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MathMath
Number TheoryNumber Theory
Axioms of Number TheoryAxioms of Number Theory
Overview
 

Number theory is a theory about the integers, a set which we call where . The operations defined on this set are addition, multiplication, and the ordering relation "less than." What follows are the basic properties of the integers


Axiom: Closure Property of Addition
  If then

Axiom: Closure Property of Multiplication
  If then

Axiom: Commutative Property of Addition
  If then

Axiom: Commutative Property of Multiplication
  If then

Axiom: Associative Property of Addition
  If then

Axiom: Associative Property of Multiplication
  If then

Axiom: Distributive Property of Multiplication over Addition
  If then

Axiom: Additive Identity Property
  is a special number with the property: If then

Axiom: Multiplicitive Identity Property
  is a special number with the property: If then

Axiom: Additive Inverse Property
  If then where In other words, there is another integer in which we'll call When you add this other integer to the origional, you get back , the additive identity. We introduce the following familiar notation: If then is written as

Axiom: Zero Property of Multiplication
  If then

Axiom: Cancelation Property of Addition
  If and then

Axiom: Cancelation Property of Multiplication
  If , and then

Axiom: Trichotomy Law
  If , then exactly one of the following statements is true: (i) (ii) (iii)

Axiom: Properties of Inequality
  (i) If and , then (ii) If , , and , then (iii) If , , and , then

Axiom: Well Ordering Property
  Every non-empty set of positive integers contains a least element. More rigorously: If then there exists an element in such that for every element in , Ths property is fundamentally important for the technique of Mathematical Induction.