
 
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

 
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 nonempty 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.

