## What Is Mathematics?: An Elementary Approach to Ideas and MethodsFor 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

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 |

293 | |

293 | |

2 Fundamental Concepts | 417 |

### Other editions - View all

What Is Mathematics?: An Elementary Approach to Ideas and Methods Richard Courant,Herbert Robbins No preview available - 1996 |

### Common terms and phrases

algebraic altitude triangle angle axioms boundary calculus circle closed curve complex numbers concept congruent conic consider construction continuous continuous function coördinates corresponding crossratio decimal defined definition denote derivative differential digits dimension distance domain ellipse equal equation Euclidean geometry Euler example Exercise existence expression fact factor Fermat Figure finite number follows formula function f(x geometry given Hence hyperbola infinite integers intersection interval intuitive inverse inverse function Jordan curve theorem Leibniz length limit mathematical mathematical induction mathematicians maxima and minima maximum minimum nested intervals nonEuclidean nth root obtain parallel plane polygon polynomial positive number primes problem projective projective geometry proof prove quadratic residue quantity radius rational numbers real numbers segment sequence side simple solution Steiner straight line surface symbol tangent tends to infinity theorem theory topological transformation variable vertices xaxis zero