What Is Mathematics?: An Elementary Approach to Ideas and Methods

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Oxford University Press, Jul 18, 1996 - Mathematics - 592 pages
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For more than two thousand years a familiarity with mathematics has been regarded as an indispensable part of the intellectual equipment of every cultured person. Today, unfortunately, the traditional place of mathematics in education is in grave danger. The teaching and learning of mathematics has degenerated into the realm of rote memorization, the outcome of which leads to satisfactory formal ability but does not lead to real understanding or to greater intellectual independence. This new edition of Richard Courant's and Herbert Robbins's classic work seeks to address this problem. Its goal is to put the meaning back into mathematics. Written for beginners and scholars, for students and teachers, for philosophers and engineers, What is Mathematics?, Second Edition is a sparkling collection of mathematical gems that offers an entertaining and accessible portrait of the mathematical world. Covering everything from natural numbers and the number system to geometrical constructions and projective geometry, from topology and calculus to matters of principle and the Continuum Hypothesis, this fascinating survey allows readers to delve into mathematics as an organic whole rather than an empty drill in problem solving. With chapters largely independent of one another and sections that lead upward from basic to more advanced discussions, readers can easily pick and choose areas of particular interest without impairing their understanding of subsequent parts. Brought up to date with a new chapter by Ian Stewart, What is Mathematics?, Second Edition offers new insights into recent mathematical developments and describes proofs of the Four-Color Theorem and Fermat's Last Theorem, problems that were still open when Courant and Robbins wrote this masterpiece, but ones that have since been solved. Formal mathematics is like spelling and grammar--a matter of the correct application of local rules. Meaningful mathematics is like journalism--it tells an interesting story. But unlike some journalism, the story has to be true. The best mathematics is like literature--it brings a story to life before your eyes and involves you in it, intellectually and emotionally. What is Mathematics is like a fine piece of literature--it opens a window onto the world of mathematics for anyone interested to view.
 

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Contents

PREFACE TO SECOND EDITION
8
EXERCISES
18
The Principle of Mathematical Induction 2
26
Introduction
29
2 Congruences
42
4 The Euclidean Algorithm
50
5 Complex Numbers
88
6 Algebraic and Transcendental Numbers
88
Curve
189
4 Schwarzs Triangle Problem
195
5 Steiners Problem
213
7 The Existence of an Extremum Dirichlets Principle
220
8 The Isoperimetric Problem
227
THE CALCULUS
255
2 The Derivative
272
5 The Fundamental Theorem of the Calculus
279

GEOMETRICAL CONSTRUCTIONS THE ALGEBRA
88
PROJECTIVE GEOMETRY AXIOMATICS NON
88
Complete Quadrilateral
88
Introduction 2 Analytic Approach 3 Geometrical
118
Topology
118
3 Other Examples of Topological Theorems
126
APPENDIX
142
FUNCTIONS AND LIMITS
149
SUPPLEMENT TO CHAPTER VI MORE EXAMPLES ON LIMITS
176
2 Example on Continuity
176
10 The Calculus of Variations
176
6 The Exponential Function and the Logarithm
285
Newtons Law of Dynamics
293
2 Orders of Magnitude
293
3 Infinite Series and Infinite Products
293
RECENT DEVELOPMENTS
293
SUPPLEMENTARY REMARKS PROBLEMS
293
SUGGESTIONS FOR FURTHER READING
293
INDEX
293
2 Fundamental Concepts
417
Copyright

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About the author (1996)

The late Richard Courant, headed the Department of Mathematicas at New York University and was Director of the Institute of Mathematical Sciences--which has subsequently renamed the Courant Institute of Mathematical Sciences. His book Mathematical Physics is familiar to every physicist, and his book Differential and Integral Calculus is acknowledged to be one of the best presentations of the subject written in modern times. Herbert Robbins is New Jersey Professor of Mathematical Statistics at Rutgers University. Ian Stewart is Professor of Mathematics at the University of Warwick, and author of Nature's Numbers and Does God Play Dice?. He also writes the "Mathematical Recreations" column in Scientific American.

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