| Fundamental of Calculus | p. 1 |
| Functions and Their Graphs | p. 1 |
| The Number e | p. 5 |
| Integral and Natural Logarithm | p. 9 |
| Harmonic Series | p. 12 |
| How to Calculate [pi] | p. 15 |
| Method of Numerical Integral | p. 15 |
| Method of Using Taylor Series | p. 17 |
| Monte Carlo Method | p. 19 |
| Best Approximations by Fractions | p. 23 |
| Best Approximation of an Irrational Number by Fractions | p. 24 |
| Frequency Ratio of the Musical Notes | p. 25 |
| Expanding Real Numbers into Continued Fractions | p. 26 |
| Calculating the Value of Logarithm | p. 30 |
| Integer Solutions of Binary Linear Equations | p. 33 |
| Sequences and Series | p. 39 |
| The Fibonacci Sequence | p. 40 |
| Harmonic Series | p. 43 |
| Questions | p. 44 |
| Prime Numbers | p. 49 |
| Discrimination and Calculation of Prime Numbers | p. 49 |
| Formula of Producing Prime Numbers | p. 53 |
| Distribution of Prime Numbers | p. 55 |
| Further Problems | p. 57 |
| The Probability | p. 61 |
| The Classical Definition of Probability | p. 61 |
| The Statistical Definition of Probability | p. 64 |
| Binomial Distribution and Poisson Distribution | p. 67 |
| Normal Distribution | p. 70 |
| The First Arc Sine Law | p. 71 |
| Geometric Transformations | p. 73 |
| Linear Transformations and Affine Transformations | p. 74 |
| Eigenvectors of Linear Transformations | p. 77 |
| Projective Transformations | p. 81 |
| Non-Euclidean Geometry | p. 83 |
| A Proof of the Fundamental Theorem in Algebra | p. 88 |
| Celestial Movement | p. 95 |
| Acceleration of the Planetary Motion | p. 96 |
| Celestial Movement under the Universal Gravitation | p. 99 |
| Mathematical Simulation of Other Physical Phenomena | p. 104 |
| Iteration(I)--Finding Solutions to Equations | p. 113 |
| Finding Roots of Equations | p. 113 |
| Iterative Solutions to Linear Equations Sets | p. 116 |
| Iterative Solutions to Nonlinear Equations Sets | p. 119 |
| Optimization | p. 125 |
| The Law of Light's Refraction | p. 125 |
| Dashing for the Optimal Point | p. 133 |
| Least Squares Method | p. 138 |
| Brachistochrone | p. 143 |
| Calculation of Time | p. 143 |
| Looking for the Brachistochrone | p. 146 |
| The Figure of the Brachistochrone | p. 147 |
| Tautochrone | p. 151 |
| Iteration(II)--Fractal | p. 159 |
| Creation Element | p. 160 |
| Iteration of Complex Function | p. 164 |
| IFS Iteration | p. 167 |
| Iteration(III)--Chaos | p. 179 |
| The Periodic Point and Periodic Orbit | p. 179 |
| Iteration of Quadratic Functions | p. 180 |
| The Feigenbaum Diagram | p. 181 |
| Properties of Chaos | p. 182 |
| Sensitivity to the Initial Values | p. 183 |
| Chaos Is Not Random | p. 183 |
| Iteration of Other Functions | p. 184 |
| Listening to Chaos | p. 185 |
| Cryptography | p. 189 |
| Monoalphabetic Cipher | p. 189 |
| Polyalphabetic Cipher | p. 191 |
| Modern Stream Cipher | p. 194 |
| Public Keys | p. 198 |
| Mechanical Proving of Elementary Geometrical Theorems | p. 203 |
| Converting Geometrical Problems to Algebraic Ones | p. 204 |
| Derivation and Proving of Algebraic Relational Expressions | p. 208 |
| Bibliography | p. 217 |
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