PROPERTIES OF DIVISION OF RATIONAL NUMBER -
Division of rational numbers follows several key properties that help understand and manipulate expressions involving division. Here are the main properties:-
1. Closure Property:-
The division of two rational numbers (except division by zero) results in another rational number. If a and b are rational numbers, then a/b is also a rational number (provided b ≠ 0).
3 2
Example:- ------ ÷ ------
4 5
3 5 (3 x 5) 15
=> ------ x ------ = ----------- = ------
4 2 (4 x 2) 8
15/8 is a rational number.
2. Identity Property:-
For any rational number a, dividing it by 1 leaves it unchanged,
a
--------- = a
1
7 7 1
Example:- ------- ÷ 1 = ------ ÷ ------
8 8 1
7 ÷ 1 7
= --------- = ------
8 ÷ 1 8
3. Inverse Property:-
Every non-zero rational number has a multiplicative inverse (reciprocal). Dividing by a number is equivalent to multiplying by its reciprocal. For a rational number a/b (where a ≠ 0 and b ≠ 0), its reciprocal is b/a.
a a
------- ÷ ------- = 1
b b
This property allows us to rewrite division as multiplication by the reciprocal:-
a c a d
------- ÷ -------- = ------- x -------
b d b c
4 4 4 7
Example.1) ------- ÷ ------ = ------ x ------ = 1
7 7 7 4
5 10 5 21
Example.2) ------- ÷ ------- = ------ x -------
14 21 14 10
(5 x 21) (1 x 3)
= ----------- = ----------
(14 x 10) (2 x 2)
3
= ------
4
4. Division by Zero is Undefined:-
Division by zero is not defined in rational numbers. For any rational number a: a/0 is undefined.
7
Example.1) ------ ÷ 0 is undefined
9
5. Distributive Property of Division over Addition and Subtraction:-
Division is not distributive over addition or subtraction in the usual sense:
a + b a b
--------- ≠ ------- + -------
c c c
However, the following property can be useful:-
a ÷ c + b ÷ c a + b
----------------- = ----------
d d x c
6. Non-commutative Property:-
Division is not commutative. This means:-
a b
-------- ≠ --------
b a
unless a = b or both a and b are reciprocals of each other.
3 2 2 3
Example.1) ------ ÷ ------ ≠ ------ ÷ ------
4 3 3 4
3 2 3 3
L.H.S = ------ ÷ ------ = ------ x -------
4 3 4 2
(3 x 3) 9
= ---------- = ------
(4 x 2) 8
2 3 2 4
R.H.S = ------ ÷ ------ = ------ x ------
3 4 3 3
(2 x 4) 8
= ---------- = ------
(3 x 3) 9
So, L.H.S ≠ R.H.S
7. Non-associative Property:-
Division is not associative. This means:-
a b
(------) ÷ c ≠ a ÷ (-------)
b c
3 2 1 3 2 1
Example:- (------ ÷ ------) ÷ ------ ≠ ------ ÷ (------ ÷ ------)
4 3 2 4 3 2
3 2 1
L.H.S = (------ ÷ ------) ÷ ------
4 3 2
3 3 1
= (------ x ------) ÷ ------
4 2 2
9 1 9 2 9
= ------ ÷ ------ = ------ x ------ = ------
8 2 8 1 4
3 2 1
R.H.S = ------ ÷ (------ ÷ ------)
4 3 2
3 2 2
= ------ ÷ (------ x -------)
4 3 1
3 4 3 3 9
= ------ ÷ ------- = ------ x ------ = ------
4 3 4 4 16
So, L.H.S ≠ R.H.S