Table Of Content

eh Rl of ts What is Menai i i “ “Approach fo 8 6 eo and 2 1 ry 4 0 bein nal Richard Courant an Herbert Robbins revised by Lall Stewart MATHEMATICS “A lucid representation of the fundamental comeepts and methods af the whole flld of mathematics... Easily understandable.” ‘Albert Einstein® ‘Witten for beginmers and scholars. for students and teachers, for philosophers and engineers, What is Mathematics? is a sparkling collection of mathematical gems that coffers an entertaining ane accessible portrait of the mathematical world. Brought up ta date with a new chapter by Tan Stewart, this second edition offers new insights into revent mathematical developments snd describes proofs of the Four-Color Theorem and Feewat’s Last Theorean, problems that wen sti open when Courant and Rubbins | ‘wrote this mastespiece, but ones that have since been solved. A marvelously lilerate stary, What is Mathematics? opens a window onto the world cof mathematics “*Pratse forthe frst edition: “Without doubt, the work will have grest influence. lt should be inthe hands of everyone, professional oc otherwise, who is interested in scientific titking.” The New York Times “A work of extranedinary perfection.” Mathematical Reviews “Excellent... Shoold prove a spurce of great pleasure and satisfaction” Journal of Appted Physics “This book is 8 work of art.” “Marston Morse —“Ihis.a-wotk of high perfoction....Iris astonishing to whut extent Whar is Mathematics? | hhas xucceeded in makiny clear by means ofthe simplest examples all the fondumnental ideas and metherts which we mathematicians consider the life blood of aur science.” Hlerman Weyl The late Richard Courant, headed the Deparunent of Mathematics at New York University sand was Director of the Institute of Mathematical Sciences, which was subseypenty renamed the Courant Iasitte of Mathematical Sciences, His book Mathematical Phyics is famifiar to every physicist, acd his book Differential and fategral Calculus is acknowledged tnbe ane of te best presentations of the subject written in modern times. Hechert Rob is New Jersey Professor of Mathématical Statist at Rutgers University. lam Stewart is Professor af Mathematics atthe Univecsity of Warwick, and author of Nanwe's Numbers and Does Gut Play Dice? He also writes the “Mathematical Recreations” coluau in Scientific ‘American. In 1985 he was swarded the Koval Society's Michael Faraday medal for signifi cant contribution 10 the public understanding of science. oon oxford Paperbacks (Onfond University Press 9 Maotest105 193! US.51995 ISBN 0-19-510519-2 WHAT IS MATHEMATICS? WHAT IS Mathematics? AN ELEMENTARY APPROACH TO IDEAS AND METHODS Second Edition BY RICHARD COURANT Law of te ourant Institute uf Mathematical Seenees Naw York University AND HERBERT ROBBINS igen Caiversity Revised by IAN STEWART Mathomatiesbattune University of Warwick New York = Qafond QXFORD UNIVERSITY PRESS 1908 Oxford University Press xiont New York Adlens Auckland Bangkok Bogoli Bombay Dyewos Aine Caleutta CapeTown Dar es Salaara Toth Florence Heng Rang. Isuanbet Kararhy MeairuChy Sawnbs Paris Singapore Tayet Tokyo Torotio ed asia comics it eel ttunta Copyright @ 1941 (renewed 1969) by Richard Courant; Revisions copyright © 1996 by Oxford University Press, Inc. ies yublisbee in (REL hy (isford Unhersity Press. 10 Shida Avenue, Now Vo, New Fark £1 Fs boed 2 an Oxford University Press payer, (Aa First putisteilan a second eeivon, 198 Oxford i x piste Irecempanic of Oxford ceray Prove All rigtis reserved: Ne_part of this publication muy be reproduced. stored ina retsewal stony, oF Laseanitted tn ary Zoran er hy shy means, elcetranie, mechanical, photrenying eecurdig, or omhersiee, ‘withont the prior permission af (xtird University Pr Lileary of Congres: Catalogang.in Prbficatina Data Courant, Wend, IRB-LOTE hat i mathematics? : an elementary approdelt viens and cues 2 hy Bohard Courant and Herbert Robbins, —2nd od. recived ty lan Sewwary, beth, Inelikles bibim-raphical rfereices aid Andes, ISBN GP 51051.2 L. Mathematics, 1. Robbins, Jortert, IL Stewart. 015. lL Tle GATTZCUN Iss Bio deze ot Buen Printed oe the Urited States of Antorics eid fee paper DEDICATED TO. ERNEST, GERTRUDE, HANS, AND LEONORE COURANT CONTENTS PRerAace 10 SecoND Eorrion PRerat To Revises Rorrions Prerace 10 Fiast Eoin How To Use Tz Book Wat is MattieManies? Cuarnen I Tae: NaTuran Nemes Inteod SI. Calculation with Integers . peeprenereneerenerrenerenns 1 1. Laws of Arithmetic. 2 The Representation of Integers. 2 ‘Computation in Systims Other thar the Devional, $2. The Infinitude of the Number System, Mathematical Induction .. 8 1. The Principle of Mathematical Induction. 2. The Arithmetival Progres- sion, 3. The Geometrical Progression. 4 The Sum of the First 7% Squares. 5 AN Important Inequality. 6 The Binomial Theo- rem. 7. Kurther Remarles on Mathematiral Induction. UPPLEMENT TO CHAPTER I. THE THEORY OF NUMBERS... . at Introduction beste tees settee BD $1. The Prime Numbers. 0... fee BL 1, Fundamental Fucts, 2. The Distribution of the Primes. a Formulas Producing Primes. b. Primesin Arithmetival Progressions. ¢, The Prime Number Theorem. d. Two Unsolved Problems Concerning Prime Num- bers. G2 Congmences 5.6. tere ects al wneral Concepts, 2. Fermat's ‘Theurem, gulratic Hesiduess 3. Pythagorean Numbers and Fermat's Last Theorem .... cee a0 4. The Euclidean Algorithm. . 2 1. General Theory. 2. Application to the Fwidamental Theoret of Arith- metic 3 Euler's ¢ Function, Fermat's Theorem Again, 4, Continued Fractions. Diophantine Equations, Csr TL, Tae Neues Syores OF MAtMeMATIOS 626s se 82 Introduction - . sei tees $1. The Rational Nouibors. Ed 1. Rational Numbers as w Device for Measuring. 2. Intrinsic Need for the Rational Numbers. Principal of Generalization. 3%, Geometrical Interpre tation of Rational Numbers $2. Incommensurable Segments, Irrational Numbers, and the Concept of Limit sees . we BB 1 Introdnetion. 2, Decimal Fractions, Infirute Decimals. i. Lint its, Infinite Geometzical Series. 4, Rational Numbers and Periodie Dect. CONTENTS mals, 5, Gouerd Definijon of Irrational Numbers by Nested Interval 1 Methods. Defining —E LN bers, Dedekind Cuts. §4. Remarks on Analytic Geametry...... sete TB 1. The Basie Principle. 2. Bquations of Lines and Curves, 4. The Mathematical Analysis of Infinity, 1. Fundatental Concepts, 2, The Denuimeranilty of the Rett bers and the Nor-Denumerability of the Continuum. 3, Cantor's "Cardinal Numbers." 4. The Indirect Method of Proof. 5. The Paradaxes of the In- finite. 6 The Foundations of Mathematics, $5. Complex Numbers... 88 1, The Origin of Complex Numbers. 2, The Geometncal Interpretation of Complex Numbers, 3, De Moivre's Formula and the Roots of Unity. 4. The Fundamental Theorem of Algebra. $8, Algebraic and Transcendental Numbers. . 1, Definition and Existence. 2. Liouville’ Theorem and the Conetiction of Transcendental Numbers Scppusmenr 10 CHAVTER Il, THE ALCRRRA OF SETS. .-. cetteeee 108 1. General Theory. 2. Applicution to Mathematical Logie, 3. An Appl: 1908 atlon 10 the Theary af Probability. Ceaprer IT. Grosermica: CONSTRLICTIONS. Tan: At Introduction cesses sees Part I. Impossibility Proofs unl Algebra... : 20 Aor Nownen Fes... 17 §1. Fundamental Geometriea! Constructions. pore 120 1. Construction of Fields and Square Root Extraction. 2. Regular Poly gous. 2 Apollonins’ Problera, 82, Constrietible Numbers and Number Fields. . wr 1, General Theory. 2. All Constructible Numbers are Algebraic. §3. The Unsolvability of the Three Greek Problems... see TB 1. Doubling the Cube. 2 A ‘Theorem on Cubie Equations. 3. Trisecting the Angle. 4. The Regular Heptagon. 3. Remarks on the Problem of Squaring the Circe, Part Il, Various Methods for Performing ConstrucWvits.....6..06 seo 110 $4. Geomemnral Transfomations Inversion... cee 140 1. General Remarks. 2 Properties of Inversion. 4. Geometzical Mruction of lnverse Poinls, 4. How to Bisect a Segment and Find the Cen: wer oF a CCE WIT Uke Compass AIOE. $8. Constructions with Other Tools, Mascheroni Constructions sith Compass ‘Alone 6 1. A Chussical Construction for Doubling the Cube. 2. Kestriction to the Gao of the Compass Alone. 3, Drawing wl Mechanical tnstni- ments, Meviouical Curves. Cyctolds, 4. Linkages, Peaucellfor’s and Hants lnversors. $6. Mare Ahout Inversions and its Applications 188 1 Invariance of Angles. Families of Circles, 2. Appiteation 10 the Prob. fem of Apollonius. 4, Repeated Reflections. ArTeR IV. PROJECTME GEOMETRY, AXIOMATICS. NOS-EVCLIDEAX GEOMETRIES .. {6 $1. Introduction, beers oe 168 CONTENTS 1. Classuieaton of Geometrical Peoperns. tnvarkince under Truusfor: nations. 2. Projective Transformations 2. Fundantental Concepts, 1. The Group of Projective Transformations. 2. Desargues’s Theorem. $8. Cross-Ratto : 1. Definition and Proof of Invarivnce. 2. Application tm tue Complete Quadritateral 4. Parallelism and Lefinity L, Punts at Infinity as “Ideal Pomis” 2 vents and: Prose tion, 3. Croas-Rano with Blenents at hf 4&5. Applications, : . an bes 1 Preinninary Henares. 2. Proof uf Desangues's Theron in ake Pane, 3, Paieats Theorem. l, Brianchon’s Theorem, 5, Remark on Duality $6. Analytic Representation : posse as L Inveudhctory Retnarks. 2 Henegeurous Coordinates. The Aigel Basis of Duality. 37, Problems on (onstnictions wth the Straightedge Alone. $8. Conies and Quadne Surfaces 1 Elemenuary Metcir Geomerry of Conkes 2 Prolectxe Properties of Conies. 3. Conios as Line Curves. 4. Paseats ant Brianchon’s General Tueorens for Canies 5. The Uyperbotald A. Axionnaticn and Nog. Euctidean Geometry inst ras De 1 The Axiomatic Method. 2. Hyperbolic Non-Buchdean Grome uy. 3. Geometry and Reality. 4. Poincant’s Model fi. Rliptir or Rie nannian Geomenry: APPrSDIX, Granta IS MORE: THAN Take Disizceass Agproact. Charree 8. Tupauocy Introduction , bee SL. Buler's Formula for Palyhedra £2. Topological Properties of Figures 1 Topwlogical Propertien 2, Connectivity $3. Other Exaniples of Tapalogical Thnoreuts, £, The dorian Curve Thearom. 2, Tie Four Cilor Problewt, 8. The t'ot- cept of Dimension. 4. 4 Festal Poi Theoreits 3 Kauty SA. The Topatogival Classification of Surfaces. 1 The Genus of a Surface. 2 The Baler face 3% One-Sided Surives, APDERDI, {The Five Color Theorem, yons. 3. The Rundaeiental TI Cuarren VE Pexcioss ast 1M Intraduction, 1, Variable aud Furenun : 1 Dettrutious ard Exaniples 2. Radian Measure of Angles 3 The Graph uf a Puntetion, livere Fintettons 4. Compotuud Rune: teristic af an Shur The Jordan Corve Theneen far Vol ‘ores of Agee CONTENTS, ens, 5, Continuity, 6. Punetions of Several Variables. 7. Funcuons and Transfonmatiows 2 Limits - Piette . : 1. The Lilt of Sequence &,, 2, Monoture Sequences, & Euler's Nu- bere, 4. The Number x, 5. Continued Fractions 8. LAmits by Continuous Approach, . 1 Introduction General Definition, 2. Remarks on the Limit Con- cept, 3. The Limit of sin rar. 4. Livaits aay — =, §4. Precise Definition of Continuity, $5. Two Putamental Theorems on Continuous Pmethons pees L. Bolzano's Theorem. 2. Proof of Rolzano's Theorem. 3, Weierstrass” Theorem on Extreme Values, 4, 4 Theorem on Sequences: Compact Sets Ps) ans $6, Same Applications of Boizano’s haerem tees o ar \. Geometrical Applications, 2. Appticution to 4 Probie in Mecianies SUPPLEMES' 10 CHAPTER VIL MORE EXAMPLES ON LIMITS AND CONTINUITY 22 $4. Examples of Limits 000000 esc seeeeeeiietenes 322 1. General Remarks. 2. The Lint of q. 3. The Limit of 4p. 4. Discon tinuous Functions as Linuts of Continuous Functions. 5, Limits by Iter tian, 42. Exampte on Continuity oar Cuavree VIL Maxina ast Miniaa, 329, Introduetion a0 §1, Problems in Elementary Geometry. 330 1. Macxineum Area of a Triungle with Two Sides Given, 2. Teros’s THo- erem. Extremum Property of Light Rays. 3. opicatons fo Probjems on Triungles. 4. Tangent Properties of Etlpse and LMyper: wen Carve, §2, A General Principal Underiying Extreme Value Probjoms, .... A. The Principle. 2. Examples. 42. Stationary Points and the Differential Cateulus 1. Extrema and Stationary Points. 2. Maxima and Minima of Functions of Several Varables, Saddle Points, 3, Minimax Points and Topol ogy. 4. The Distance tron a Point te a Surtare. SA. Schward’s Telangle Prablen siete settee 1 Sehwarz’s Proof. 2 Another Proof. 3. Obtuse Triangles. 4, Triangles Formed by Light Rays. 5, Remarks Concerning Problems of Refiection and Ergodic Motion. $5. Steers Probfent wo ieee eecetees 1. Probiew and Solution. 2, Analysis of Ue Alternatives 3, A Comple- nientary Problen, 4. Remarks and Exercises. 5. Generatizalion to the Street Network Frobtew, Extrema und Inequalities 1, The Amthmeueal and Goomtrienl Mean af Twa Positive Quantities, 2. Generalization to 1 Variables. 3. The Method of Least Squares. §7. The Existence of an Extrentum. Dirichiot's Principle 6 246 3B 36 68