Commutative Property

What Is Commutative Property?

There are four basic operations in Mathematics namely addition, subtraction, multiplication, and division. These four operations are the most fundamental constituents of today’s world of Mathematics. These operations obey five fundamental properties namely closure property, commutative property, associative property, distributive property and identity property. All the four basic operations may not obey all the properties. However, each Mathematical operation obeys a few properties based on several conditions. The commutative property is also known popularly as commutative law states that the result of a mathematical operation between the operands remains unaffected by changing the order in which the operation is performed. 

Commutative Property Definition:

Commutative law states that the result obtained by performing a Mathematical operation on any number of operands is the same even when the order of the operands is reversed. Mathematically, commutative property definition can be explained as follows. If ‘c’ and ‘d’ are two numbers, then for any operation performed between ‘c’ and ‘d’ is said to be commutative if and only if the result obtained when ‘c’ is the first operand and ‘d’ is the second operand is equal to the result obtained when ‘d’ is the first operand and ‘c’ is the second. 

Commutative Property of Addition:

Commutative property of addition states that the sum of any two numbers remains the same even when the position of the numbers are interchanged. For example, if ‘E’ and ‘F’ are any two real numbers, then according to commutative property of addition, it is true that

E + F = F + E

Let us take an example of any two numbers, say 9 and 23. It is true that the sum of 9 and 23 is equal to 32 irrespective of the direction of addition. i.e. 9 + 23 = 23 + 9 = 32

So addition operation satisfies commutative law. 

Commutative Property of Subtraction:

Commutative property does not hold good for subtraction because the value of the difference between the numbers depends on the direction in which the numbers are subtracted. For example, let us consider the subtraction of 9 and 23. 

The difference between 9 and 23 when 9 is subtracted from 23 is 14 and the difference between 9 and 23 when 23 is subtracted from 9 is - 14.

So, 23 - 9 ≠ 9 - 23

Commutative Property of Multiplication:

The multiplication operation obeys commutative law. Commutative property of multiplication states that the product of any two numbers remains the same even if we change the direction in which the multiplication is performed. If ‘G’ and “H’ are the two numbers, then the commutative property of multiplication states that:

G x H = H x G

For example, if a = 8 and b = 4 are the two numbers, then the value of 4 times 8 and 8 times 4 is the same and is equal to 32. 

Commutative Property of Division:

Commutative law is not true for division. The quotient obtained is dependent on the values in place of numerator and denominator. So, the answer does not remain the same when numerator and denominator are interchanged. Hence, commutative law of division does not exist.

Commutative Property Example Problems:

  1. State whether the following statements are true or false. Give appropriate reasons to substantiate your answer.  (Hint: Recall what is commutative property)

    1.  7 + 8 = 8 + 7

    2. 8 - 7 = 7 - 8


Solution:

  1. 7 + 8 = 8 + 7 is true

Because the sum obtained is equal to 15 in both the cases according to the commutative property of addition.

  1. 8 - 7 = 7 - 8 is not true 

Because the commutative property of subtraction does not exist. 8 - 7 = 1 and 7 - 8 = - 1.

Fun Quiz:

  1. Fill in the blanks with appropriate answers.

  • 4 + 7 = ___ + 4

  • 1.5 x ___ = 3.4 x 1.5

  1. Match the expressions in column A with the appropriate statements in column B.

Column A

Column B

M + N 

M x N

N x M

Not a commutative property definition

Subtraction of 2 numbers

N + M

Non-Commutative

A ÷  B


FAQ (Frequently Asked Questions)

1. What is commutative property?

Commutative property or commutative law states that the result of a mathematical operation remains the same even when the order of the operands are reversed. Commutative property does not hold good for all the four basic Mathematical operations. However, it is true for addition and multiplication operations. The sum of two numbers and the product of two numbers remain the same even if the order of the numbers are interchanged. However, the difference between the two numbers and quotients of two numbers are not the same when the order of the numbers is changed. So, subtraction and division operations do not satisfy the commutative law. 

For example: 

6 + 3 = 9 and 3 + 6 = 9

6 x 3 = 18 and 3 x 6 = 18

6 - 3 = 3 and 3 - 6 = - 3

6 ÷ 3 = 2 and 3 ÷ 6 = ½ 


2. What are the fundamental properties of basic Mathematical operations?

There are four basic Mathematical operations namely addition, subtraction, multiplication and division. All these operations may obey one or more fundamental properties or laws. The fundamental laws in Mathematics are:

  • Commutative law: The result remains the same even when the operands are swapped. Commutative law is true for addition and multiplication only. 

  • Associative law: The result of the operation does not depend on the grouping. This law also applies to addition and multiplication only.

  • Distributive law: This law indicates the distribution of addition over multiplication or multiplication over addition.