Properties of Addition

The following are the properties of addition for real numbers. Some textbooks list just a few of them, others list them all. They may have slightly different names in your textbook.

PROPERTIES OF ADDITION
Identity Property

There is a unique real number 0 such that for every real number a,

a + 0 = a and 0 + a = a

Zero is called the identity element of addition.

Commutative Property

For all real numbers a and b,

a + b = b + a

The order in which you add two numbers does not change the result.

Associative Property

For all real numbers a, b, and c,

(a + b) + c = a + (b + c)

When you add three real numbers, the grouping (or association) of the numbers does not change the result.

Property of Opposites

For all real number a, there is a unique real number –a such that

a + (–a) = 0 and (–a) + a = 0

A number and its opposite are called additive inverses of each other because their sum is zero.

Property of Opposite of a Sum

For all real numbers a and b,

–(a + b) = (–a) + (–b)

The opposite of a sum of real numbers is equal to the sum of the opposites.