The Project Gutenberg EBook #38769: A Course Of Pure Mathematics
The Project Gutenberg EBook of A Course of Pure Mathematics, byG. H. (Godfrey Harold) HardyThis eBook is for the use of anyone anywhere at no cost and withalmost no restrictions whatsoever. You may copy it, give it away orre-use it under the terms of the Project Gutenberg License includedwith this eBook or online at www.gutenberg.netTitle: A Course of Pure MathematicsThird EditionAuthor: G. H. (Godfrey Harold) HardyRelease Date: February 5, 2012 [EBook #38769]Language: EnglishCharacter set encoding: ISO-8859-1*** START OF THIS PROJECT GUTENBERG EBOOK A COURSE OF PURE MATHEMATICS ***
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A COURSEOFPURE MATHEMATICS
CAMBRIDGE UNIVERSITY PRESSC. F. CLAY, ManagerLONDON: FETTER LANE, E.C. 4NEW YORK : THE MACMILLAN CO.BOMBAY CALCUTTA MACMILLAN AND CO., Ltd. MADRASTORONTO : THE MACMILLAN CO. OFCANADA, Ltd.TOKYO : MARUZEN-KABUSHIKI-KAISHAALL RIGHTS RESERVED
A COURSEOFPURE MATHEMATICSBYG. H. HARDY, M.A., F.R.S.FELLOW OF NEW COLLEGESAVILIAN PROFESSOR OF GEOMETRY IN THE UNIVERSITYOF OXFORDLATE FELLOW OF TRINITY COLLEGE, CAMBRIDGETHIRD EDITIONCambridgeat the University Press1921
First Edition 1908Second Edition 1914Third Edition 1921
PREFACE TO THE THIRD EDITIONNo extensive changes have been made in this edition. The most important are in §§ 80–82, which I have rewritten in accordance with suggestionsmade by Mr S. Pollard.The earlier editions contained no satisfactory account of the genesis ofthe circular functions. I have made some attempt to meet this objectionin § 158 and Appendix III. Appendix IV is also an addition.It is curious to note how the character of the criticisms I have had tomeet has changed. I was too meticulous and pedantic for my pupils offifteen years ago: I am altogether too popular for the Trinity scholar ofto-day. I need hardly say that I find such criticisms very gratifying, as thebest evidence that the book has to some extent fulfilled the purpose withwhich it was written.G. H. H.August 1921EXTRACT FROM THE PREFACE TO THESECOND EDITIONThe 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 aproof of Weierstrass’s theorem concerning points of condensation; in Chapter IV an account of ‘limits of indetermination’ and the ‘general principle ofconvergence’; in Chapter V a proof of the ‘Heine-Borel Theorem’, Heine’stheorem 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 sectionon differentials. I have also rewritten in a more general form the sectionswhich deal with the definition of the definite integral. In order to findspace 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 ofminor alterations.G. H. H.October 1914EXTRACT FROM THE PREFACE TO THE FIRSTEDITIONThis book has been designed primarily for the use of first year studentsat the Universities whose abilities reach or approach something like what isusually described as ‘scholarship standard’. I hope that it may be useful toother classes of readers, but it is this class whose wants I have consideredfirst. It is in any case a book for mathematicians: I have nowhere madeany attempt to meet the needs of students of engineering or indeed anyclass of students whose interests are not primarily mathematical.I regard the book as being really elementary. There are plenty of hardexamples (mainly at the ends of the chapters): to these I have added,wherever space permitted, an outline of the solution. But I have done mybest to avoid the inclusion of anything that involves really difficult ideas.For instance, I make no use of the ‘principle of convergence’: uniformconvergence, double series, infinite products, are never alluded to: andI prove no general theorems whatever concerning the inversion of limit 2f 2fand. In the last two chapters Ioperations—I never even define x y y xhave occasion once or twice to integrate a power-series, but I have confinedmyself to the very simplest cases and given a special discussion in eachinstance. Anyone who has read this book will be in a position to read withprofit Dr Bromwich’s Infinite Series, where a full and adequate discussionof all these points will be found.September 1908
CONTENTSCHAPTER IREAL 14.15.16.17.18.19.Rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Irrational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Relations of magnitude between real numbers . . . . . . . . . . . . . . . . .Algebraical operationswith real numbers . . . . . . . . . . . . . . . . . . . . . The number 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Quadratic surds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The continuous real variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Sections of the real numbers. Dedekind’s Theorem . . . . . . . . . . . .Points of condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Weierstrass’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .131416182122262930323434Decimals, 1. Gauss’s Theorem, 6. Graphical solution of quadraticequations, 22. Important inequalities, 35. Arithmetical and geometrical means, 35. Schwarz’s Inequality, 36. Cubic and other surds, 38.Algebraical numbers, 41.CHAPTER IIFUNCTIONS OF REAL e idea of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The graphical representation of functions. Coordinates . . . . . . . .Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Algebraical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Transcendental functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Graphical solution of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4346485053566067
CONTENTSSECT.viiiPAGE31.32.33.Functions of two variables and their graphical representation . .Curves in a plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Loci in space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68697175Trigonometrical 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 IIICOMPLEX ements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The quadratic equation with real coefficients . . . . . . . . . . . . . . . . . .Argand’s diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .De Moivre’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Rational functions of a complex variable . . . . . . . . . . . . . . . . . . . . . .Roots of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .819296100101104118121Properties 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 pointsshould be concyclic, 116. Complex functions of a real variable, 116.Construction of regular polygons by Euclidean methods, 120. Imaginarypoints and lines, 124.CHAPTER IVLIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE50.51.52.Functions of a positive integral variable . . . . . . . . . . . . . . . . . . . . . . . 128Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129Finite and infinite classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
85–86.87–88.Properties possessed by a function of n for large values of n . . .Definition of a limit and other definitions . . . . . . . . . . . . . . . . . . . . .Oscillating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .General theorems concerning limits . . . . . . . . . . . . . . . . . . . . . . . . . . .Steadily increasing or decreasing functions . . . . . . . . . . . . . . . . . . . .Alternative proof of Weierstrass’s Theorem . . . . . . . . . . . . . . . . . . .nThe limit of x. . . . . 1 n.The limit of 1 nSome algebraical lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The limit of n( n x 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Infinite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The infinite geometrical series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The representation of functions of a continuous real variable bymeans of limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The bounds of a bounded aggregate . . . . . . . . . . . . . . . . . . . . . . . . . . .The bounds of a bounded function . . . . . . . . . . . . . . . . . . . . . . . . . . . .The limits of indetermination of a bounded function . . . . . . . . . .The general principle of convergence . . . . . . . . . . . . . . . . . . . . . . . . . .Limits of complex functions and series of complex terms . . . . . .Applications to z n and the geometrical series . . . . . . . . . . . . . . . . .Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83185188190 xnOscillation of sin nθπ, 144, 146, 181. Limits of nk xn , n x, n n, n n!,,n! m nx , 162, 166. Decimals, 171. Arithmetical series, 175. Harmonicalnseries, 176. Equation xn 1 f (xn ), 190. Expansions of rational functions, 191. Limit of a mean value, 193.CHAPTER VLIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS ANDDISCONTINUOUS FUNCTIONS89–92.Limits as x or x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
xCONTENTSSECT.PAGE93–97.Limits as x a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .98–99.Continuous functions of a real variable . . . . . . . . . . . . . . . . . . . . . . . .100–104. Properties of continuous functions. Bounded functions. Theoscillation of a function in an interval . . . . . . . . . . . . . . . . . . . .105–106. Sets of intervals on a line. The Heine-Borel Theorem . . . . . . . . . .107.Continuous functions of several variables . . . . . . . . . . . . . . . . . . . . . .108–109. Implicit and inverse functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .200210215223228229233Limits and continuity of polynomials and rational functions, 204, 212.xm am, 206. Orders of smallness and greatness, 207. Limit ofLimit ofx asin x, 209. Infinity of a function, 213. Continuity of cos x and sin x, 213.xClassification of discontinuities, 214.CHAPTER VIDERIVATIVES AND erivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .General rules for differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Derivatives of complex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The notation of the differential calculus . . . . . . . . . . . . . . . . . . . . . . .Differentiation of polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Differentiation of rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . .Differentiation of algebraical functions . . . . . . . . . . . . . . . . . . . . . . . .Differentiation of transcendental functions . . . . . . . . . . . . . . . . . . . .Repeated differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .General theorems concerning derivatives. Rolle’s Theorem . . . .Maxima and minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Integration. The logarithmic function . . . . . . . . . . . . . . . . . . . . . . . . .Integration of polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Integration of rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .237244246246249251253255258262264274277281281
xiCONTENTSSECT.PAGE132–139. Integration of algebraical functions. Integration by rationalisation. Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .140–144. Integration of transcendental functions . . . . . . . . . . . . . . . . . . . . . . . .145.Areas of plane curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .146.Lengths of plane curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .286298302304308Derivative of xm , 241. Derivatives of cos x and sin x, 241. Tangent andnormal to a curve, 241, 257. Multiple roots of equations, 249, 309. Rolle’sTheorem 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 ofreduction, 314.CHAPTER VIIADDITIONAL THEOREMS IN THE DIFFERENTIAL AND 156–161.162.163.164.Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Taylor’s Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Applications of Taylor’s Theorem to maxima and minima . . . . .Applications of Taylor’s Theorem to the calculation of limits . .The contact of plane curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Differentiation of functions of several variables . . . . . . . . . . . . . . . .Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Definite Integrals. Areas of curves . . . . . . . . . . . . . . . . . . . . . . . . . . . .Alternative proof of Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . .Application to the binomial series . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Integrals of complex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Newton’s method of approximation to the roots of equations, 322. Series for cos x and sin x, 325. Binomial series, 325. Tangent to a curve,331, 346, 374. Points of inflexion, 331. Curvature, 333, 372. Osculating319324326327330335342347367368369370
xiiCONTENTSconics, 334, 372. Differentiation of implicit functions, 346. Fourier’s integrals, 355, 360. The second mean value theorem, 364. Homogeneous functions, 372. Euler’s Theorem, 372. Jacobians, 374. Schwarz’s inequality forintegrals, 378. Approximate values of definite integrals, 380. Simpson’sRule, 380.CHAPTER VIIITHE CONVERGENCE OF INFINITE SERIES AND INFINITE INTEGRALSSECT.PAGE165–168. Series of positive terms. Cauchy’s and d’Alembert’s tests of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .169.Dirichlet’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .170.Multiplication of series of positive terms . . . . . . . . . . . . . . . . . . . . . .171–174. Further tests of convergence. Abel’s Theorem. Maclaurin’s integral test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .P s175.The seriesn.176.Cauchy’s condensation test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .177–182. Infinite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .183.Series of positive and negative terms . . . . . . . . . . . . . . . . . . . . . . . . . .184–185. Absolutely convergent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .186–187. Conditionally convergent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .188.Alternating series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .189.Abel’s and Dirichlet’s tests of convergence . . . . . . . . . . . . . . . . . . . .190.Series of complex terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .191–194. Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .195.Multiplication of series in general . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .P k nThe seriesn r and allied series, 385. Transformation of infinite integralsbysubstitutionand integration by parts, 404, 406, 413. The seriesPPan cos nθ,an sin nθ, 419, 425, 427. Alteration of the sum of a seriesby rearrangement, 423. Logarithmic series, 431. Binomial series, 431, 433.Multiplication of conditionally convergent series, 434, 439. Recurring series, 437. Difference equations, 438. Definite integrals, 441. Schwarz’sinequality for infinite integrals, 35
CONTENTSxiiiCHAPTER IXTHE LOGARITHMIC AND EXPONENTIAL FUNCTIONS OF A 16.The logarithmic function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The functional equation satisfied by log x . . . . . . . . . . . . . . . . . . . . .The behaviour of log x as x tends to infinity or to zero . . . . . . . .The logarithmic scale of infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The number e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The exponential function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The general power ax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The exponential limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The logarithmic limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Common logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Logarithmic tests of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The exponential series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The logarithmic series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The series for arc tan x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The binomial series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Alternative development of the theory . . . . . . . . . . . . . . . . . . . . . . . .Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Integrals containing the exponential function, 460. The hyperbolic functions, 463. Integrals of certain algebraical functions, 464. Euler’s constant, 469, 486. Irrationality of e, 473. Approximation to surds by the binomial theorem, 480. Irrationality of log10 n, 483. Definite integrals, 491.CHAPTER XTHE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, ANDCIRCULAR FUNCTIONS217–218.219.220.221.Functions of a complex variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Curvilinear integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Definition of the logarithmic function . . . . . . . . . . . . . . . . . . . . . . . . .The values of the logarithmic function . . . . . . . . . . . . . . . . . . . . . . . .495496497499
231.The exponential function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The general power az . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The trigonometrical and hyperbolic functions . . . . . . . . . . . . . . . . .The connection between the logarithmic and inverse trigonometrical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .232.The exponential series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .233.The series for cos z and sin z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .234–235. The logarithmic series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .236.The exponential limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .237.The binomial series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .505507512518520522525529531534The functional equation satisfied by Log z, 503. The function ez , 509.Logarithms to any base, 510. The inverse cosine, sine, and tangent ofa complex number, 516. Trigonometrical series, 523, 527, 540. Roots oftranscendental equations, 534. Transformations, 535, 538. Stereographicprojection, 537. Mercator’s projection, 538. Level curves, 539. Definiteintegrals, e proof that every equation has a root . . . . . . . . . . . . . . .A note on double limit problems . . . . . . . . . . . . . . . . . . . . . . . .The circular functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The infinite in analysis and geometry . . . . . . . . . . . . . . . . . . .545553557560
CHAPTER IREAL VARIABLES1. Rational numbers. A fraction r p/q, where p and q are positive or negative integers, is called a rational number. We can suppose(i) that p and q have no common factor, as if they have a common factorwe can divide each of them by it, and (ii) that q is positive, sincep/( 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 arithmeticalrules for the manipulation of rational numbers. The examples which followdemand 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 coursemeaningless).2. If λ, m, and n are positive rational numbers, and m n, thenλ(m2 n2 ), 2λmn, and λ(m2 n2 ) are positive rational numbers. Hence showhow to determine any number of right-angled triangles the lengths of all ofwhose sides are rational.3. Any terminated decimal represents a rational number whose denominator contains no factors other than 2 or 5. Conversely, any such rational numbercan 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 simpleseries as follows:11,21,12,31,22,13,41,32,23,14,.Show that p/q is the [ 12 (p q 1)(p q 2) q]th term of the series.[In this series every rational number is repeated indefinitely. Thus 1 occurs1 2 3as 1 , 2 , 3 , . . . . We can of course avoid this by omitting every number which hasalready occurred in a simpler form, but then the problem of determining theprecise position of p/q becomes more complicated.]1
[I : 2]2REAL VARIABLES2. The representation of rational numbers by points on a line.It is convenient, in many branches of mathematical analysis, to make agood 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 areillustrations and nothing more, and are employed merely for the sake ofclearness of exposition. This being so, it is not necessary that we shouldattempt any logical analysis of the ordinary notions of elementary geometry; we may be content to suppose, however far it may be from the truth,that we know what they mean.Assuming, then, that we know what is meant by a straight line, asegment of a line, and the length of a segment, let us take a straight line Λ,produced indefinitely in both directions, and a segment A0 A1 of any length.We call A0 the origin, or the point 0, and A1 the point 1, and we regardthese points as representing the numbers 0 and 1.In order to obtain a point which shall represent a positive rationalnumber r p/q, we choose the point Ar such thatA0 Ar /A0 A1 r,A0 Ar being a stretch of the line extending in the same direction along theline as A0 A1 , a direction which we shall suppose to be from left to rightwhen, as in Fig. 1, the line is drawn horizontally across the paper. Inorder to obtain a point to represent a negative rational number r s,A sA 1A0A1AsFig. 1.it is natural to regard length as a magnitude capable of sign, positive ifthe length is measured in one direction (that of A0 A1 ), and negative ifmeasured in the other, so that AB BA; and to take as the pointrepresenting r the point A s such thatA0 A s A s A0 A0 As .
[I : 3]REAL VARIABLES3We thus obtain a point Ar on the line corresponding to every rationalvalue of r, positive or negative, and such thatA0 Ar r · A0 A1 ;and if, as is natural, we take A0 A1 as our unit of length, and writeA0 A1 1, then we haveA0 Ar r.We shall call the points Ar the rational points of the line.3. Irrational numbers. If the reader will mark off on the line allthe points corresponding to the rational numbers whose denominators are1, 2, 3, . . . in succession, he will readily convince himself that he can coverthe line with rational points as closely as he likes. We can state this moreprecisely as follows: if we take any segment BC on Λ, we can find as manyrational points as we please on BC.Suppose, for example, that BC falls within the segment A1 A2 . It isevident that if we choose a positive integer k so thatk · BC 1, (1)and divide A1 A2 into k equal parts, then at least one of the points ofdivision (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 partsinto which A1 A2 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 canfind 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 aswe please. We may express this by saying that BC includes infinitely manyrational points. The assumption that this is possible is equivalent to the assumption of what isknown as the Axiom of Archimedes.
[I : 3]REAL VARIABLES4The meaning of such phrases as ‘infinitely many’ or ‘an infinity of ’, in suchsentences as ‘BC includes infinitely many rational points’ or ‘there are an infinityof rational points on BC’ or ‘there are an infinity of positive integers’, will be
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