Associative Property

Nov 03, 2022, 16:45 IST

Introduction to Associative Property

Associative property explains that addition and multiplication of numbers are possible regardless of how they are grouped. By grouping we mean the numbers which are given inside the parenthesis (). Suppose you are adding three numbers, say 2, 5, 6, altogether. Then even if we group the numbers in addition procedures such as 4 + (2 + 3) or (2 + 3) + 4, in both the ways the result will be the same. The same rule applies to multiplication, i.e., 2 x (5 x 6) = (2 x 5) x 6. This property is almost similar to commutative property, where only two numbers are used.

Table of content for Associative property

i. Introduction to Associative Property

ii. Associate Property Definition

iii. Associative Property for Addition

iv. Associate Property of Multiplication

v. Associate Property of Rational Number

vi. Examples

Associate Property Definition

Associative because the name implies, means grouping. Basic mathematical operations which may be performed using associate property are addition and multiplication. This is often normally applicable to quite 2 numbers.

As in case of Commutative property, the order of grouping does not matter in Associative property. It will not alter the result. The grouping of number can be done in parenthesis irrespective of the order of terms. Thus, the associative law expresses that it doesn’t make a difference which part of the operation is carried out first; the answer will be the same.

Associative Property for Addition

The addition follows associative property i.e. no matter how numbers are parenthesized the ultimate sum of the numbers are going to be an equivalent. Associative property of addition states that:

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

Let us say, we want to add 3+2+4. It can be seen that the answer is 9. Now, let us group the numbers; put 3 and 2 in the bracket. We get,

= (3 + 2) + 4 = 5 + 4 = 9

Now, let’s regroup the terms like 2 and 4 in brackets:

= 3 + (2 + 4) = 9

It can be seen that the sum in both cases are the same. This is the associative property of addition

Examples of Associative property

(1) 3+ (1+5) = (3+1)+5

3+6 = 4+5

9 = 9

L.H.S = R.H.S

(2) 4 + (-2+2) = [4 + (-6)] + 2

4 + (-4) = [4-6] + 2

4-4 = -2+2

0 = 0

L.H.S = R.H.S

Associate Property of Multiplication

Rule for the associative property of multiplication is:

(a b) c = a (b c)

On solving 1×3×2, we get 6 as a product. Now as in addition, let’s group the terms:

⇒ (1 × 3) × 2 = 3 × 2 = 6

After regrouping,

⇒ 1 × (3 × 2) = 1 × 6 =6

Products will be the same.

Thus, addition and multiplication are associative in nature but subtraction and division are not associative.

For example, divide 100 ÷ 10 ÷ 5

⇒ (100 ÷ 10) ÷ 5 ≠ 100 ÷ (10 ÷ 5)

⇒ (10) ÷ 5 ≠ 100 ÷ (2)

⇒ 2 ≠ 50

Subtract, 4 − 2 − 1

⇒ (4 − 2) − 1 ≠ 4 − (2 − 1)

⇒ (2) – 1 ≠ 4 − (1)

⇒ 1 ≠ 3

Hence, proved the associative property is not applicable for subtraction and division methods.Rational numbers follow the associative property for addition and multiplication.