Reciprocals of Numbers

Clearly 3 x 1/3 = 1

1/3 is called the reciprocal of 3 and 3 is called the reciprocal of 1/3.

One number is the reciprocal of another if their product is 1.

For example, the reciprocal of 5/6 is 6/5 since 5/6 × 6/5 is 1.

In general:

The reciprocal of a fraction is obtained by interchanging the numerator and the denominator, i.e. by inverting the fraction.

Example 1:

Find the reciprocal of 20.

Solution:

Reciprocal of 20 is 1/20.

Example 2:

Find the reciprocal of 3/7.

Solution:

Reciprocal of 3/7 is 2 1/3.

We will learn the reciprocal of a rational number.

For every non-zero rational number a/b there exists a rational number b/a such that

a/b × b/a = 1 = b/a × a/b

The rational number b/a is called the multiplicative inverse or reciprocal of a/b and is denoted by (a/b)^-1.

The reciprocal of 12 is 1/12

The reciprocal of 5/16 is 16/5.

The reciprocal of 3/4 is 4/3 i.e., (3/4)^-1 = 4/3.

The reciprocal of -5/12 is 12/-5 i.e., (-5/12)^-1 = 12/-5.

The reciprocal of (-14)/17 is 17/-14 i.e., (-17)/14.

The reciprocal of -8 is 1/-8 i.e., (-1)/8.

The reciprocal of -5 is 1/-5, since -5 × 1/-5 = -5/1 × 1/-5 = -5 × 1/-5 × 1 = 1.

Note: The reciprocal of 1 is 1 and the reciprocal of -1 is -1. 1 and -1 are the only rational numbers which are their own reciprocals. No other rational number is its own reciprocal.

We know that there is no rational number which when multiplied with 0, gives 1. Therefore, rational number 0 has no reciprocal or multiplicative inverse.

Solved example on reciprocal of a rational number:

Problem 1 :

Write the reciprocal of each of the following rational numbers?

(i) 5

(ii) -15

(iii) 7/8

(iv) -9/13

(v) 11/-19

Solution:

(i) The reciprocal of 5 is 1/5 i.e., (5)^-1 = 1/5.

(ii) The reciprocal of -15 is 1/-15 i.e., (-15)^-1 = 1/-15.

(iii) The reciprocal of 7/8 is 8/7 i.e., (7/8)^-1 = 8/7.

(iv) The reciprocal of -9/13 is 13/-9 i.e., (-9/13)^-1 = 13/-9.

(v) The reciprocal of 11/-19 is -19/11 i.e., (11/-19)^-1 = -19/11.

Problem 2 :

Find the reciprocal of 3/7 × 2/11.

Solution:

3/7 × 2/11

= (3 × 2)/(7 × 11)

= 6/77

Therefore, the reciprocal of 3/7 × 2/11 = Reciprocal of 6/77 = 77/6.

Problem 3 :

Find the reciprocal of -4/5 × 6/-7.

Solution:

-4/5 × 6/-7

= (-4 × 6)/(5 × -7)

= -24/-35

= 24/35

Therefore, the reciprocal of -4/5 × 6/-7 = Reciprocal of 24/35 = 35/24.