Content introduction:

This book is written for beginners and experts, students and teachers, philosophers and engineers. This book is a classic of mathematics. It collects many shining mathematical treasures and gives a set of interesting and simple pictures of the mathematical world. This book has been handed down to this day, and a new chapter has been added by I Stewart. This second edition expounds the latest development of mathematics from a new angle, and describes the proof of the four-color theorem and Fermat's last theorem. These problems were not solved when Courant and Robin wrote this book, but they have been solved now.

This book is a world-famous popular mathematics reading. It brings together many classic mathematical treasures and profoundly and vividly expounds the basic concepts and methods in the whole mathematical field. No matter who is a math major or who is willing to think about math, you can read this book. Especially for middle school math teachers, college students and high school students, it is an excellent reference book.

Directory:

What is mathematics?

Chapter 65438 +0 natural number

introduce

Calculation of 1 integer

Infinite mathematical induction of 2-number system

Chapter 65438 +0 Supplementary Number Theory

introduce

1 prime number

2 congruence

Pythagoras number and Fermat's last theorem

4 Euclid's stage division

Chapter II Number System in Mathematics

introduce

1 rational number

2 the concept of irrational numbers and the limit of incommensurable line segments

3 Overview of Analytic Geometry

4 Infinite mathematical analysis

5 complex number

6 algebraic numbers and transcendental numbers

Chapter II Supplementary Set Algebra

Chapter III Algebra of Geometric Drawing Number Domain

introduce

Proof and algebra of 1 partial impossibility

1 basic geometric drawing

2 graphable numbers and number fields

Three unsolvable Greek problems

The second part of various drawing methods

4 geometric transformation inversion

5 Drawing with other tools Only Macheroni of compasses is used for drawing.

6 Talk about inversion sentence and its application again.

Chapter 4 Axiomatic System of Projective Geometry Non-Euclidean Geometry

1 Introduction

2 Basic concepts

3 cross ratio

4 Parallel and Infinite

5 application

6 Analytic representation

Draw with only one ruler.

8 Quadratic curves and quadric surfaces

Axiomatic system and non-Euclidean geometry

Appendix Geometry in High Dimensional Space

Chapter 5 Topology

introduce

Euler formula of 1 polyhedron

Topological properties of 2 graphs

3 Other examples of topological theorems

Topological classification of 4 surfaces

appendix

Chapter 6 Functions and Limitations

introduce

1 variables and functions

2 limit

3 Limit of continuous approximation

4 the precise definition of continuity

5 Two basic theorems about continuous functions

6 Some applications of Bulcha's Nobel Theorem

Chapter 6 supplements some examples of limit and continuity.

1 limit example

2 examples of continuity

Chapter 7 minimax and minimax

introduce

1 Problems in Elementary Geometry

2 the general principle of the basic extreme value problem

3 stagnation and differentiation

4 Schwartz's triangle problem

5 Steiner problem

6 extreme sum inequality

Existence of Dirichlet's Principle of 7 Extremes

8 isoperimetric problem

9 the relationship between Steiner problem and isoperimetric problem of extreme value problem with boundary conditions

10 variational method

1 1 the experimental solution of the soap film experiment of the minimum problem

Chapter VIII Calculus

introduce

1 integral

2 derivative

3 difference method

Mark of Leibniz and "infinitesimal"

Basic theorem of calculus

6 exponential function and logarithmic function

7 differential equation

Chapter VIII Supplement

1 main content

Two orders of magnitude

3 Infinite Series and Infinite Product

4. Obtain the prime number theorem by statistical method.

Chapter IX Latest Progress

1 formula for generating prime numbers

Goldbach conjecture and twin prime numbers

3 Fermat's last theorem

4 continuum hypothesis

Five symbols in set theory

6 Four-color Theorem

7 Hausdorff dimension and fractal

8 knots

9 a problem in mechanics

10 Steiner problem

1 1 soap film and minimum surface

12 nonstandard analysis

The appendix supplements the questions and exercises.

Arithmetic and algebra

Analytic geometry

geometric figure

Projective geometry and non-Euclidean geometry

analysis situs

Function, Limit and Continuity

Maxim and minimax

calculus

comprehensive