# Multiplicative Inverse of a Complex Number

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• 0:04 Multiplicative Inverse
• 0:47 Complex Numbers
• 1:12 Inverse of a Complex Number
• 2:31 Lesson Summary

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Lesson Transcript
Instructor: Usha Bhakuni

Usha has taught high school level Math and has master's degree in Finance

In this lesson, you'll learn about the multiplicative inverse of a complex number and how to find it. The process is explained with the help of relevant examples.

## Multiplicative Inverse

A multiplicative inverse is a number that, when multiplied by the given number, yields 1. So, how do you find the multiplicative inverse of any number? It's simple. Given a non-zero number a, its multiplicative inverse can be found out by solving for x as seen in the following equation.

So we can see that the multiplicative inverse of a non-zero number a would be its reciprocal, 1 over a. For example, the multiplicative inverse of 8 would be 1 over 8, as you can see below.

Note that the number zero is non-invertible as its inverse 1 over 0 is undefined.

## Complex Numbers

The thing that is most complex about complex numbers is that they consist of an imaginary part. A complex number is of the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as the square root of -1. For example, 3 + 5i is a complex number. The image below shows the imaginary part i as the square root of -1.

## Inverse of a Complex Number

As we saw just a moment ago, the multiplicative inverse of a number is basically its reciprocal. The same rule applies in the case of complex numbers. For example, the multiplicative inverse of 8 + 4i would be 1 over 8 plus 4i, which you can see play out here.

However, in order to fully complete this we have to rationalize the denominator. So, what are the steps for the rationalization of a complex number fraction? For a fraction with a complex number denominator, rationalize the denominator to convert the fraction into the form x + yi, where x and y are real numbers.

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