Commutative Property

In mathematical computation, commutative property or commutative law explains that order of terms doesn’t matters while performing an operation. Although the official use of this law began at the end of the 18th century, it was known even in the ancient era. This property is applicable only for two operations: addition and multiplication. Suppose two numbers A and B on addition gives a sum C, then if we interchange the position of A and B, the result will be C only, such as; A + B = B + A = C.  For example, 4 + 3 = 7 = 3 + 4; here, whether 3 come before or after the plus sign, the sum of 4 and 3 will be always 7 irrespective of their order. Apart from commutative, there are two more major properties addition and multiplication of integers, they are associative and distributive. Let us discuss all the three properties or laws here.

 

Commutative Property

What is Commutative, Associative and Distributive Property?

There are three major properties used in the mathematical or arithmetic operations followed by integers. They are:

  1. Commutative Law
  2. Associative Law
  3. Distributive Law

Let us discuss all the three laws or properties here.

What is Commutative Law?

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 and Multiplication

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

Hence,

  • A + B = B + A
  • A.B = B.A

Associative  Law of Addition and Multiplication

According to the associative law, regardless of how the numbers are grouped, you can add or multiply them. 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

Distributive Law 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:

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

Non-Commutative Law

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, 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.

Examples

Example 1: Which of the following follows commutative law?

  1. ×  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 = 0.17

=> 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)

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

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