what is mathematics An Elementary Approach to Ideas and Methods

【实例简介】
本书详细介绍了数学的各个分支。对于有需要了解数学方法的程序员有很大的帮助。
WHAT IS MATHEMATICS? WHAT IS Mathematics 2 AN ELEMENTARY APPROACH TO IDEAS AND METHODS Second edition BY RICHARD COURANT Courant Institute of mathematical scienees New York truiversity AND HERBERT ROBBⅠNs Revised by IAN STEWART Mathematics Institute University of Warwick New york Orford OXFORD UNIVERSITY PRESS 1996 Oxford University Press Oxford New York Athens Auckian Bangkok Bogota bombay B Delli Flor ng Istanbul Karac hi Kuala Lumpur Madras Madrid Melb(name MexicoCity Nairobi Pinis Singapore [lI 2s Berlin ibadi Copyright o 194I(renewed 1969)by Richard Courant; Revisions copyright 1996 by Oxford University Press, Inc First published in i!Il by Oxford [niversity Press Inc 98 Sadist)n Avenue. New York. New York 1(M16 First issued as an Oxford ('iniversity Press paperback. 1978 First pilblislel as a second edition. 1991 Oxford is a registered trademark of Oxford (niversity I'ress All rights reserved. No part of this publication may be reproduced stored in a retrieval systen, or transmitte, in any form or by any means, clectronic, mechanical, pintorojying, recording, or otherwise without the pnor permission of Oxford tniversity Press Library of Congress CataloginginPublication Data (ourant Rinard, 1888-1972 by Richard(ourant and Herbert Robbins --2nd ed /revis( UF thods wliat is mathematics?: all elenentary approach t ideas and ill p.e'll. includes bibliographical references and inde n Stewart SBN(-15105192 I. Mathematics. 1. Robbins, Herbert. II. Stewart. lan, 14-4 III. Title QA:32C6194651dc209乐n I'rititedi in the ['nited Statts of Aerica o icici fre DEDICATED TO ERNEST, GERTRUDE, HANS, AND LEONORE COURANT CONTENTS PREFACE TO SECOND EDITION PREFACE TO REVISED EMITIONS PREFACE TO FIRST EDITION HOw TO USE THE BOOK WILAT IS MATHEMATIC S? CHAPTER I. THE NATURAL NUMBERS Introduction sl. Calculation with Integers 1. Laws of Arithmetic. 2. The Representation of Integers. 3 Computation in Systems Other than the decimal 82. The Infinitude of the Number System. Mathematical Induction 1. The Principle of Mathematical Induction. 2. The Arithmetical Progres- sion. 3. The Geometrical Progression. 4. The Sum of the First n quares. 5. An Important Inequality. 6. The Binomial Theo- rem. 7. Further Remarks on Mathematical Induction SUPPLEMENT TO CHAPTER I. THE THEORY OF NUMBERS Introduction §1. The Prime Numbers 222 I. Fundamental Facts. 2. The Distribution of the Primes. a formulas Producing Primes, b. Primes in Arithmetical Progressions. c. The Prime Number Theorem. d. Two Unsolved Problems Concerning Prime Num §2. Congruences l. General Concepts. 2. Fermat,'s Theorem. 3. Quadratic Residues. 83, Pythagorean Numbers and Fermats Last Theorem 40 s4. The Euclidean algorithm 1. General Theory. 2. Application to the Fundamental Theorem of Arith- metic. 3. Euler's o Function. Fermat's Theorem Again, 4. Continued Fractions. Diophantine equations CHAPTER IL. THE NUMBER SYSTEM OF MATHEMATICS 52 Introduction 1. The rational numbers 1, Rational Numbers as a Device for measuring. 2. Intrinsic Need for the Rational Numbers. Principal of Generalization. 3. Geometrical Interpre tation of Rational Numbers $2. Incommensurable segments, Irrational Nimbers, and the concept of 58 Introductio. 2. Decimal Fractions, Infinite Decimals its. Infinite Geometrical Series. 4. Rational numbers and periodic deci CONTENTS mals. 5. General Definition of Irrational Numbers by Nested Intervals. 6. Altemative Methods of Defining Irrational Num bers. Dedekind Cuts $3. Remarks on Analytic geometry 72 1. The Basic Principle. 2. Equations of Lines and curves. $4. The Mathematical Analysis of Infinity 1. Fundamental Concepts. 2. The Denumerability of the Rational Num bers and the non - Denumerability of the continuum. 3. Cantors Cardinal Numbers 4. The Indirect Method of Proof. 5. The Paradoxes of the ln finite. 6. The Foundations of Mathematics §5. Complex Numbers 88 1. The Origin of Complex Numbers. 2. The Geometrical Interpretation of Complex Numbers. 3. De Moivre's Formula and the roots of Cnity. 4. The Fundamental Theoren of Algebra. 86. Algebraic and Transcendental Numbers Definition and Existence. 2. Liouville's Theorem and the construction of Transcendental Numbers SUPPLEMENT TO CHAPTER II. THE ALGEBRA OF SETS 108 1. General Theory. 2. Application to Mathematical logic. 3. An Appli cation to the Theory of Probability. ChaPTeR II. GEOMETRICAL CONSTRUCTIONS. THE ALGEBRA OF NUMBER FIELDS,.. 117 Introduction 7 Part I. Impossibility Proofs and algebra.., .,..,.. 120 $1. Fundamental Geometrical Constructions 120 1. Construction of Fields and Square Root Extraction. 2. Reglar Poly- gons. 3. Apollonius"Problem. 92. Constructible Numbers and Number Fields ,,127 1. General Theory. 2. All Constructible Numbers are algebraic. The Regular Heptagon. 5. Remarks on the Problem of quaring the circle Part Il. Various Methods for Performing Constructions 140 $4. Geometrical Transformations. Inversion 140 1. General Remarks. 2. Properties of Inversion. 3. Geometrical Con struction of Inverse Points. 4. How to Bisect a segment and find the cen ter of a circle with the c Alone $5. Constructions with Other Tools. Mascheroni Constructions with Compass Alone 146 1. A Classical Construction for Doubling the Cube. 2. Restriction to the Use of the Compass Alone. 3. Drawing with Mechanical Instal ments. Mechanical Curves. Cycloids. 4. Linkages. Peaucellier's and Harts inversors S6. More aboli Inversions and its Applications 158 1. Invariance of Angles. Families of Circles. 2. Application io the Prob lem of apollonius. 3. Repeated reflections CHaPtEr I. PROJECTTVE GEOMETRY. AXIOMATICS. NON-ET'CLIDEAN GEOMETRIES §1. Introduction l66 CONTENTS 1. Classification of (icoltetrical properies. invariance alder transfor mations. 2. Projective transformations s2. Fundamental Concepts 168 1. The (iroup of Projective Transformations. 2. Desargues s Thorem 53. (ross Ratio 47 Definition and proof of invariance Application to the Complete Quadrilateral $4. Parallelism and Infinity 1. Points at Infinity as ldeal Points. 2. ldeal Elements and projec. tion. 3. Cross Rato with Elements at Infinity. $5. Applications 1. Prelininary Reimarks. 2. Proof of Desargues's Theorent in tite Plane. 3. Pascals Theor(ln. 1. Brianchon's Theore. 3. Remark on Duality 86. Analytic Representation 91 1. Intruhictor Remarks Hoinoge'lteous (ordinates. The algebraic Basis of Duality s7. Problems on Constructions with the Straightedge alone 8. Conics and quadric Surfaces 198 I. Elementary Metric Geometry of (onics. 2. Projective Properties of Conics. 3. Conics as Line Curves. Pas als anld Brianchon's General Theorems for Conics. 3. The Hyperboloid $9. Axiomatics and Non Euclidean (geometry The Axiomatic Method. 2. Hyperbolic Non- Euclidean (ieoine 3. Geometry and Reality. + Poincare s Model. 5. Elliptic or Rie niannian Geometry APPE.NDEX. GEOMETRY IN MORE: THAN THREE IIMENSICINS 27 1. IntrodhictiolL. 2. Analytic Approach. 3. (ieoluetrical or(oinbinatorial Approuch CIAPTEⅴ.TPLAκ Introduction Sl. Euler s Formula for Polyhedra 236 $2. Topological Properies of Figures 241 I. Topological Properties 2. Connectivity S&. (ther Exantples of Toplogical Theorems I. The ordan curve theore. 2. The Four ( olor problen, The colt cept of Dimentsion. 1. A Fixed F'oilll Theorell. 5 Knots K1. Tie Topological (lassification of Surfaces 256 I. The Genus of i Surface. 2. The Euler (haracteristic of a ir face, +3. (hle. Side d surfi AIlENDIN ag-1 I. The Five Colr Theoren. 2 2 The ordan ( e The orel for Polv gons. 33. The Fundaniental The oreint of Algebra CILAPTE VI. FT'NTR)\S AN, ListIT\ IntroductioN 2)2 l.Ⅴ arial and Firlctioll 27 I. Definitions anld Examples 2. Radiat Measure of Argles Giraptt of Function, Inverse Functions 4. (olmpou Func CONTENTS tions. 5, Continuity Functions of Several variables, 7. Functions and transformations. §2. Limits 289 I. The limit of a sequence a, 2. Monotone Sequences. 3. Enler s Nu ber e. 4. The Nunber I. 5. Continued fractions $3. Limits by Continuous Approach 303 1. Introduction. General Definition. 2. Rentarks on the Limit Con cept.3.T" he Limit of sin x/x∵.4. Linits as r→x $4. Precise Definition of Continuity 310 $5. Two Fundamental Theorems on Continuous Functions 312 I. Bolzano's theoren. 2. Proof of bolzano's Theorem Weierstrass Theore on Extreme values. 4. A Theorem on Sequences. copact Sets 66. Some applications of Botzano's Thoerem 317 I. Geometrical Applicatiols. 2. Application to a Problent in Mechanics St'PPLEMENT TO CHAPTER VI. MORE EXAMPLES ON LIMITS AND CONTINIETY 22 §1. Examples of limit 322 1. General Remarks. 2. The linit of d". 3. The Limit of " p. 4. Discon tinuous Functions as Linus of Continuous Functions. 5. Limits by itera- lion $2, Example on Continuity 327 CHAPTER VIL. MAXIMA AND MINIMA 329 Introduction Sl. Problems in Elementary Geometry 30 1. Maxintum area of a Triangle with Two Sides given. 2. Herons Tho- crem. Extremum Property of Light Rays. 3. Applications to Problens on Triangles. 4. Tangent Properties of Elipse and llyper bola. Corresponding Extremum Properties. 5. Extreme distances to a 〔 riven curve $2, A General Principal Underlying Extreme value Problems 338 1. The Principle. 2. Examples $3. Stationary points and the Differential calcultis 1. Extrema and Stationary Points. 2. Maxima and Minima of functions of Several Variables. Saddle points, 3. Minimax Points and Topol- ogy. 4. The distance front a point to a Surface 84. Schwarz's Triangle probleln 346 1. Schwarz's Proof. 2. Another Proof. 3. Obtuse Triangle 4. Triangles Formed by light Rays. 5. Remarks Concerning Problems of Refection and ergodic motion §5. Steiners problen 1. Problen and Solution. 2. Analysis of the Alternatives. 3. A Comple mentary Problem. 4. Remarks and Exercises. 5. Generalization to the Street network problen 6. Extrema and lnequalities I, The Arithmetical and Geometrical Mean of two Positive Quantities, 2. generalization to 2 variables. 3. The Method of Least Squares S7. The existence of an Extremun. Dirichlet, s Principle 366 【实例截图】
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