This page will help you in learning about the Reciprocal of a fraction or Multiplicative Inverse of a fraction. This will give the definition, explain the rules and steps that one should follow while solving problems on Multiplicative Inverse, Know From where Multiplicative Inverse is Derived, and few solved examples on finding the Multiplicative Inverse.

Also, Refer: Properties of Multiplication

## What is meant by Multiplicative Inverse?

When the product of two fractions or a fraction and a natural number is 1, then one of them is called the reciprocal or multiplicative inverse of the other.

In simple words, reciprocal is defined as the inverse of a value or a given number. Consider N as a real number, then its reciprocal will be 1/N. This means, here we just convert the given number into an upside-down form. The multiplicative inverse of any given number is represented by 1/N or N^{-1}

The Multiplicative Inverse of any given number is obtained in such a way that when it is multiplied with the original number the value is equally identity by 1. In other words, we can say that it is a method of dividing a number by its own to generate identity 1, which means N/N = 1.

So we can say that whenever a number is multiplied by its own multiplicative inverse the resultant value will always be equal to 1. Now that we have some idea about multiplicative inverse let’s try to define this.

### Multiplicative Inverse – Definition

Reciprocal or Multiplicative Inverse means an expression or a number which when multiplied by another expression or a number, gives 1 as its result. For any number ‘a’, the reciprocal will be 1/a or a^{-1} (Inverse of a).

Therefore, a reciprocal or Multiplicative Inverse is just a flipped number that can bring back the original number.

#### Multiplicative Inverse of Natural Numbers

Consider x as any natural number (except 0) then the multiplicative inverse of x will be 1/x.

For example, consider a natural number 5 then the multiplicative inverse (or) reciprocal for 5 will be 1/5. Whereas the reciprocal of 0 will give an infinite value because 1/0 = ∞.

#### Multiplicative Inverse of Fraction Numbers

The reciprocal or multiplicative inverse for any given fraction number can be obtained by simply interchanging the numerator and the denominator values.

**Example**: Find the reciprocal of (4/5)

Solution:

To find the multiplicative inverse we need to follow the following steps.

The reciprocal of 4/5 is 5/4.

Or else we can also use the formula, N = 1/N, where 4/5 = (1/4)/(1/5)

Therefore, the reciprocal of a fraction number 4/5 is 5/4.

#### Multiplicative Inverse of Mixed Fraction Numbers

The reciprocal or multiplicative inverse of a mixed fraction number can be found by converting the given mixed fraction number to a fraction number and then we need to do the operation.

**Example: **Find the reciprocal of 4 (1/2)

Solution:

To find the multiplicative inverse for a mixed fraction number we need to follow few steps.

First, we need to convert the given mixed fraction number to a fraction number.

4 (1/2) = (9/2)

The reciprocal or multiplicative inverse of 9/2 is 2/9.

#### Multiplicative Inverse of Decimal Numbers

The reciprocal or multiplicative inverse of a decimal number is the same as it is for a number defined by one over the number.

**Example: **Find the multiplicative inverse for a decimal number (0.75)

Solution:

Method 1:

The multiplicative inverse of a natural number, x=1/x

So in the same way the multiplicative inverse of a decimal number, 0.75= 1/0.75

Method 2:

We also have an alternate method to find the multiplicative inverse of a decimal number as below.

We will consider the same example for this method (0.75)

First of all, we need to check whether the given decimal number can be converted into a fractional number. So here our decimal number is 0.75 which can be written as 3/4

Now, we need to find the multiplicative inverse of 3/4 which gives us 4/3

Now when you verify both the solutions from the above two methods the results are the same.

This means, 1/0.75 = 1.33 and 4/3 = 1.33.

#### Examples Problems on Multiplicative Inverse

Let us see few examples of the multiplicative inverse.

**Example 1: **Find the multiplicative inverse for following Integers

(i) 2 (ii) 9 (iii) -5

Solution:

Multiplicative Inverse of any given integer x will be 1/x.

**(i)** Multiplictiave inverse of 2 will be 1/2

**(ii)** Multiplictiave inverse of 9 will be 1/9

**(iii)** Multiplictiave inverse of -5 will be -1/5

**Example 2: **Find the multiplicative inverse for the following Fractions

(i) 11/33 (ii) 5/4

Solution:

Multiplicative Inverse of any given Fraction x/y will be y/x.

**(i)** Multiplictiave inverse of 11/44 will be 44/11

**(ii)** Multiplictiave inverse of 5/3 will be 3/5.

**Example 3: **Find the multiplicative inverse for the following Mixed Fractions.

(i) 3(2/3) (ii) 4(1/2)

Solutions:

**(i)** Mixed fraction 3(2/3) can be written as Fraction (9/2)

So now the Multiplictiave inverse of 9/2 will be 2/9

This means Multiplictiave inverse of Mixed fraction 3(2/3) will be (2/9)

**(ii)** Mixed fraction 4(1/2) can be written as Fraction (9/2)

So now the Multiplictiave inverse of 9/2 will be 2/9

This means Multiplictiave inverse of Mixed fraction 4(1/2) will be (2/9)

### FAQs on Multiplicative Inverse

**1. How to find the Multiplicative Inverse?**

Converting the given number upside down is known as the Multiplicative Inverse. Whenever a Number is multiplied by its multiplicative inverse the result is 1.

**2. What is the Multiplicative Inverse of 0 (Zero)?**

Number zero (0) will not have a multiplicative inverse. Because, if any multiplicative inverse number is multiplied by 0, it will not give the product as 1. It will result in zero, According to our Multiplictiave inverse when the reciprocal number is multiplied result should be 1.

**3. What is the Multiplicative Inverse of Infinity (∞)?**

The multiplicative inverse of infinity will always be zero(0). This means that 1/∞ will be equal to 0. It is noted that the multiplicative inverse of infinity is zero exactly, which means not infinitesimal.