这是一本人人都能读的数学书，将为你开启一扇认识数学世界的窗口。无论你是初学者还是专家，学生还是教师，哲学家还是工程师，通过这本书，你都将领略到数学之美，最终迷上数学。

本书是世界著名的数学科普读物。它荟萃了许多数学的奇珍异宝，对数学世界做了生动而易懂的描述。内容涵盖代数、几何、微积分、拓扑等领域，其中还穿插了许多相关的历史和哲学知识。

本书不仅是数学专业人员的必读之物，也是任何愿意做科学思考者的优秀读物。对于中学数学教师、高中生和大学生来说，这都是一本极好的参考书。

PREFACE TO SECOND EDITION

PREFACE TO REVISED EDITIONS

PREFACE TO FIRST EDITION

How TO USE THE BOOK

WHAT IS MATHEMATICS?

CHAPTER Ⅰ. THE NATURAL NUMBERS

 Introduction

 1. Calculation with Integers

 1. Laws of Arithraetic. 2. The Representation of Integers. 3. Computation in Systems Other than the Decimal.

 2. The Infinitude of the Number System, Mathematical Induction

 1. The Principle of Mathematical .Induction. 2. The Arithmetical Progression. 3. The Geometrical Progression. 4. The Sum of the First n Squares. 5. An Important Inequality. 6. The Binomial Theorem. 7. Further Remarks on Mathematical Induction.

 SUPPLEMENT TO CHAPTER I. THE THEORY OF NUMBERS

 Introduction

 1. The Prime Numbers

 1. Fundamental Facts. 2. The Distribution of the Primes. 3. Formulas Producing Primes. b. Primes in Aritlunetical Progressions. c. The Prime Number Theorem. d. Two Unsolved Problems Concerning Prime Numbers.

 2. Congruences

 1. General Concepts. 2. Fermat's Theorem. 3. Quadratic Residues.

 3. Pythagorean Numbers and Fermat's Last Theorem

 4. The Euclidean Algorithm

 1. General Theory. 2. Application to the Fundamental Theorem of Arithmetic. 3. Euler's Function. Fermat's Theorem Again. 4. Continued Fractions. Diophantine Equations.

CHAPTER Ⅱ. THE NUMBER SYSTEM OF MATHEMATICS

 Introduction

 1. The Rational Numbers

 1. Rational Numbers as a Device for Measuring. 2. Intrinsic Need for the Rational Numbers. Principal of Generation. 3. Geometrical Interpretation of Rational Numbers.

 2. Incommensurable Segments, Irrational Numbers, and the Concept of Limit

 1. Introduction. 2. Decimal Fractions. Infinite Decimals. 3. Limits. Infinite Geometrical Series. 4. Rational Numbers and Periodic Deci- maiN. 5. General Definition of Irrational Numbers by Nested

 Intervals 6. Alternative Methods of Defining Irrational Numbers. Dedekind Cuts.

 3. Remarks on Analytic Geometry

 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 Numbers and the Non-Denumerability of the Continuum. 3. Cantor's "Cardinal Numbers." 4. The Indirect Method of Proof. 5. The Paradoxes of the Infinite. 6. The Foundations of Mathematics.

 5. Complex Numbers

 1. The Origin of Complex Numbers. 2. The Geometrical Interpretation of Complex Numbers. 3. De Moivre's Formula and the Roots of Unity. 4. The Fundamental Theorem of Algebra.

 6. Algebraic and Transcendental Numbers

 1. Definition and Existence. 2. Liouville's Theorem and the Construction of Transcendental Numbers.

 SUPPLEMENT TO CHAPTER II. THE ALGEBRA OF SETS

 1. General Theory. 2. Application to Mathematical Logic. 3. An Application to the Theory of Probability.

CHAPTER Ⅰ. GEOMETRICAL CONSTRUCTIONS. THE ALGEBRA OF NUMBER FIELDS

 Introduction

 Part Ⅰ. Impossibility Proofs and Algebra

 1. Fundamental Geometrical Constructions

 1. Construction of Fields and Square Root Extraction. 2. Regular Polygons. 3. Apollonius' Problem.

 2. Constructible Numbers and Number Fields

 1. General Theory. 2. All Constructible Numbers are Algebraic.

 3. The Unsolvability of the Three Greek Problems

 1. Doubling the Cube. 2. A Theorem on Cubic Equations. 3. Trisecting the Angle. 4. The Regular Heptagon. 5. Remarks on the Problem of Squaring the Circle.

 Part Ⅱ. Various Methods for Performing Constructions

 4. Geometrical Transformations. Inversion

 1. General Remarks. 2. Properties of Inversion. 3. Geometrical Constrnction of Inverse Points. 4. How to Bisect a Segment and Find the Center of a Circle with the Compass Alone.

 5. Constructions with Other Tools. Mascheroni Constructions with Compass Alone

 1. A Classical Construction for Doubling the Cube. 2. Restriction to the Use of the Compass Alone. 3. Drawing with Mechanical Instruments. Mechanical Curves. Cycloids. 4. Linkages. PeauceUier's and Hart's Inversors.

 6. More About Inversions and its Applications

 1. Invariance of Angles. Families of Circles. 2. Application to the Problem of Apollonius. 3. Repeated Reflections.

CHAPTER Ⅳ. PROJECTIVE GEOMETRY. AXIOMATICS. NON-EucLIDEAN GEOMETRIES .

 1. Introduction

 ……

CHAPTER Ⅴ TOPOLOGY

CHAPTER Ⅵ FUNCTIONS AND LIMITS

CHAPTER Ⅶ MAXIMA AND MINIMA

CHAPTER Ⅷ THE CALCULUS

CHAPTER Ⅸ RECENT DEVELOPMENTS

APPENDIX: SUPPLEMENTARY REMARKS, PROBLEMS, AND EXERCISES

SUGGESTIONS FOR FURTHER READING

SUGGESTIONS FOR ADDITIONAL READING

INDEX