Commutative Property

The Commutative property states that the result of an operation between two numbers remains the same, regardless of their position. For example, 2 + 3 is the same as 3 + 2, and 4 × 5 is the same as 5 × 4.

This property is satisfied by the addition (+) and multiplication (×) operations.

Examples:

• 4 + 7 = 7 + 4 = 11
• 5 + 4 = 4 + 5 = 9
• 4 × 7 = 7 × 4 = 28
• 5 × 4 = 4 × 5 = 20

Commutative Property of Addition

For the addition of two numbers, X and Y, the commutative property for the addition formula is given below:

X + Y = Y + X

Following are some examples for the commutative property for addition:

• 4 + 9 = 9 + 4
• (-12) + 5 = 5 + (-12)
• 8/9 + 1/3 = 1/3 + 8/9

Commutative Property of Multiplication

For multiplication of two numbers X and Y, the commutative property for the addition formula is given below:

X × Y = Y × X

Following are some examples for the commutative property for addition:

• 4 × 9 = 9 × 4
• (-12) × 5 = 5 × (-12)
• (8 / 9) × (1 /3) = (1 / 3) × (8 / 9)

Non-Commutative Operations—Division and Subtraction

Non-Commutative operations refers to those operations that do not follow the commutative property and changing the order of the numbers in the operations changes the result of the operation. The arithmetic operators subtraction and division does not satisfy the commutative property as changing the order of the operands changes the result of the expression, and this can be explained by the given examples.

For two operands R and S,

• R - S ≠ S - R
• R / S ≠ S / R

Example 1: Suppose we take two numbers, 12 and 3, then dividing 12 by 3 and dividing 3 by 12 gives the separate results, i.e.,

• 12/3 = 4
• 3/12 = 1/4

Example 2: Let's take two numbers, 15 and 35, then subtract 15 from 35, and subtract 35 from 15 gives the separate results, i.e.,

• 35 - 15 = 20
• 15 - 35 = -20

Commutative Property vs. Associative Property

The differences between commutative property and associative property are explained in the table below.

Commutative Law Worksheet