Some properties listed here below will be of good use in solving problems.

**Properties
of Rational Numbers**

Some properties
listed here below will be of good use in solving problems.

** **

__1. Closure
property/law for the collection ____ℚ____ of rational numbers__

i) Closure property for Addition

For any two
rational numbers ** a** and

ii) Closure property for Multiplication

For any two
rational numbers ** a** and

**Illustration**

**Try this**

The closure property on integers holds for subtraction and not for
division. What about rational numbers? Verify.

**Solution:**

Let 0 and 1/2 be two rational numbers 0 – 1/2 is a rational
number

∴ Closure property for subtraction holds for rational numbers.

But consider the two rational number 5/2 and 0.

5/2 ÷ 0 = 5 / [2 × 0] = 5
/ 0

Here denominator = 0 and it is not a rational number.

∴ Closure property is not true for division of rational numbers.

__2. Commutative
property/law for the collection ____ℚ____ of rational numbers__

i) Commutative property for Addition

For any two
rational numbers ** a** and

ii) Commutative property for Multiplication

For any two
rational numbers ** a** and

**Illustration**

Here, we
find that *a* +
*b* = *b*
+
*a* and hence addition is commutative.

Further,

Here, we
find that *a* ×
*b* = *b*
×
*a* and hence multiplication is commutative

**Solution:**

**Is 3/5 – 7/8 = 7/8 – 3/5 ?**

LHS = 3/5 + 7/8 = [ (3 × 8) – (7 × 5) ] / 40 = [24 – 35] / 40 = −11/40

RHS = 7/8 − 3/5 = [ (7 × 5) – (3 × 8) ] / 40 = [35 – 24] / 40 =
11/40

LHS ≠ RHS

∴ 3/5 ÷ 7/8 ≠ 7/8 – 3/5

∴ Subtraction of rational numbers is not commutative.

**(ii) Is 3/5 ÷ 7/8 = 7/8 – 5/3 ? **

LHS = 3/5 ** ÷ ** 7/8 = 3/5 × 8/7 = 24/35

RHS = 7/8 ** ÷ ** 5/3 = 7/8 × 3/5 = 21/40

∴ LHS ≠ RHS

∴ 3/5 ÷ 7/8 ≠ 7/8 ÷ 5/3

∴ Commutative property not hold good for division of rational
numbers.

__3.
Associative property/law for the collection ____ℚ____ of
rational numbers____ __

i) Associative property for Addition

For any three
rational numbers ** a**,

ii) Associative property for Multiplication

For any three
rational numbers ** a**,

**Illustration**

Take rational
numbers *a* , *b*, *c* as *a* = −1/2 , *b* = 3/5 and *c* =
−7/10

(3) and (4) shows that

(*a* ×
*b*) × *c*
=
*a* × (*b* × *c*) is true for rational
numbers. Thus, the associative property is true for addition and multiplication
of rational numbers.

**Try this**

Check whether associative
property holds for subtraction and division.

**Solution:**

Consider the rational numbers 2/3, 1/2 and 3/4

To verify (2/3 – 1/2) – 3/4 = 2/3 – (1/2 – 3/4)

LHS = (2/3 – 1/2) – 3/4 = ( [(2 × 2) – (1 × 3)] / 6 ) – (3/4)

= ([4 – 3]/6) – 3/4 = 1/6 – 3/4 = [(1 × 2) – (3 × 3)] / 12 = [2
– 9] / 12 = – 7 / 12

RHS = 2/3 – (1/2 – 3/4) = 2/3 – ( [2 – 3] / 4) = ( 2/3 − (−1/4) )

= 2/3 + 1/4 = [ (2 × 4) +
( 1 × 3) ] / 12 = [ 8 + 3] / 12 = 11 / 12

LHS ≠ RHS

∴ ( 2/3 – 1/2) – 3/4 ≠ 2/3 − (1/2 − 3/4)

∴ Associative property not holds for subtraction of rational
numbers

Also to verify (2/3 ÷ 1/2) ÷ 3/4 = 2/3 ÷ (1/2 ÷ 3/4)

LHS = (2/3 ÷ 1/2) ÷
3/4 = (2/3 × 2/1) ÷ 3/4

= 4/3 ÷ 3/4 = 4/3 × 4/3 =
16 / 9

RHS = 2/3 ÷ (1/2 ÷ 3/4) = 2/3 ÷ ( 1/2 × 4/3) = 2/3 ÷
( 2/3 )

= 2/3 × 3/2 = 1

LHS ≠ RHS

ie. (2/3 ÷ 1/2) ÷ 3/4 ≠ 2/3 ÷ ( 1/2 ÷ 3/4)

∴ Associative property does not hold for division of rational
numbers.

__4. Identity
property/law for the collection ____ℚ____ of rational numbers__

i) Identity property for Addition

For any rational
number ** a**, there exists a unique rational number

**0 **+** ***a*** **=** ***a*** ***=*** 0 **+** ***a**.*

ii) Identity property for Multiplication

For any rational
number ** a**, there exists a unique rational number

**1 **×** ***a*** ***=*** ***a*** ***=*** ***a*** **×** 1**.

**Illustration**

Take *a* = 3/-7 that is, *a* = -3/7 . Now -3/7 + 0 = -3/7 = 0 + -3/7 (Isn’t it?)

Hence, 0
is the additive identity for −3/7

Also, -3/7
× 1 = - 3/7 = 1 × -3/7 (Isn’t it?)

Hence, 1
is the multiplicative identity for −3/7

** **

__5. Inverse
property/law for the collection ____ℚ____ of rational numbers__

i) Additive Inverse property

For any rational
number ** a**, there exists a unique rational number

ii) Multiplicative Inverse property

For any rational
number ** b**, there exists a unique rational number 1/

**Illustration**

** **

__6. Distributive
property/law for the collection ____ℚ____ of rational numbers__

Multiplication
is distributive over addition for the collection of rational numbers. For any three
rational numbers ** a**,

**Illustration**

(1) and (2)
shows that *a* ×
(*b* + *c*
) =
(*a* × *b*)
+
(*a* × *c*)
.

Hence, multiplication
is distributive over addition for the collection **ℚ** of rational
numbers.

Tags : Numbers | Chapter 1 | 8th Maths , 8th Maths : Chapter 1 : Numbers

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