Share
Notifications

View all notifications

Properties of Rational Numbers - Associativity

Login
Create free account


      Forgot password?

notes

1) Whole number : 

Operation Numbers Remarks 
Addition 3 + (5 + 2) = 10 and
(3 + 5) + 2 = 10 
For any three whole  numbers a, b and c  
a + (b + c) = (a + b) + c
Addition is Associative .
Subtraction 3 - (2 - 1) = 2 but (3 - 2 )- 1 = 0 
For any three whole  numbers a, b and c  
a - (b - c) ≠ (a - b)- c 
Subtraction is not Associative .
Multiplication - 7 × (2 × 5) = (7 × 2) × 5 = 70 
For any three whole  numbers a, b and c  
a × (b × c)  = (a × b) × c
Multiplication is  Associative
Division 3 ÷ (5 ÷ 10) = 6 but
(3 ÷ 5) ÷ 10 = 0.06
For any three whole  numbers a, b and c 
 a ÷  (b ÷  c) ≠  (a ÷  b) ÷  c
Division  is not Associative .

2 ) Integers :

Operation Numbers  Remarks
Addition (-3) + (5 + 1) = 3 and
(-3 + 5) + 1 = 3
For any three integers a, b and c 
a + (b + c) = (a + b) + c
Addition is Associative .
Subtraction -  5 – (7 – 3) = 1 but (5 – 7) – 3 = -5 
For any three integers a, b and c 
a - (b - c) ≠  (a - b) - c 
Subtraction is not Associative .
Multiplication

5 × [(–7) × (– 8) = 280 and  
[5 × (–7)] × (– 8) = 280 
For any three integers a, b and c 
a × (b × c) = (a × b) × c 

Multiplication is Associative .
Division [(–10) ÷ 2] ÷ (–5) =25  but  
(–10) ÷ [2 ÷ (– 5)] = 1 
For any three integers a, b and c 
a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c 
Division is not Associative .

3) Rational numbers - 

(a) Addition :

we have `-2/3  + [3/5 + (-5/6)] = -2/3 + (-7/30) = -27/30 = -9/10`

`[-2/3 + 3/5] + (-5/6) = -1/15 + (-5/6) = -27/30 = -9/10`

we say that addition is associative for rational numbers. That is, for any three rational numbers a, b and c,  
a + (b + c) = (a + b) + c. 

(b) Subtraction :

We have `-2/3 - [-4/5 -1/2] = 19/30 ,
[2/3 - (-4/5)] - 1/2 = 26/30` 
Subtraction is not associative for rational numbers.
That is, for any three rational numbers a, b and c,  
a - (b - c) ≠ (a - b) - c. 

(c) Multiplication : 
We have `-7/3 xx (5/4 xx 2/9) = (-7/3 xx 5/4) xx 2/9 = -35/54`
We observe that multiplication is associative for rational numbers. That is for any three rational numbers a, b and c, 
a × (b × c) = (a × b) × c.

(d) Division :
Let us see if `1/2 ÷ [-1/3 ÷ 2/5] = [1/2 ÷ (-1/3)] ÷ 2/5 `

We have,  LHS  = `1/2 ÷ [-1/3 ÷ 2/5]  = -30 / 10`

RHS =` [1/2 ÷ (-1/3)] ÷ 2/5  = -15/4`

We say that division is not associative for rational numbers.
That is, for any three rational numbers a, b and c,  
a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c.

Shaalaa.com | Associative Property Rational Number

Shaalaa.com


Next video


Shaalaa.com


Associative Property Rational Number [00:09:29]
S
Series 1: playing of 1
0%


S
View in app×