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Identity Property

One final property of addition that will be very useful in algebra is the Identity Property, which says that for any number a, the following are always true:

a + 0 = a
0 + a = a 

The Identity Property of Addition says that a number does not change its identity when 0 is added. For example, 12 + 0 = 12. 0 + 17 = 17. Or, if someone is given zero dollars, the amount of money he has does not change.

Using the Properties of Addition

These properties can be used in any order. Right now, they are useful because they make it easier to add long strings of numbers. Later, they will help us to solve algebraic equations, which we will discuss in Inverse Operations.

Examples

Here are some examples to show how these properties can make mental math easier:

Example 1.12 + 67 + 98 = ?
Commutative Property: 12 + 67 + 98 = 12 + 98 + 67
12 + 98 + 67 = 110 + 67 = 177

Example 2.(13 + 21) + (9 + 5) + 5 = ?

Associative Property: (13 + 21) + (9 + 5) + 5 = 13 + (21 + 9) + (5 + 5)
13 + (21 + 9) + (5 + 5) = 13 + 30 + 10 = 53
Example 3.54 + 17 + 6 + 12 + 3 + 18 = ?

Commutative Property: 54 + 17 + 6 + 12 + 3 + 18 = 54 + 6 + 17 + 3 + 12 + 18
Associative Property: 54 + 6 + 17 + 3 + 12 + 18 = (54 + 6) + (17 + 3) + (12 + 18)
(54 + 6) + (17 + 3) + (12 + 18) = 60 + 20 + 30 = 110

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