1. Lecture 1: Construction & Extension: Story of Numbers Addressed by Z.Liu

2. 0 Outline Complex Numbers Real Numbers Rational Numbers Integers Natural Numbers

3. 1 Motivation Peano Axioms Construction Binary Operations

4. 1.0 Motivation

5. 1.1 Peano Axioms P 1P 2P 3P 4

6. 1.2 Construction Set Theory Relations & Functions Preliminaries Axiom of Infinity Linear Ordering & Well Ordering Construction Mathematical Induction Ordinals & Transfinite Induction Induction

7. 1.2.0 Preliminaries Cartesian Product RelationFunction

8. 1.2.1 Construction

9. 1.2.2 Induction Natural Numbers Ordinal Numbers Mathematical Induction Transfinite Induction

10. 1.3 Binary Operation AdditionMultiplicationExponentiation

11. 2 Integers MotivationConstruction Algebraic Structure Modular Arithmetic

12. 2.0 Motivation

13. 2.1 Construction

14. 2.2 Algebraic Structure Group Monoid Semigroup Field Division Ring Or Integral Domain Ring Or Commutative Ring

15. 2.3 Modular Arithmetic

16. 3 Rational Numbers Motivation Construction & Quotient Field Extension & Vector Space p-adic Numbers

17. 3.0 Motivation

18. 3.1 Construction

19. 3.2 Extension

20. 3.3 p-adic Numbers

21. 4 Real Numbers Motivation Cardinal Number Metric Space & Topology Measure Theory

22. 4.0 Motivation Dedekind Completion Cauchy Completion

23. 4.1 Cardinal Number

24. 4.2 Metric Space & Topological Space Inner Product Space Normed Space Metric Space Topological Space

25. 4.3 Measure Theory

26. 5 Complex Numbers MotivationConstruction Linear Representation Complex Analysis

27. 5.0 Motivation

28. 5.1 Construction

29. 5.2 Linear Representation

30. 5.3 Complex Analysis Differentiation Integral Laurent Series

31. Thank you For your attention!