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# Properties of Rational Numbers - Associativity

#### 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 = 0.06For 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 = 3For 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 -

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.

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Associative Property Rational Number [00:09:29]
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