Commutative Property

In mathematics, commutative property or commutative law explains that order of terms doesn’t matter while performing arithmetic operations. This property is applicable only for addition and multiplication processes. Thus, it means we can change the position or swap the numbers when adding or multiplying any two numbers. For example: 1+2 = 2+1 and 2 x 3 = 3 x 2. 

Commutative Property:

A + B = B + A   (Addition)

A x B = B x A    (Multiplication)

 

Commutative Property

Definition

As we already discussed in the introduction, as per the commutative property or commutative law, when two numbers are added or multiplied together, then change in their positions does not change the result. This is one of the major properties of integers.

Examples

  • 2+3 = 3+2 = 5
  • 2 x 3 = 3 x 2 = 6
  • 5 + 10 = 10 + 5 = 15
  • 5 x 10 = 10 x 5 = 50

History

Although the official use of commutative property began at the end of the 18th century, it was known even in the ancient era.

The word, Commutative, originated from the French word ‘commute or commuter’ means to switch or move around, combined with the suffix ‘-ative’ means ‘tend to’. Therefore, the literal meaning of the word is tending to switch or move around. It states that if we swipe the positions of the integers, the result will remain the same.

Commutative Property of Addition

According to this property, when we add two integers, the answer will remain unchanged even if the position of the numbers are changed. Let A and B be the two integers, then;

A + B = B + A

Examples:

  • 1 + 2 = 2 + 1 = 3
  • 3 + 8 = 8 + 3 = 11
  • 12 + 5 = 5 + 12 = 17

Commutative Property of Multiplication

As per this property, when we multiply two integers, the answer we get after multiplication will remain the same, even if the position of the integers are interchanged. Let A and B be the two integers, then;

× B = B × A

Examples: 

  • 1 × 2 = 2 × 1 = 2
  • 3 × 8 = 8 × 3 = 24
  • 12 × 5 = 5 × 12 = 60

Other Properties

The other major properties of addition and multiplication are:

Now, observe the other properties as well here:

Associative Property of Addition and Multiplication

According to the associative law, regardless of how the numbers are grouped, you can add or multiply them together, the answer will be the same. In other words, the placement of parentheses does not matter when it comes to adding or multiplying.

Hence,

  • A + (B + C) = (A + B) + C
  • A.(B.C) = (A.B).C

Examples: 

  • 1 + (2+3) = (1+2) + 3 → 6
  • 3 x (4 x 2) = (3 x 4) x 2 → 24

Distributive Property of Multiplication

The distributive property of Multiplication states that multiplying a sum by a number is the same as multiplying each addend by the value and adding the products then.

According to the Distributive Property, if a, b, c are real numbers, then:

a x (b + c) = (a x b) + (a x c)

Example:

  • 2 x (5 x 8) = (2 x 5) + (2 x 8) → 80

There are certain other properties such as Identity property, closure property which are introduced for integers.

Non-Commutative Property

Some operations are non-commutative. By non-commutative, we mean the switching of the order will give different results. The mathematical operations, subtraction and division are the two non-commutative operations. Unlike addition, in subtraction switching of orders of terms results in different answers.

Example: 4 – 3 = 1 but 3 – 4 = -1 which are two different integers.

Also, the division does not follow the commutative law. That is,

6 ÷ 2 = 3

2 ÷ 6 = 1/3

Hence, 6 ÷ 2 ≠  2 ÷ 6

Important Note: Commutative property works for addition and multiplication only but not for subtraction and division.

Solved Examples

Example 1: Which of the following obeys commutative law?

  1. 3 ×  12
  2. 4 + 20
  3. 36 ÷ 6
  4. 36 – 6
  5. -3 × 4

Solution: Options 1, 2 and 5 follow the commutative law

Explanation:

  1. 3 × 12 = 36 and

       12 x 3 = 36

=> 3 x 12 = 12 x 3 (commutative)

  1. 4 + 20 = 24 and

     20 + 4 = 24

     => 4 + 20 = 20 + 4 (commutative)

  1. 36 ÷ 6 = 6 and

     6 ÷ 36 = 0.167

=> 36 ÷ 6 ≠ 6 ÷ 36  (non commutative)

  1. 36 − 6 = 30 and 

      6 – 36 = – 30

=> 36 – 6 ≠ 6 – 36  (non commutative)

  1. −3 × 4 = -12 and

       4 x -3 = -12

=> −3 × 4 = 4 x -3  (commutative)

Q.2: Prove that a+ b = b+a if a = 10 and b = 9.

Sol: Given that, a = 10 and b = 9 

LHS = a+b = 10 + 9 = 19   ……(1)

RHS = b + a = 9 + 10 = 19 ……(2)

By equation 1 and 2 we get;

LHS = RHS

Hence, proved.

To solve more problems on properties of math, download BYJU’S – The Learning App from Google Play Store and watch interactive videos.

Frequently Asked Questions – FAQs

What is commutative property? Give examples.

In Mathematics, a commutative property states that if the position of integers are moved around or interchanged while performing addition or multiplication operations, then the answer remains the same.
Examples are:
4+5 = 5+4 and 4 x 5 = 5 x 4
9 + 2 = 2 + 9 and 9 x 2 = 2 x 9

What is commutative property of addition?

When two numbers are added together, then if we swap the positions of numbers, the sum of the two remains the same.
For example, 3+4 = 4+3 = 7

What is the commutative property of multiplication?

When two numbers are multiplied together and if we interchange their positions, then the product of the two remains the same.
For example, 5 x 3 = 3 x 5 = 15

What are the major four properties in Maths?

The four major properties in Maths are:
Identity property, commutative property, associative property and Distributive property

What is the difference between commutative and associative property?

Commutative property holds regardless of order of numbers while addition or multiplication. Whereas associative property holds regardless of grouping of numbers.

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