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.
|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
Subtraction and division are not
For example, 5 -3 =2, but 3 - 5 = 2.
Likewise, 6 Ã· 2 = 3, but 2
Ã· 6 = 1/ 3.
||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
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
(Addition Property of 0)
(Multiplication Property of 1)
|The sum of a number and 0 is the
||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
product of a number and 0 is 0.
3 Â· 0 = 0
a Â· 0 = 0
|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
The opposite of a number is also
called the additive inverse of the
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