# Reciprocal of a Rational  Number

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:

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.

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.

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.

Rational Numbers

What is Rational Numbers?

Is Every Rational Number a Natural Number?

Is Zero a Rational Number?

Is Every Rational Number an Integer?

Is Every Rational Number a Fraction?

Positive Rational Number

Negative Rational Number

Equivalent Rational Numbers

Equivalent form of Rational Numbers

Rational Number in Different Forms

Properties of Rational Numbers

Lowest form of a Rational Number

Standard form of a Rational Number

Equality of Rational Numbers using Standard Form

Equality of Rational Numbers with Common Denominator

Equality of Rational Numbers using Cross Multiplication

Comparison of Rational Numbers

Rational Numbers in Ascending Order

Representation of Rational Numbers on the Number Line

Addition of Rational Number with Same Denominator

Addition of Rational Number with Different Denominator

Properties of Addition of Rational Numbers

Subtraction of Rational Number with Different Denominator

Subtraction of Rational Numbers

Properties of Subtraction of Rational Numbers

Simplify Rational Expressions Involving the Sum or Difference

Multiplication of Rational Numbers

Properties of Multiplication of Rational Numbers

Rational Expressions Involving Addition, Subtraction and Multiplication

Division of Rational Numbers

Rational Expressions Involving Division

Properties of Division of Rational Numbers

To Find Rational Numbers