Commutative property
The commutative property states that the order in which two numbers are added or multiplied does not change the result. The same cannot be said about division and subtraction.
Commutative property of addition
The commutative property of addition can be expressed as follows.
a + b = b + a
The commutative property of addition dictates that changing the order in which two numbers are added does not change the sum.
Examples
| Ex. 1 | Ex. 2 |
| 7 + 9 = 16 |  +  = 
|
| 9 + 7 = 16 | + =
|
One way to visualize the commutative property of addition is to use objects to represent an addition problem, as in the figure below.
Regardless whether 3 stars are added to 4, or 4 stars are added to 3, the result is still 7 stars.
Commutative property of multiplication
The commutative property of multiplication states that changing the order in which two numbers are multiplied does not change the product. It can be expressed as follows.
a × b = b × a
Examples
| Ex. 1 | Ex. 2 |
| 8 × 5 = 40 | 4 × 1.3 = 5.2 |
| 5 × 8 = 40 | 1.3 × 4 = 5.2 |
The commutative property of addition can be visualized using an array to represent a multiplication problem. The figure below depicts the multiplication problems 4 × 2 and 2 × 4 using different arrays.
Regardless whether we think of the problem as 4 groups of 2, or 2 groups of 4, there are still a total of 8 circles.