Commutative Property

Have you ever thought why the answer changes when you change the order of numbers in division and subtraction, but not in addition and multiplication? Let’s help you with this question. It’s all about the Commutative Property of addition and multiplication. 

In this blog, you will learn what is commutative property and how it is applied to arithmetic operations of multiplication and addition. 

What is the Commutative Property? 

Let’s break down the term to understand what is the commutative property. The term ‘commutative’ is rooted in commute, which means to move or change the position. It shows that the commutative property in math revolves around changing the position of numbers in operation (addition or subtraction). 

In mathematical terms, if we change the position of numbers (to be added or multiplied), it will not change the result. This specific characteristic of an arithmetic operation is known as the commutative property or commutative law.   

You will learn the commutativity of multiplication and addition separately in detail. 

Formula for Commutative Property

A simple formula for commutative property is used to make it easy to solve. Let’s understand it with an example.

Consider we have two numbers A and B. The commutative property formula for these numbers will be expressed as;

The formula shows that changing the position of numbers in addition and multiplication does not change the answer, while in division and subtraction, it will change the answer. 

Commutative Property of Addition

The commutative property of addition states that the result of two different numbers will remain the same even if the order of the numbers in addition is changed. For two different numbers A and B, the commutative property for addition will be written as;

Let’s look at the example of a commutative property of addition. 

Example:

Suppose we have two different numbers 10 and 14. We will prove the commutative property of addition. 

First, we will add the numbers. 

Now we will change the order of numbers and add again.

We noticed that the answer remained the same with changing the order of numbers. 

So, the commutative property of addition proved.

Commutative Property of Multiplication

The commutative property of multiplication states that the order of two numbers at the time of multiplication does not impact the result. 

For two numbers ‘A’ and ‘B’, multiplication commutative property will be represented as;

We will understand it through the commutative property of multiplication example. 

Example: 

If we have two numbers, 3 and 7. We will check the commutative property.

First, multiply the two numbers.

Now we will change the order of numbers

The answer didn’t change with a change in the order of numbers.

Hence, it proves the commutative property or commutative law of multiplication.

Commutative Property in Subtraction

The commutative property can not be applied in subtraction. Let’s understand it with the example. 

Example: 

Suppose we have two numbers 8 and 3.

Subtract the numbers. 

Now change the order of numbers and see the result.

The above calculation shows that

So, the commutive property does not apply to subtraction. 

Commutative Property of Division

Same as subtraction, the commutative property does not apply to division. We will understand the commutative property of division with an example. 

Example:  

Consider two numbers 3 and 9. 

First, divide the numbers.

Now change the order of numbers. 

From the equation, we calculated that

So, it is proved that commutative property can not be applied to division.