EXISTENCE OF MULTIPLICATIVE INVERSE PROPERTY OF RATIONAL NUMBER -
The existence of the multiplicative inverse property states that for every non-zero rational number, there exists another rational number, called its multiplicative inverse or reciprocal, such that their product is the multiplicative identity, which is 1.
Definition:-
Finding the Multiplicative Inverse:-
p
If a = ------, where p and q are integers and a ≠ 0, the
q
q
multiplicative inverse of a is ------
p
This is because -
P q (p X q) 1
----- X ------ = ---------- = ------ = 1
q p (q X p) 1
Another way of Understanding-
For any non-zero rational number a, there exists a number b such that: -
a × b = b × a = 1
Every nonzero rational number a/b has its multiplicative inverse b/a.
Thus, (a/b × b/a) = (b/a × a/b) = 1
b/a is called the reciprocal of a/b. Clearly, zero has no reciprocal.
Reciprocal of 1 is 1 and the reciprocal of (-1) is (-1) For example:-
1) Positive Rational Number :-
3
Let, a = ------
5
3 5
The multiplicative inverse of ------ is ------
5 3
3 5 (3 X 5) 1
So, ------ X ------ = ---------- = ------ = 1
5 3 (5 X 3) 1
2) Negative Rational Number:-
- 3
Let, a = ------
7
- 3 7
The multiplicative inverse of ------ is ------
7 - 3
- 3 7 {(-3) X 7} - 21 - 1
So, ------ X ------ = ------------- = ------- = ------ = 1
7 - 3 {7 X (-3)} - 21 - 1
3) Whole Number:-
7
Let, a = 7 (which can be written as ------)
1
1
The multiplicative inverse of 7 is ------
7
7 1 7
So, ------ X ------ = ------ = 1
1 7 7
4) Reciprocal of 5/7 is 7/5, since (5/7 × 7/5) = (7/5 × 5/7) = 1
5) Reciprocal of -8/9 is -9/8, since (-8/9 × -9/8) = (-9/8 × -8/9 ) =1
6) Reciprocal of -3 is -1/3, since
{-3 × (-1)/3} = {-3/1 × (-1)/3} = {(-3) × (-1)}/(1 × 3) = 3/3 = 1
and (-1/3 × (-3)) = (-1/3 × (-3)/1) = {(-1) × (-3)}/(3 × 1) = 1
Note:-
Denote the reciprocal of a/b by (a/b)-1 Clearly (a/b)-1 = b/a
In all these cases, the product of a rational number and its multiplicative inverse is 1, demonstrating the existence of the multiplicative inverse property for rational numbers.