Multiplicative Inverse: Definition, Property & Examples

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Lesson Transcript
Instructor: Karin Gonzalez

Karin has taught middle and high school Health and has a master's degree in social work.

In this lesson, we will cover the definition of the multiplicative inverse, as well as its property. We will review some examples of the property so that we can gain a better understanding of the material. Following the lesson will be a brief quiz.

What Is the Multiplicative Inverse?

A multiplicative inverse is a reciprocal. What is a reciprocal? A reciprocal is one of a pair of numbers that when multiplied with another number equals the number 1. For example, if we have the number 7, the multiplicative inverse, or reciprocal, would be 1/7 because when you multiply 7 and 1/7 together, you get 1!


Let's look at a couple examples before proceeding with the lesson.

Example 1:

What is the multiplicative inverse of 15? In other words, which number when multiplied with 15 would give us the number 1 as a result? Let's solve this in an algebraic way, with x being the unknown multiplicative inverse.

15 * x = 1
x = 1/15

That's it! It was really that simple! The multiplicative inverse of a number is that number as the denominator and 1 as the numerator. When we multiply 15 and 1/15, we get 1.

Example 2:

What is the multiplicative inverse of 1/4? Now this example is a little different because we are beginning with a fraction. Let's again solve this algebraically, with x being the unknown multiplicative inverse of 1/4.

1/4 * x = 1
x = 1 / (1/4)
(1/1) / (1/4) = (1/1) * (4/1) = 4

Remember that when you divide fractions, you must flip the numerator and denominator of the second fraction and then multiply. We got 4 as the multiplicative inverse of 1/4. Makes sense, right?

So, the conclusion that we can draw from these two examples is that when you have a whole number, the multiplicative inverse of that number will be that number in fraction form with the whole number as the denominator and 1 as the numerator. When you have a fraction with 1 as the numerator, the multiplicative inverse of that fraction will simply be the denominator of the fraction.

Multiplicative Inverse of More Complicated Fractions

You may be thinking, that's just way too easy! What about when we have a fraction like 4/5? What would be the multiplicative inverse of that? Well, let's solve it algebraically, with x being the unknown multiplicative inverse.

4/5 * x = 1
1 / (4/5) = x
(1/1) * (5/4) = x
5/4= x

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