Suppose that x, y, and z are numbers.
The Commutative Laws
The Commutative Law of Addition
The Commutative Law of Multiplication
The Associative Laws
The Associative Law of Addition
The Associative Law of Multiplication
The Distributive Law
The Identity Properties
The Zero Property
There exists a number, 
, such that 
The Multiplicative Identity Property
There exists a number, 
, such that for any number 
, 
Multiplying one by a number results in that number.
To be complete, we state also that 
.
Inverse Properties
The Additive Inverse
For any number 
, there exists a number x such that 
The Multiplicative Inverse
If 
is any number except 0, there exists a number x such that 
Laws of Equality
The Reflexive Property:
The Symmetric Property:
If 
, then 
The Transitive Property:
If 
, and 
, then 
If 
, then 
If 
, then 
Laws of Inequality
If 
, then 
If 
and 
, then 
If 
and 
, then 
