I Arithmetic

1 What is a Number?


1.1 Various Numerals to Represent

2 Arithmetic in Different Bases

3 Arithmetic in Euclid’s Elements

4 Gauss?Advent of Modern Number Theory


4.1 Number Theory of Gauss


4.2 Cryptography


4.3 Complex Numbers


4.4 Application of Number Theory ? Construction of Septadecagon


4.5 How Did Gauss Do It?


4.6 Equations over Finite Fields*


4.7 Law of Quadratic Reciprocity*


4.8 Cubic Equations*


4.9 Riemann Hypothesis*

5 Numbers beyond Rationals


5.1 Arithmetic of Rational Numbers


5.2 Real Numbers

II Geometry

6 Basic Geometry

7 Greece: Beginning of Theoretical Mathematics

8 Euclid: The Founder of Pure Mathematics


8.1 Some Comments on Euclid’s Proof

9 Famous Problems from Greek Geometry

III Contributions of Some Prominent Mathematicians

10 Fibonacci’s Time and Legacy


10.1 Liber Abaci


10.2 Liber Quadratorum


10.3 Equivalent Formulations of the Problems

11 Solution of the Cubic


11.1 Introduction


11.2 History

12 Leibniz, Newton, and Calculus


12.1 Differential Calculus


12.2 Integral Calculus


12.3 Proof of FTC


12.4 Application of FTC

13 Euler and Modern Mathematics


13.1 Algebraic Number Theory


13.2 Analytical Number Theory


13.3 Euler’s Discovery of eπi + 1 = 0


13.4 Graph Theory and Topology


13.5 Traveling Salesman Problem


13.6 Planar Graphs


13.7 Euler-Poincare Characteristic


13.8 Euler Characteristic Formula

14 Non-European Roots of Mathematics

15 Mathematics of the 20th Century*


15.1 Hilbert’s 23 Problems

1 Riemann Hypothesis

2 Poincare Conjecture

3 Birch & Swinnerton-Dyer (B&S-D) Conjecture

15.2 Fermat’s Last Theorem

15.3 Miscellaneous