In mathematical computation, commutative property explains that order of terms doesn’t matters while performing an operation. That is, suppose two numbers A and B on addition gives a sum C, then:

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.

Although the official use of commutative property or commutative law began at the end of 18th century, the use of commutative property was known even in the ancient era. This usage has now been organized properly with rules and is known as commutative law or commutative property.

## What is Commutative, Associative and Distributive Property?

**Commutative Property**

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, literal meaning of word is tending to switch or move around.

The commutative property states that if we swipe the positions of the numbers result will remain the same.

**Associative Property**

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.

**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:

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

**Non-Commutative Property**

Some operations are non-commutative. By non-commutative we mean the switching of 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 property. That is,

\(6~ ÷ ~2\) = \(3\)

\(2~ ÷~ 6\) = \(\frac{1}{3}\)

\(\Rightarrow~ 6 ~÷ ~2~ ≠~ 2 ~÷ ~6\)

**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?

- \(3~\times~ 12\)
- \(4 ~+~ 20\)
- \(36 ~÷~ 6\)
- \(36 ~- ~6\)
- \(-3~\times~4\)

**Solution: **Options 1, 2 and 5 follows the commutative law

**Explanation:**

- 3 × 12 = 36 and

12 x 3 = 36

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

- 4 + 20 = 24 and

20 + 4 = 24

=> 4 + 20 = 20 + 4 (commutative)

6 ÷ 36 = 0.17

=> 36 ÷ 6 ≠ 6 ÷ 36 (non commutative)

6 – 36 = – 30

=> 36 – 6 ≠ 6 – 36 (non commutative)

- −3 × 4 = -12 and

4 x -3 = -12

=> −3 × 4 = 4 x -3 (commutative)

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