The Magic of Commutative and Associative Property

by ContributedPost
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If in a rack 6 pairs of shoes are arranged in 2 rows. Add them from the top row and then add them from the bottom row. The answer you got is 6 only. This explains the commutative property of addition. Formally you can define this property as the sum of the numbers remains the same even after interchanging their places. This property is applicable for multiplication also.

Ex: 3+4 = 4+3 = 7

What is Associative Property?

The associative property is very similar to the commutative property. But here, you are changing the position of the parenthesis instead of numbers. So, Associative property can be defined as the property in which the final answer remains the same even after changing the position of the parenthesis. Ex: 3+ (4+7) = (3+4) + 7 =14

 Applications of Commutative and Associative Properties in Basic Mathematical Operations

Addition:

  • Commutative property: As per the definition result remains unchanged even after changing the position of the numbers. Ex: 3+6+9 = 18 and 6+9+3 = 18
  • Associative property: As per the definition result remains unchanged even after changing the position of the parenthesis in the given equation.

Ex: take x=1, y=1 and z=0 then x+(y+z) = 1+(1+0) = 2, (x+y)+z = (1+1)+0 = 2

In both cases, the result was the same. Hence Both the properties are applicable for addition.

Subtraction:

  • Commutative property:

To satisfy the requirements of commutative property, We have to check whether the final result remains the same or not. Let us take an example

3 – 2 – 1 = 0 but 2 – 1 – 3 = -2. We got 2 different answers here after interchanging the values.

  • Associative property: To satisfy the requirements of this property, We have to check here whether the final result remains the same or not after interchanging the parenthesis.

Ex: 4 – (2 – 1) = 3 but (4 – 2) -1 = 1. Here also we got two different answers.

So both commutative and associative properties are not applicable in subtraction.

Multiplication:

  • Commutative property: As per the definition result remains unchanged even after changing the position of the numbers. Ex: 3 4 = 12 and 4 3 = 12.
  • Associative property: As per the definition result remains unchanged even after changing the position of the parenthesis in the given equation. Ex: (4 1)2 = 8 and 4(1 2) = 8

Since the result remained undisturbed in both the cases, the Commutative and associative property applies to multiplication.

Division:

  • Commutative property:To satisfy the requirements of commutative property, We have to check here whether the final result remains the same or not. Let us take an example, 12 = 0.5 and 21 = 2. We got two different answers here after interchanging the values.
  • Associative property: As per the definition result remains unchanged even after changing the position of the parenthesis in the given equation.

Ex: 1 (2 4) = 2 and (1 2) 4 = ⅛ = 0.125 Here also we got two different answers.

Therefore commutative and associative properties are not applicable in division.

The division and subtraction operations are also known as non commutative operations.

Mathematical Illustrations:

  1. Verify x – (y+z) = (x – y) + z

Solution: Let x=1, y=0 and z=2

Then, x – (y+z) = 1 – (0+2) = 1-2 = -1 And (x – y) + z = (1-0) + 2 = 1+2 = 3

x – (y+z)  (x – y) + z

  1. Verify x(yz) = xyz

Solution: Let x=1, y=2 and z=3

Then, x(yz) = 1(2 3) = 6 and xyz = 1  2  3 = 6

x(yz) = xyz

Have you understood the concepts of commutative and associative property? If you still have doubts visit cuemath website for more information.

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