The Commutative Property states that the sum of two integers remains the same even if the order of the integers is changed.

If a and b are any two integers, then a + b = b + a

Similarly, the product of two integers remains the same even if the order of the integers is changed.

If a and b are any two integers, then a x b = b x a

Commutative property is defined as the property of changing the order of the mathematical expression so that the end result remains the same. The Commutative property is true for both addition and multiplication. Subtraction and Multiplication are not commutative.

Examples:
  • 5 + 8 = 8 + 5 = 13
  • 5 x 8 = 8 x 5 = 40
In the above case, the sum or product of the two numbers does not change even if we change the order in which the numbers are added or multiplied.
The sum of two numbers does not change even if we change the order of summation of the numbers.

a + b = b + a

In order to verify this property, let us consider some pair of integers and apply the commutative property of addition.

Consider the following:

a b a + b b + a
5 7 5 + 7 = 12 7 + 5 = 12

Here, 5 and 7 are integers. Now, if we add these two integers, the sum is 5 + 7 = 12. 12 is also an integer. If we interchange 5 and 7, the sum of both the integers remains the same. 7 + 5 = 12.

a b a + b b + a
-5 0 -5 + 0 = -5 0 + (-5) = -5

Here, -5 and 0 are integers. Now, if we add these two integers, the sum is -5 + 0 = -5. -5 is also an integer. If we interchange -5 and 0, the sum of both the integers remains the same. 0 + (-5) = -5.

a b a + b b + a
-5 -7 -5 + (-7) = -12 (-7) + (-5) = -12

Here, -5 and -7 are integers. Now, if we add these two integers, the sum is -5 + (-7) = -12. -12 is also an integer. If we interchange -5 and -7, the sum of both the integers remains the same. -7 + (-5) = -12.

Given below are some examples on the commutative property of addition.

Example 1:

Write the complete equation by using the commutative property of addition
Commutative Property of Addition Example
Solution:

Commutative Property of Addition states that
a + b = b + a
Here, we have used the commutative property of addition to swap 5 and 4. So, the answer is 5 + 4 = 4 + 5

Example 2:

Rewrite the expression 5 + 8 + 10 using the commutative law of addition

Solution:

Given expression = 5 + 8 + 10 = 23
We can write the given expression by using commutative property of addition as follows:
10 + 8 + 5 = 23
8 + 10 + 5 = 23
10 + 5 + 8 = 23
8 + 5 + 10 = 23
5 + 10 + 8 = 23
So, any of the above expressions would be correct.

Example 3:

In the below solution, swap 74 + 28 with 28 + 74 and find the solution?
74 + 28 + 30 = (74 + 28) + 30
= 102 + 30
= 132
Solution:

We have to swap 74 + 28 with 28 + 74
So, (74 + 28) + 30 = (28 + 74) + 30
= 102 + 30
= 132
The product of two numbers do not change even if we change the order of multiplication of the numbers.

a x b = b x a

In order to verify this property, let us consider some pair of integers and apply the commutative property of multiplication.

Consider the following:

a b a x b b x a
5 7 5 x 7 = 35 7 x 5 = 35

Here, 5 and 7 are integers. Now, if we multiply these two integers, the product is 5 x 7 = 35. 35 is also an integer. If we interchange 5 and 7, the product of both the integers remains the same. 7 x 5 = 35

a b a x b b x a
-5 0 -5 x 0 = 0 0 x (-5) = 0

Here, -5 and 0 are integers. Now, if we multiply these two integers, the product is -5 x 0 = 0. 0 is also an integer. If we interchange -5 and 0, the product of both the integers remains the same. 0 x (-5) = 0

a b a x b b x a
-5 -7 (-5) x (-7) = 35 (-7) x (-5) = 35

Here, -5 and -7 are integers. Now, if we multiply these two integers, the product is (-5) x (-7) = 35. 35 is also an integer. If we interchange -5 and -7, the product of both the integers remains the same. (-7) x (-5) = 35
Given below are some examples on commutative property of multiplication.

Example 1:

Write the equation corresponding to the picture shown below:

Commutative Property of Multiplication Example

Solution:

First, let us see graphic on the left. It has 3 square boxs with 4 flowers in each. So, we can write it as 3 x 4 = 12

Now, coming to the graphic on the left, we have 4 square boxes with 3 flowers each. So, we can write it as 4 x 3 = 12

Using the Commutaive property of Multiplication
a x b = b x a
In this case, 3 x 4 = 4 x 3 = 12

Example 2:

Rewrite the expression 6 x 9 x 11 using the commutative law of multiplication.

Solution:

Given expression is 6 x 9 x 11 = 594
We can write the given expression by using the commutative property of multiplication as follows:
11 x 9 x 6 = 594
9 x 11 x 6 = 594
11 x 6 x 9 = 594
9 x 6 x 11 = 594
6 x 11 x 9 = 594
So, any of the above expressions would be correct.

Example 3:

In the solution given below, swap 75 x 29 with 29 x 75 and find the solution?
75 x 29 x 31
= (75 x 29) x 31
= 2175 x 31
= 67425
Solution:

We have to swap 75 x 29 with 29 x 75
So, (75 x 29) x 31 = (29 x 75) x 31
= 2175 x 31
= 67425