A Course of Pure Mathematics by G. H. Hardy

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.]