• Associativity of Addition:

The sum of three natural numbers remains the same though the grouping of numbers is changed. We can say that if we will add any three natural numbers; whether we will add the numbers at first two positions first and then add the third one to it or if we add the numbers at second and third positions first then adding the remaining number the answer will be same in both the cases.

General Property:

For all a, b, c belongs to N

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

Example 1: 

Let a = 7, b = 5 and c = 2

(7 + 5 ) + 2 = 7 + ( 5 + 2 )

12 + 2 = 7 + 7

14 = 14

LHS = RHS

Therefore, we see above the sum of three natural numbers 7, 5, 2 has not changed and remained the same even when a grouping of these numbers is changed.

Example 2 :

Let a = 9, b = 6 and c = 8

(9 + 6) + 8 = 9 + (6 + 8)

15 + 8 = 9 + 14

23 = 23

LHS =  RHS

Therefore, we see above when we added three natural numbers 9, 6, 8, their sum has not changed and remained the same even when they are regrouped.

Example 3: 

Let a = 11, b = 8 and c = 16

(11 + 8) + 16 = 11 + (8 + 16)

19 + 16 = 11 + 24

35 = 35

LHS = RHS

Therefore, we see above the sum of three natural numbers 11, 8, 16  has not changed and remained the same even when they are regrouped in a different way.

Example 4:

Let a = 100, b= 45 and c = 90

(100 + 45) + 90 = 100 + (45 + 90)

145 + 90 = 100 + 135

235 = 235

LHS = RHS

Therefore, we see above that the sum of three natural numbers 100, 45,90 is the same and there is no change even when a grouping of these numbers is changed.

• Associativity of Multiplication:

The product of three natural numbers remains the same though the grouping of numbers is changed. We can say that if we will multiply any three natural numbers, whether we’ll first multiply the first two numbers then the product will be multiplied by the third one or we’ll multiply the last two numbers first and then multiply the product with the number at the first position; the answer will be same in both the cases.

General Property: For all a, b, c belongs to N

(a × b) × c = a × (b × c)

Example 1.

Let a = 5, b = 3 and c = 12

( 5 × 3 ) × 12 = 5 × ( 3 × 12 )

15 × 12 = 5 × 36

180 = 180

LHS = RHS

Here on the left-hand side we first multiplied 5 and 3 then we multiplied the product 15 with 12, we got the answer 180. On the right-hand side, we see that we multiplied 3 with 12 to get 36 and multiplied 36 with5, again we got 180 as the answer.

Example 2 :

Let a = 9, b = 7 and c = 2

(9 × 7) × 2 = 9 × (7 ×2)

63 × 2 = 9 × 14

126 = 126

LHS =  RHS

Here on the left-hand side we first multiplied 9 and 7 to get 63 then we multiplied the product 63 with 2, we got the answer 126. On the right-hand side, we see that we multiplied 7 with 2 to get 14 and multiplied 14 with 9, again we got 126 as the final result.

Example 3 :

Let a = 10, b = 15 and c = 30

(10 × 15) × 30 = 10 × (15 × 30)

150 × 30 = 10 × 450

4500 = 4500

LHS = RHS

Here on the left-hand side we first multiplied 10 and 15  then we multiplied the result 150 with 30, to get the answer 4500.  On the right-hand side, we see that we multiplied 15 with 30 to get 450 and multiplied 450 with 10, again we got 4500.

Example 4: 

Let a = 100, b = 25 and c = 11

(100 × 25) × 11 = 100 × (25 × 11)

2500 × 11 = 100 × 275

27500 = 27500

LHS = RHS

Here on the left-hand side we first multiplied 100 and 25 then we multiplied the product 2500 with 11, we got the answer 27500. On the right-hand side, we multiplied 25 with 11 to get 275 and multiplied 275 with 100, again we got 27500 as the final result.

General Property: For all a, b, c belongs to N

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

and

(a × b) × c = a × (b × c)

So, the set of natural numbers, N is associative both under addition and multiplication.

Here we have explained above the Definition with Examples for the Associative Property of Natural Numbers in detail. Winaum Learning offers an interesting way to teach Associative Property with Math Games, Activities, Puzzles, Quizzes. Book a free trial.