Brainflux: Axioms of Number Theory



$Number Theory$
	$Axioms of Number Theory$


	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

then


	Axiom: Closure Property of Multiplication

then


	Axiom: Commutative Property of Addition

then


	Axiom: Commutative Property of Multiplication

then


	Axiom: Associative Property of Addition

then


	Axiom: Associative Property of Multiplication

then


	Axiom: Distributive Property of Multiplication over Addition

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

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

then


	Axiom: Cancelation Property of Addition

and

then


	Axiom: Cancelation Property of Multiplication

and

then


	Axiom: Trichotomy Law

, 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

such that for every element

Ths property is fundamentally important for the technique of Mathematical Induction.