In this chapter we will introduce the concept of reciprocal of rational number with examples.

We will also tell you the method to find the reciprocal of any given rational number.

## Multiplicative inverse of rational number

The reciprocal of any rational number is also called** multiplicative inverse**.

It is produced by** switching the position of numerator and denominator of given rational number**.

Let a/b is the given rational number.

To get the reciprocal, **change the numerator and denominator position**.

Hence, the reciprocal is b/a.

### Features of reciprocal number

The **reciprocal when multiplied with the original rational number produce number 1**.**For example;**

Let 11/3 be the given rational number.

It’s reciprocal is given as 3/11.

Multiply both the numbers we get;

\mathtt{\Longrightarrow \ \frac{11}{3} \times \frac{3}{11}}\\\ \\ \mathtt{\Longrightarrow 1}

**Conclusion**

Multiplication of rational number with its reciprocal always results in number 1.

### Multiplicative inverse of Negative Rational numbers

The **reciprocal of negative rational number is also a negative rational number**.

Note that while taking the reciprocal only the position of numerator and denominator change, **the sign of the number remains the same**.**For example**;

Find the reciprocal of -6/11.

Interchanging the numerator and denominator numbers, we get -11/6.

Hence, \mathtt{\frac{-6}{11}} reciprocal is \mathtt{\frac{-11}{6}} .

### Does every rational number has multiplicative inverse ?

NO !!

Rational number 0 does not have any multiplicative inverse.

Let me help you understand this.

Number 0 can be written as \mathtt{\frac{0}{1}} .

Now if you interchange the position of numerator & denominator, you will get \mathtt{\frac{1}{0}} .

The any number with 0 in denominator is “not defined” .

\mathtt{\frac{1}{0} \Longrightarrow not\ defined}

Hence, rational number 0 has no reciprocal.

### Inverse of Reciprocal

When we take the inverse of reciprocal, we will return back to the original number.

For example;

Let a/b be the given rational number.

The reciprocal of a/b is b/a.

Now if you take reciprocal of b/a, you will get a/b, which is an original number.

### Rational number that are equal to their reciprocal

There are **only two rational numbers 1 & -1 whose value is equal to its reciprocal**.**For example;**

Consider rational number 1.

Number 1 can be written as \mathtt{\frac{1}{1}}

For reciprocal, let us interchange the position of numerator and denominator.

We get;

Hence, reciprocal of 1 is the same number 1.

Similar is the case with -1.**Conclusion**

The number 1 & -1 have value equal to its reciprocal.

I hope you understood the above concepts. Let us solve some problems for better clarity.

## Reciprocal of Rational number – Solved problems

(01) Find the reciprocal of below rational numbers.

(i) 5/3

(ii) -2/7

(iii) -7

(iv) 0

(v) 6/-13

(vi) 100/21

(vii) -11

**Solution**

We know that reciprocal can be found by switching the position of numerators and denominators.**(i) 5/3**

Reciprocal is 3/5

**(ii) -2/7**

Reciprocal is -7/2**(iii) -7**

The given number can be written as -7/1.

So the reciprocal is -1/7.**(iv) 0**

Reciprocal of number 0 is not defined.**(v) 6/-13**

Reciprocal is -13/6**(vi) 100/21**

Reciprocal is 21/100**(vii) -11 **

The number can be written as -11/1.

Reciprocal is -1/11.

(02) Find the reciprocal of given rational numbers.

(i) \mathtt{\frac{2}{3} \times \frac{7}{9}}

(ii) \mathtt{\frac{-1}{5} \times \frac{-6}{13}}

(iii) \mathtt{\frac{-11}{3} \times \frac{1}{4}}