The term inverse in mathematics means a reciprocal quantity or mathematical expression which is the output of inversion. The term reciprocal should not be interchanged with reciprocation, which is related to geometry.

In this module, we will learn what is multiplicative inverse and what is additive inverse and solve some examples related to them.

Multiplicative inverse and additive inverse will help you solve problems of algebra. While multiplicative inverse is used to nullify the effect of any number, the additive inverse is very helpful in getting rid of unwanted terms.

Let us now discuss both types in detail.

**Multiplicative Inverse**

A number which after multiplying by the number gives the result as 1 is known as the multiplicative inverse of that number.

The definition of multiplicative inverse can be put into another way i.e when the product of two numbers is one, both the numbers are said to be the multiplicative inverse of each other. It is commonly called the reciprocal of the number in consideration and is denoted by 1/a.

Example: The multiplicative inverse of 11 is 1/11. Why? 11 * 1/11 = 1. Let us check out another example of the multiplicative inverse. The multiplicative inverse of 4/5 is 5/4 because the product of 4/5 and 5/4 is 1.

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**Some Important Points of Multiplicative Inverse**

- The multiplicative inverse of the number 0 cannot be defined.
- We can obtain the multiplicative inverse of a fraction by reversing the order of the numerator and the denominator of the given fraction.
- The multiplicative inverse of the number 1 is the number itself.
- To find the multiplicative inverse of a mixed fraction, we should first convert the given mixed fraction into an improper fraction and then reverse the order of the numerator and the denominator of the improper fraction.
- We can find the multiplicative inverse of real numbers as well as complex numbers.

**Additive Inverse**

You have a glass of lukewarm water. You add a certain amount of hot water which makes the temperature of the water rise and now you cannot consume it. To nullify its effect, you add the same amount of cold water.

A similar thing applies in the case of the additive inverse. To put it forward in a layman’s language, the additive inverse is any number that is added with a number and results in zero.

Example: The additive inverse of 11 is -11. Why? 11 + (-11) = 0. Let us check out another example of the additive inverse. The additive inverse of 4/5 is -4/5 because the sum of 4/5 and -4/5 is zero.

**Some Important Points of Additive Inverse**

- The additive inverse of any number can have positive values as well as negative values.
- It is important to note that the additive inverse of the number 0 is the number itself.
- The additive inverse of a negative number can never be negative. Similarly, the additive inverse of a positive number can never be positive.

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**Some Solved Examples of Additive Inverse and Multiplicative Inverse**

1. What is the additive inverse of the number 16/35?

Solution: To find the additive inverse of a given number, it should satisfy the given equation:

x + (-x) = 0.

From the given equation, we can deduce that the additive inverse of 16/35 is -16/35.

2. What is the multiplicative inverse of the number 16/35?

Solution: To find the multiplicative inverse of a given number, it should satisfy the given equation: x + 1/x = 1.

From the given equation, we can deduce that the multiplicative inverse of 16/35 is 35/16.

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