The commutative property of addition states that the order in which two numbers are added does not change the sum. Given two addends, a and b, it doesn't matter whether a is added to b or b is added to a. In either case, the sum will be the same. This can be written in the form of an addition sentence as:

a + b = b + a

One way to visualize the commutative property of addition is to use a set of objects. The stars in the figure below are the same object. The difference in color represents the two addends.

As can be seen from the figure, regardless of whether 3 is added to 4 or 4 is added to 3, the result is still the same. As long as the value of the addends remains unchanged, for any amount of addends, they can be added in any order to achieve the same sum.

Examples

Confirm that the order in which we add the following examples does not matter:

1. 7 + 9 + 2 + 8 = ? ; 9 + 7 + 8 + 2= ?

(7 + 9) + (2 + 8) = 16 + 10 = 26

(9 + 2) + (8 + 7) = 11 + 15 = 26

In both examples, the order in which the numbers are added does not affect the outcome. You can add the numbers however you want, but for the purpose of this example, certain groupings were chosen to also demonstrate the associative property of addition, which allows us to group the addends in an addition sentence however we want.

The commutative property applies to the addition of any type of number, not just whole numbers. Below is an example demonstrating the commutative property in a fraction addition problem.

2. ;   Learning the various properties of addition is important because it helps to form a foundation for learning more complex mathematical concepts in the future. Particularly in algebra, moving variables around as well as regrouping them (associative property) is often a necessary aspect of solving an algebraic equation.