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

0 Outline Complex Numbers Real Numbers Rational Numbers Integers Natural Numbers

1 Motivation Peano Axioms Construction Binary Operations

1.0 Motivation

1.1 Peano Axioms P 1P 2P 3P 4

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

1.2.0 Preliminaries Cartesian Product RelationFunction

1.2.1 Construction

1.2.2 Induction Natural Numbers Ordinal Numbers Mathematical Induction Transfinite Induction

1.3 Binary Operation AdditionMultiplicationExponentiation

2 Integers MotivationConstruction Algebraic Structure Modular Arithmetic

2.0 Motivation

2.1 Construction

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

2.3 Modular Arithmetic

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

3.0 Motivation

3.1 Construction

3.2 Extension

3.3 p-adic Numbers

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

4.0 Motivation Dedekind Completion Cauchy Completion

4.1 Cardinal Number

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

4.3 Measure Theory

5 Complex Numbers MotivationConstruction Linear Representation Complex Analysis

5.0 Motivation

5.1 Construction

5.2 Linear Representation

5.3 Complex Analysis Differentiation Integral Laurent Series

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