# CLASS-7EXISTENCE OF MULTIPLICATIVE INVERSE  PROPERTY OF RATIONAL NUMBER

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.