Properties of Numbers and Definitions
There are several properties that will help you simplify calculations with
real numbers. Here is a summary of the properties we will use most often
in algebra. In the following examples, a, b, and c represent real numbers.
Commutative Property
Commutative Property
of Addition 
Commutative Property
of Multiplication 
When you add two numbers,
regardless of the order, the sum
is the same. 
When you multiply two numbers,
regardless of the order, the
product is the same. 
5 + 7 = 7 + 5 a + b = b + a 
5 Â· 7 = 7 Â· 5
a Â· b = b Â· a 
Note
Subtraction and division are not
commutative.
For example, 5 3 =2, but 3  5 = 2.
Likewise, 6 Ã· 2 = 3, but 2
Ã· 6 = 1/ 3.
Associative Property
Associative Property
of Addition 
Associative Property of Multiplication 
When you add numbers, regardless
of how you group (or associate)
them, the sum is the same. 
When you multiply numbers, regardless of how you
group (or associate) them, the product is the same. 
2 + (3 + 4) = (2 + 3) + 4 a + (b + c) = (a + b) + c 
2 Â· (3 Â· 4) = (2 Â· 3) Â· 4
a Â· (b Â· c) = (a Â· b) Â· c 
Distributive Property
This property allows us to convert a product into an equivalent sum.
2 Â· (3 + 4) = 2 Â· 3 + 2 Â· 4
a Â· (b + c) = a Â· b + a Â· c
Identities
Additive Identity
(Addition Property of 0)

Multiplicative Identity
(Multiplication Property of 1)

The sum of a number and 0 is the
number itself. 
The product of a number and 1 is
the number itself. 
3 + 0 = 3 a + 0 = a
0 is called the additive identity. 
3 Â· 1 = 3
a Â· 1 = a
1 is called the multiplicative identity. 
Multiplication Property of 0 The
product of a number and 0 is 0.
3 Â· 0 = 0
a Â· 0 = 0 
Inverses
Additive Inverse
(Opposite) 
Multiplicative Inverse
(Reciprocal)

Example 
The sum of a number and its opposite is 0. 
The product of a number and its
reciprocal is 1. 
The opposite of 7 is 7 The opposite of 7 is 7. 
7 + (7) = 0 a + (a) = 0 

The reciprocal of 7 is
.
The reciprocal of
is 7. 
The opposite of a number is also
called the additive inverse of the
number.
The reciprocal of a number is
also called the multiplicative
inverse of the number.
Zero does not have a reciprocal
because you may not divide by
zero.
