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 · 7 = 7 · 5
a · b = b · a |
Note
Subtraction and division are not
commutative.
For example, 5 -3 =2, but 3 - 5 = 2.
Likewise, 6 ÷ 2 = 3, but 2
÷ 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 · (3 · 4) = (2 · 3) · 4
a · (b · c) = (a · b) · c |
Distributive Property
This property allows us to convert a product into an equivalent sum.
2 · (3 + 4) = 2 · 3 + 2 · 4
a · (b + c) = a · b + a · 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 · 1 = 3
a · 1 = a
1 is called the multiplicative identity. |
| Multiplication Property of 0 The
product of a number and 0 is 0.
3 · 0 = 0
a · 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.
|
