Lecture 1:

Construction & Extension:Story of Numbers

Addressed by Z.Liu

0 OutlineComplex Numbers

Real Numbers

Rational Numbers

Integers

Natural Numbers

1 Natural Numbers

Motivation

PeanoAxioms

Construction

BinaryOperations

1.0 Motivation

54321

1.1 Peano Axioms

P 1

P 2

P 3

P 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 PreliminariesAxiom of Extensionality

Axiom of Pairing

Axiom of Separation

Axiom of Union

Axiom of Power Set

Axiom of Infinity

Axiom of Replacement

Axiom of Regularity

Axiom of Choice

CartesianProduct

Relation Function

1.2.1 Construction

1.2.2 Induction

NaturalNumbers

OrdinalNumbers

MathematicalInduction

TransfiniteInduction

1.3 Binary Operation

Addition Multiplication Exponentiation

2 Integers

Motivation

ConstructionAlgebraicStructure

ModularArithmetic

2.0 Motivation

2.1 Construction

-1=0-1 =1-2 =2-3

2.2 Algebraic Structure

Group

Monoid

Semigroup

Field

DivisionRingOr

Integral Domain

RingOr

Commutative Ring

2.3 Modular Arithmetic

3 Rational Numbers

Motivation

Construction&

Quotient Field

Extension&

Vector Space

p-adicNumbers

3.0 Motivation

3.1 Construction

0.5=1/2 =2/4 =3/6 =4/8

3.2 Extension

3.3 p-adic Numbers

4 Real Numbers

Motivation

CardinalNumber

Metric Space&

Topology

MeasureTheory

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

Motivation

ConstructionLinear

Representation

ComplexAnalysis

5.0 Motivation

5.1 Construction

5.2 Linear Representation

5.3 Complex Analysis

Differentiation

IntegralLaurentSeries

Thank youFor your attention!