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Multiplicative Inverse 4. Division of Rational Numbers
p q We know that division is an inverse operation
Consider two rational numbers and where
q p of multiplication. So, we can do the process of
division in the following steps.
p ¹ 0 and q ¹ 0. We find their product by the
Step 1 : Find the multiplicative inverse of
following formula.
di vi sor.
p q pq
´ = =1 Step 2 : Mul ti ply it by the div i dend, ac cord ing
q p pq
to the rule of mul ti pli ca tion, i.e.
We can notice that the product of these two p r p s ps
¸ = ´ =
rational numbers is 1. Hence, two rational q s q r qr
p q
numbers and are known as multiplicative
q p
Ex am ple 9. Sim plify
inverse of each other and 1 is called the 8 16 4 æ 6 ö
ç
multiplicative identity. (i) - ¸ (ii) - ¸ - ÷
3 3 5 è 25 ø
1 1 3 7
For Ex am ple, 2 and , - 5 and - , and 3 æ 6ö
2 5 7 3 (iii) ¸ - ÷
ç
5 è 5 ø
etc. all are multi pli ca tive in verse of each other.
So lu tion :
Ex am ple 8. Find the multi pli ca tive in verse of
8 16
the fol low ing ra tio nal numbers. (i) - ¸
3 3
3 11
(i) -4 (ii) (iii) - 8 3 -1
5 9 = - ´ =
3 16 2
So lu tion : To find the multi pli ca tive in verse of 4 æ 6 ö
(ii) - ¸ - ÷
ç
-4 , write the nu mer a tor as de nom i na tor and 5 è 25 ø
de nom i na tor as numerator. 4 æ 25 ö
ç
1 = - ´ - ÷
Multiplicative inverse of -4 is - 5 è 6 ø
4 - ( 4 ) ´ -25 )
(
æ 1 ö =
Check : (-4 ) ´ - ÷ =1 5 ´ 6
ç
è 4 ø - ( 2 ) ´ - )
5
(
=
Multiplicative Multiplicative inverse 3
3 5 11 9 10
inverse of is of - is - . =
5 3 9 11 3
3 æ 6ö
Check : Check : (iii) ¸ - ÷
ç
3 5 15 5 è 5 ø
´ = = 1 - 11 ´ - 9 = 99 = 1
5
5 3 15 9 11 99 3 æ ö
= ´ - ç ÷
5 è ø
6
Note :
(
3 ´ -5) -1
p = =
For any non-zero rational number the 5 ´ 6 2
q
q Finding Reciprocal of a Rational Number
rational number is called its
3
p Consider a non-zero rational number which
reciprocal. 7
is made up of two integers 3 as numerator and 7
The number 0 has no reciprocal.
as denominator. If we interchange the integers
The multiplicative inverse of a non-zero in numerator and denominator, we get another
rational number is its reciprocal.
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Mathematics In Focus - 7
