ].

0 1 2

Fig. 15c.

0 1 2

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 =

p (1 + p 2 )/(1 + q 2 ) 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 discon-

tinuous, 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

p (1 + p 2 )/(1 + q 2 ) 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, ex-

cept the one point (1,1). Again, if p < q,

p (1 + p 2 )/(1 + q 2 ) > p/q; if p > q,

p (1 + p 2 )/(1 + q 2 ) < 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,

p (1 + p 2 )/(1 + q 2 ) 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 un-

known 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 equa-

tion (1).

Examples XVII. 1. The quadratic equation ax 2 + 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 = x 2 , y = −(2bx + c)/a.

But the most elementary method is probably to draw the circle

a(x 2 + y 2 ) + 2bx + c = 0,

whose centre is (−b/a,0) and radius { √ b 2 − ac}/a. The abscissae of its inter-

sections with the axis of x are the roots of the equation.

2. Solve by any of these methods

x 2 + 2x − 3 = 0, x 2 − 7x + 4 = 0, 3x 2 + 2x − 2 = 0.

3. The equation x m + ax + b = 0. This may be solved by constructing

the curves y = x m , y = −ax − b. Verify the following table for the number of

[II:31] FUNCTIONS OF REAL VARIABLES 68

roots of

x m + ax + b = 0 :

(a)m even

(

b positive, two or none,

b negative, two;

(b)m odd

(

a positive, one,

a negative, three or one.

Construct numerical examples to illustrate all possible cases.

4. Show that the equation tanx = ax+b has always an infinite number of

roots.

5. Determine the number of roots of

sinx = x, sinx =

1

3 x,

sinx =

1

8 x,

sinx =

1

120 x.

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

x − a =

1

2 π sin

2

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 rela-

tion 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 vari-

able. 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 par-

allel 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 = ρ 2 ,

or

x 2 + y 2 + z 2 + 2Fx + 2Gy + 2Hz + C = 0,

where F 2 + G 2 + H 2 − C > 0, represents a sphere; and so on. For proofs

of these propositions we must again refer to text-books of Analytical Ge-

ometry.

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 no-

tation 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 = x 2 , sinx, acos 2 x + bsin 2 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,

y 5 −y−x = 0 or x 5 +y 5 −ay = 0, it is known to be impossible to express y

explicitly as an algebraical function of x. If

x 2 + y 2 + 2Gx + 2Fy + C = 0,

y can indeed be so expressed, viz. by the formula

y = −F +

√ F

2

− x 2 − 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 = acost, y = asint.

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 x 2 + y 2 = a 2 , 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, x 2 − y 2 = 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+t 2 ), y/a = (1−t 2 )/(1+t 2 ),

when t varies through all real values?

33. Loci in space. In space of three dimensions there are two fun-

damentally 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 = ρ 2 , 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. ∗

1000

2000

3000

4000

5000

5000

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 semi-

vertical angle of the cone. The equation of the cone (which must be regarded as

extending both ways from its vertex) is x 2 + y 2 − z 2 tan 2 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 x 2 = z 2 tan 2 α.

That is to say, the equation of the surface generated by the revolution of the

curve y = 0, x 2 = z 2 tan 2 α round the axis of z is derived from the second of

these equations by changing x 2 into x 2 + y 2 . 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

p

x 2 + y 2 = 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 tanαcost, y = z tanαsint,

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 PQ, 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 Pq = h). Unfasten the end Q of the rod PQ and turn PQ

about P until Q can be fastened to the third circular rod in the position Q 0 .

The angle qOQ 0 = α in the figure is evidently given by

l 2 − h 2 = qQ 02 =

? 2asin

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

3sinx + 4cosx, sin

?

π

√ 2 sinx

?

.

(Math. Trip. 1896.)

4. Draw the graphs of the functions

sinx(acos 2 x + bsin 2 x),

sinx

x

(acos 2 x + bsin 2 x),

? sinx

x

? 2

.

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

6. Draw the graphs of the functions

(i) arccos(2x 2 − 1) − 2arccosx,

(ii) arctan

a + x

1 − ax

− arctana − arctanx,

where the symbols arccosa, arctana 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 x 3 +px+q = 0 are the abscissae of the points of

intersection (other than the origin) of the parabola y = x 2 and the circle

x 2 + y 2 + (p − 1)y + qx = 0.

9. The roots of x 4 +nx 3 +px 2 +qx+r = 0 are the abscissae of the points

of intersection of the parabola x 2 = y −

1

2 nx and the circle

x 2 + y 2 + ( 1

8 n

2

−

1

2 pn +

1

2 n + q)x + (p − 1 −

1

4 n

2 )y + r = 0.

10. Discuss the graphical solution of the equation

x m + ax 2 + bx + c = 0

[II:33] FUNCTIONS OF REAL VARIABLES 77

by means of the curves y = x m , y = −ax 2 − bx − c. Draw up a table of the

various possible numbers of roots.

11. Solve the equation secθ + cosecθ = 2 √ 2; and show that the equation

secθ + cosecθ = c has two roots between 0 and 2π if c 2 < 8 and four if c 2 > 8.

12. Show that the equation

2x = (2n + 1)π(1 − cosx),

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 xsinx = 1 has four roots between −π and π.

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

(1) cotx + x −

3

2 π = 0,

(2) x 2 + sin 2 x = 1,

(3) tanx = 2x/(1 + x 2 ),

(4) sinx − x +

1

6 x

3

= 0,

(5) (1 − cosx)tanα − x + sinx = 0.

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

the values α, β, γ is

α (x − b)(x − c)

(a − b)(a − c)

+ β

(x − c)(x − a)

(b − c)(b − a)

+ γ

(x − a)(x − b)

(c − a)(c − b)

.

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

when x = a 1 , a 2 , … a n , the values α 1 , α 2 , … α n .

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 −

x 2

x + (x − 1)

r

2 − x

3

[II:33] FUNCTIONS OF REAL VARIABLES 78

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

1

6 ,

1

3 ,

1

2 ,

2

3 ,

5

6 , 1, and use tables. For which of these values is the formula exact?]

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 = sinx + siny, z =

sinxsiny, z = sinxy, z = sin(x 2 + y 2 )?

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 ax 2 +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 mea-

sured by any irrational number which is defined by any combination of square

roots, however complicated. Thus

4

v

u

t

s

17 + 3 √ 11

17 − 3 √ 11

−

s

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 should begin by constructing

√ 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 x 2 −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 = ρ 2

(or x 2 +y 2 +2gx+2fy +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, x 2 + y 2 + 2gx + 2fy + 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

p (x

1 − x 2 ) 2 + (y 1 − y 2 ) 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 con-

struct 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, that is to say of the construction 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, Squaring 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

ax 2 +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 NM 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 length, are rational lengths.

25. Constructions for

3

√ 2. O is the vertex and S the focus of the parabola

y 2 = 4x, and P is one of its points of intersection with the parabola x 2 = 2y.

Show that OP 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

(x 2 + y 2 )x − y 2 = 0

(the Cissoid of Diocles). Sketch the curve. Take along the axis of y a length

OD = 2. Let AD cut the 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 num-

ber’ 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 A 0 P along the line Λ

of Chap. I. It may be regarded as the mark of a point, viz. the point P

whose distance from A 0 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 PQ. 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 dis-

placements as equivalent whose lengths and senses are the same. Then the

displacement is completely specified by the length PQ = 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 PQ = [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 sym-

bol [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 PQ 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 PQ; (ii) the direction

of the displacement, which is determined by the angle which PQ 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

A

P

Q

R

S

B

O X

Y

Fig. 19.

in magnitude, direction, and sense. In other words, if PQ 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 PQ and RS as

equivalent, and write

PQ = 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 PQ, the

sense of motion from O to A being the same as that from P to Q. Then

PQ 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 = PQ = RS = [x,y].

35. Equivalence of displacements. Multiplication of displace-

ments by numbers. If ξ and η are the coordinates of P, and ξ 0 and η 0

those of Q, it is evident that

x = ξ 0 − ξ, y = η 0 − η.

The displacement from (ξ,η) to (ξ 0 ,η 0 ) is therefore

[ξ 0 − ξ,η 0 − η].

It is clear that two displacements [x,y], [x 0 ,y 0 ] are equivalent if, and

only if, x = x 0 , y = y 0 . Thus [x,y] = [x 0 ,y 0 ] if and only if

x = x 0 , y = y 0 . (1)

The reverse displacement QP would be [ξ −ξ 0 ,η −η 0 ], and it is natural

to agree that

[ξ − ξ 0 ,η − η 0 ] = −[ξ 0 − ξ,η 0 − η],

QP = −PQ,

these equations being really definitions of the meaning of the symbols

−[ξ 0 − ξ,η 0 − η], −PQ. 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 defi-

nition which enables us to attach any meaning to the expressions

PQ + P 0 Q 0 , [x,y] + [x 0 ,y 0 ].

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 QQ 1 be drawn equal and parallel to P 0 Q 0 , so that the result of

successive displacements PQ, P 0 Q 0 on a particle at P is to transfer it first

to Q and then to Q 1 then we should define the sum of PQ and P 0 Q 0 as

being PQ 1 . If then we draw OA equal and parallel to PQ, and OB equal

and parallel to P 0 Q 0 , and complete the parallelogram OACB, we have

PQ + P 0 Q 0 = PQ 1 = OA + OB = OC.

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

coordinates of B are x 0 , y 0 , then those of the middle point of AB are

1

2 (x + x

0 ), 1

2 (y + y

0 ), and those of C are x + x 0 , y + y 0 . Hence

[x,y] + [x 0 ,y 0 ] = [x + x 0 ,y + y 0 ], (3)

which may be regarded as the symbolic definition of addition of displace-

ments. We observe that

[x 0 ,y 0 ] + [x,y] = [x 0 + x,y 0 + y]

= [x + x 0 ,y + y 0 ] = [x,y] + [x 0 ,y 0 ]

[III:36] COMPLEX NUMBERS 85

P

Q

O

A

B

C

Q 1

Q 2

P ′

Q ′

Fig. 20.

In other words, addition of displacements obeys the commutative law ex-

pressed in ordinary algebra by the equation a+b = b+a. This law expresses

the obvious geometrical fact that if we move from P first through a dis-

tance PQ 2 equal and parallel to P 0 Q 0 , and then through a distance equal

and parallel to PQ, we shall arrive at the same point Q 1 as before.

In particular

[x,y] = [x,0] + [0,y]. (4)

Here [x,0] denotes a displacement through a distance x in a direction par-

allel to OX. It is in fact what we previously denoted by [x], when we were

considering only displacements along a line. We call [x,0] and [0,y] the

components of [x,y], and [x,y] their resultant.

When we have once defined addition of two displacements, there is no

further difficulty in the way of defining addition of any number. Thus, by

[III:36] COMPLEX NUMBERS 86

definition,

[x,y] + [x 0 ,y 0 ] + [x 00 ,y 00 ] = ([x,y] + [x 0 ,y 0 ]) + [x 00 ,y 00 ]

= [x + x 0 ,y + y 0 ] + [x 00 ,y 00 ] = [x + x 0 + x 00 ,y + y 0 + y 00 ].

We define subtraction of displacements by the equation

[x,y] − [x 0 ,y 0 ] = [x,y] + (−[x 0 ,y 0 ]), (5)

which is the same thing as [x,y] + [−x 0 ,−y 0 ] or as [x − x 0 ,y − y 0 ]. In

particular

[x,y] − [x,y] = [0,0].

The displacement [0,0] leaves the particle where it was; it is the zero

displacement, and we agree to write [0,0] = 0.

Examples XX. 1. Prove that

(i) α[βx,βy] = β[αx,αy] = [αβx,αβy],

(ii) ([x,y] + [x 0 ,y 0 ]) + [x 00 ,y 00 ] = [x,y] + ([x 0 ,y 0 ] + [x 00 ,y 00 ]),

(iii) [x,y] + [x 0 ,y 0 ] = [x 0 ,y 0 ] + [x,y],

(iv) (α + β)[x,y] = α[x,y] + β[x,y],

(v) α{[x,y] + [x 0 ,y 0 ]} = α[x,y] + α[x 0 ,y 0 ].

[We have already proved (iii). The remaining equations follow with equal

ease from the definitions. The reader should in each case consider the geometrical

significance of the equation, as we did above in the case of (iii).]

2. If M is the middle point of PQ, then OM =

1

2 (OP + OQ).

More

generally, if M divides PQ in the ratio µ : λ, then

OM =

λ

λ + µ

OP +

µ

λ + µ

OQ.

3. If G is the centre of mass of equal particles at P 1 , P 2 , …, P n , then

OG = (OP 1 + OP 2 + ··· + OP n )/n.

[III:36] COMPLEX NUMBERS 87

4. If P, Q, R are collinear points in the plane, then it is possible to find

real numbers α, β, γ, not all zero, and such that

α · OP + β · OQ + γ · OR = 0;

and conversely. [This is really only another way of stating Ex. 2.]

5. If AB and AC are two displacements not in the same straight line, and

α · AB + β · AC = γ · AB + δ · AC,

then α = γ and β = δ.

[Take AB 1 = α·AB, AC 1 = β ·AC. Complete the parallelogram AB 1 P 1 C 1 .

Then AP 1 = α · AB + β · AC. It is evident that AP 1 can only be expressed in

this form in one way, whence the theorem follows.]

6. ABCD is a parallelogram. Through Q, a point inside the parallelogram,

RQS and TQU are drawn parallel to the sides. Show that RU, TS intersect

on AC.

A B T

D C U

R S

Q

Fig. 21.

[Let the ratios AT : AB, AR : AD be denoted by α, β. Then

AT = α · AB, AR = β · AD,

AU = α · AB + AD, AS = AB + β · AD.

Let RU meet AC in P. Then, since R, U, P are collinear,

AP =

λ

λ + µ

AR +

µ

λ + µ

AU,

[III:37] COMPLEX NUMBERS 88

where µ/λ is the ratio in which P divides RU. That is to say

AP =

αµ

λ + µ

AB +

βλ + µ

λ + µ

AD.

But since P lies on AC, AP is a numerical multiple of AC; say

AP = k · AC = k · AB + k · AD.

Hence (Ex. 5) αµ = βλ + µ = (λ + µ)k, from which we deduce

k =

αβ

α + β − 1 .

The symmetry of this result shows that a similar argument would also give

AP 0 =

αβ

α + β − 1

AC,

if P 0 is the point where TS meets AC. Hence P and P 0 are the same point.]

7. ABCD is a parallelogram, and M the middle point of AB. Show that

DM trisects and is trisected by AC. ∗

37. Multiplication of displacements. So far we have made no

attempt to attach any meaning whatever to the notion of the product of two

displacements. The only kind of multiplication which we have considered

is that in which a displacement is multiplied by a number. The expression

[x,y] × [x 0 ,y 0 ]

so far means nothing, and we are at liberty to define it to mean anything

we like. It is, however, fairly clear that if any definition of such a product

is to be of any use, the product of two displacements must itself be a

displacement.

We might, for example, define it as being equal to

[x + x 0 ,y + y 0 ];

∗ The two preceding examples are taken from Willard Gibbs’ Vector Analysis.

[III:37] COMPLEX NUMBERS 89

in other words, we might agree that the product of two displacements was

to be always equal to their sum. But there would be two serious objections

to such a definition. In the first place our definition would be futile. We

should only be introducing a new method of expressing something which

we can perfectly well express without it. In the second place our definition

would be inconvenient and misleading for the following reasons. If α is a

real number, we have already defined α[x,y] as [αx,αy]. Now, as we saw

in § 34, the real number α may itself from one point of view be regarded

as a displacement, viz. the displacement [α] along the axis OX, or, in our

later notation, the displacement [α,0]. It is therefore, if not absolutely

necessary, at any rate most desirable, that our definition should be such

that

[α,0][x,y] = [αx,αy],

and the suggested definition does not give this result.

A more reasonable definition might appear to be

[x,y][x 0 ,y 0 ] = [xx 0 ,yy 0 ].

But this would give

[α,0][x,y] = [αx,0];

and so this definition also would be open to the second objection.

In fact, it is by no means obvious what is the best meaning to attach to

the product [x,y][x 0 ,y 0 ]. All that is clear is (1) that, if our definition is to

be of any use, this product must itself be a displacement whose coordinates

depend on x and y, or in other words that we must have

[x,y][x 0 ,y 0 ] = [X,Y ],

where X and Y are functions of x, y, x 0 , and y 0 ; (2) that the definition

must be such as to agree with the equation

[x,0][x 0 ,y 0 ] = [xx 0 ,xy 0 ];

[III:38] COMPLEX NUMBERS 90

and (3) that the definition must obey the ordinary commutative, distribu-

tive, and associative laws of multiplication, so that

[x,y][x 0 ,y 0 ] = [x 0 ,y 0 ][x,y],

([x,y] + [x 0 ,y 0 ])[x 00 ,y 00 ] = [x,y][x 00 ,y 00 ] + [x 0 ,y 0 ][x 00 ,y 00 ],

[x,y]([x 0 ,y 0 ] + [x 00 ,y 00 ]) = [x,y][x 0 ,y 0 ] + [x,y][x 00 ,y 00 ],

and

[x,y]([x 0 ,y 0 ][x 00 ,y 00 ]) = ([x,y][x 0 ,y 0 ])[x 00 ,y 00 ].

38. The right definition to take is suggested as follows. We know that,

if OAB, OCD are two similar triangles, the angles corresponding in the

order in which they are written, then

OB/OA = OD/OC,

or OB · OC = OA · OD. This suggests that we should try to define

multiplication and division of displacements in such a way that

OB/OA = OD/OC, OB · OC = OA · OD.

Now let

OB = [x,y], OC = [x 0 ,y 0 ], OD = [X,Y ],

and suppose that A is the point (1,0), so that OA = [1,0]. Then

OA · OD = [1,0][X,Y ] = [X,Y ],

and so

[x,y][x 0 ,y 0 ] = [X,Y ].

The product OB · OC is therefore to be defined as OD, D being obtained

by constructing on OC a triangle similar to OAB. In order to free this

[III:38] COMPLEX NUMBERS 91

O A

B

C

D ′

D

Fig. 22.

definition from ambiguity, it should be observed that on OC we can de-

scribe two such triangles, OCD and OCD 0 . We choose that for which the

angle COD is equal to AOB in sign as well as in magnitude. We say that

the two triangles are then similar in the same sense.

If the polar coordinates of B and C are (ρ,θ) and (σ,φ), so that

x = ρcosθ, y = ρsinθ, x 0 = σ cosφ, y 0 = σ sinφ,

then the polar coordinates of D are evidently ρσ and θ + φ. Hence

X = ρσ cos(θ + φ) = xx 0 − yy 0 ,

Y = ρσ sin(θ + φ) = xy 0 + yx 0 .

The required definition is therefore

[x,y][x 0 ,y 0 ] = [xx 0 − yy 0 ,xy 0 + yx 0 ]. (6)

We observe (1) that if y = 0, then X = xx 0 , Y = xy 0 , as we desired;

(2) that the right-hand side is not altered if we interchange x and x 0 , and

y and y 0 , so that

[x,y][x 0 ,y 0 ] = [x 0 ,y 0 ][x,y];

[III:39] COMPLEX NUMBERS 92

and (3) that

{[x,y] + [x 0 ,y 0 ]}[x 00 ,y 00 ] = [x + x 0 ,y + y 0 ][x 00 ,y 00 ]

= [(x + x 0 )x 00 − (y + y 0 )y 00 ,(x + x 0 )y 00 + (y + y 0 )x 00 ]

= [xx 00 − yy 00 ,xy 00 + yx 00 ] + [x 0 x 00 − y 0 y 00 ,x 0 y 00 + y 0 x 00 ]

= [x,y][x 00 ,y 00 ] + [x 0 ,y 0 ][x 00 ,y 00 ].

Similarly we can verify that all the equations at the end of § 37 are

satisfied. Thus the definition (6) fulfils all the requirements which we made

of it in § 37.

Example. Show directly from the geometrical definition given above that

multiplication of displacements obeys the commutative and distributive laws.

[Take the commutative law for example. The product OB ·OC is OD (Fig. 22),

COD being similar to AOB. To construct the product OC ·OB we should have

to construct on OB a triangle BOD 1 similar to AOC; and so what we want to

prove is that D and D 1 coincide, or that BOD is similar to AOC. This is an

easy piece of elementary geometry.]