数学实验MATHEMATICS EXPERIMENTS

Owing to the advent of computers, experiments are becoming an increasingly important part of mathematics. This book provides guidance to students doing experiments in mathematics. The aim is to stimulate interest in mathematics through examples and experiments.
Each experiment in the book starts with an interesting problem. The students are expected to work with these problems on computers, try to find the solutions themselves, and experience the scientific exploration in the process.
The problems which the authors have chosen cover a wide spectrum in mathematics, ranging from calculus, number theory, coding and probability to geometry and chaos. They are introduced in a simple way and yet show great depth. The discussions are thorough but not lengthy.
This book is useful not only to mathematics students, but also to students in all areas of sciences who are interested in learning some of the mathematical tools. It provides a hands-on approach to the most fundamental issues in mathematics — an approach which may help to revolutionize the teaching of mathematics.

1 Fundamental of Calculus
　1.1 Functions and Their Graphs
　1.2 The Number e
　1.3 Integral and Natural Logarithm
　1.4 Harmonic Series
2 How to Calculate π
　2.1 Method of Numerical Integral
　2.2 Method of Using Taylor Series
　2.3 Monte Carlo Method
3 Best Approximations by Fractions
　3.1 Best Approximation of an Irrational Number by Fractions
　3.2 Frequency Ratio of the Musical Notes
　3.3 Expanding Real Numbers into Continued Fractione
　3.4 Calculating the Value of Logarithm
　3.5 Integer Solutions of Binary Linear Equations
4 Sequences and Series
　4.1 The Fibonacci Sequence
　4.2 Harmonic Series
　4.3 Questions
5 Prime Numbers
　5.1 Discrimination and Calculation of Prime Numbers
　5.2 Formula of Producing Prime Numbers
　5.3 Distribution of Prime Numbers
　5.4 Further Problems
6 The Probability
　6.1 The Classical Definition of Probability
　6.2 The Statistical Definition of Probability
　6.3 Binomial Distribution and Poisson Distribution
　6.4 Normal Distribution
　6.5 The First Arc Sine Law
7 Geometric Transformations
　7.1 Linear Transformations and Affine Transformations
　7.2 Eigenvectors of Linear Transformations
　7.3 Projective Transformations
　7.4 Non-Euclidean Geometry
　7.5 A Proof of the Fundamental Theorem in Algebra
　Celestial Movement
　8.1 Acceleration of the Planetary Motion
　8.2 Celestial Movement under the Universal Gravitation
　8.3 Mathematical Simulation of Other Physical Phenomena
9 Iteration(I)-Finding Solutions to Equations
　9.1 Finding Roots of Equations
　9.2 Iterative Solutions to Linear Equations Sets
　9.3 Iterative Solutions to Nonlinear Equations Sets
10 Optimization
　10.1 The Law of Light's Refraction
　10.2 Dashing for the Optimal Point
　10.3 Least Squares Method
11 Brachistochrone
　11.1 Calculation of Time
　11.2 Looking for the Brachistochrone
……
12 Ineration(II)-Fractal
13 Iteration(III)-Chaos
14 Cryptography
15 Mechanical Proving of Elementary Geometrical Theorems
Bibliography