# Multiplicative Inverse Definition

The **multiplicative inverse** of a number is the number that gives “**1**” as the answer when the **original number** is multiplied by it. **For example**, when we multiply **2** by **1/2**, we get 1 as the output which means 1/2 is the **multiplicative** inverse of 2 and vice versa.

Generally, the word **inverse** means **opposite** to the original thing. In Mathematics, the **multiplicative inverse **is the **reciprocal** of the **original number** because it is considered the opposite of the original one. We can also say that if we **divide** the original number by “**1**”, it will give the multiplicative inverse of that number.

**Note:**

Read about Reciprocal and Division here.

**Table of Contents**

## Representation of Multiplicative Inverse

In Mathematics, the multiplicative inverse of the number is represented by putting “**-1**” as the exponent of the number. **For example**, if we have a number “**n**”, the multiplicative inverse of this number will be denoted as “**n-1**” or “**1/n**”.

**Note:**

Read about **Exponent** here.

### How to find the Multiplicative Inverse?

Finding this term is not a difficult task because you only have to be proficient in basic Mathematics. Here we have listed methods to find the multiplicative inverse of different types of numbers.

### For Natural Numbers

If you have a “**Natural Number**”, you can find the inverse of this number in a **single step**. You only have to **reciprocal** the original number to get its inverse.

**Note:**

Read about **Natural Number **here.

**Example 1: **

Find the multiplicative inverse of “5”.

**Solution: **

As “5” is the natural number, we only have to find its reciprocal. To do so, we have to divide this term by 1.

Multiplicative Inverse = 1/5

For confirmation, we have to multiply the original number by its reciprocal. By doing so, we have found “1” as the answer that indicates that both numbers are inverse to each other.

### For Fractions

Finding the multiplicative inverse of the fraction is a little different from the above method. Here are the steps that you have to follow in this regard.

- Write the
**original fraction** - Make the
**numerator**of the original fraction denominator of a new fraction - Similarly, write the
**denominator**of the original fraction as the numerator of the new fraction - The fraction you have written now will be the multiplicative inverse of the original fraction

**Example 2: **

Find the multiplicative inverse of 2/3.

**Solution: **

Let us find the multiplicative inverse of this fraction using the above steps.

First, we have to switch the places of the numerator and denominator with each other in the new fraction.

Here is what we have found as the result.

Multiplicative Inverse = 3/2

By multiplying the original fraction with this one, we found that it gives “1” as the output. It confirms that both numbers are multiplicative inverse for each other.

### For Mixed Fractions

The method to find the **multiplicative inverse** of **mixed** fractions is similar to the above method. The only additional step is the **conversion** of the mixed fraction into a **simple fraction** (**Proper** and **Improper** fraction).

When you have converted the mixed fraction into a simple fraction, you can follow the above steps to find its inverse.

**Note:**

Read about **Mixed**, **Proper**, and **Improper** **Fractions** here.

### Fun Facts about Multiplicative Inverse

- The reciprocal of 1 is always 1.
- “0” is the only whole number that has no multiplicative inverse.
- The result of the multiplication of the number with its multiplicative inverse is always “1”.
- The multiplicative inverse of a mixed fraction can’t be found without simplifying the fraction.
- The inverse of a proper fraction will always be an improper fraction.

### FAQ's

**What is multiplicative inverse?**

It is a number that gives “1” as the result when multiplied by the original number.

**Can we find the multiplicative inverse of a complex number?**

Yes, we can find the inverse of a complex number but we have to first solve it with its conjugate.

**Is there any formula to find the multiplicative inverse?**

No, there is no formula but it is a simple process in which you only have to reciprocal the original number.

**What is the difference between multiplicative and additive inverse?**

The additive inverse is the number that gives “0” as the output when added to a number. On the other hand, the multiplicative inverse is the number that gives “1” as the output when multiplied.

**Does “0” has a multiplicative inverse?**

No, “0” has no multiplicative inverse.

**What does inverse mean in Mathematics?**

In Mathematics, Inverse means the opposite or reciprocal of any number.