Multiplicative Inverse of a Complex Number

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.


Multiplicative inverse equation

Multiplicative Inverse 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.


Mulitplicative Inverse

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.


Imaginary part of a complex number

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.


Multiplicative inverse of 8+4i

Multiplicative inverse of 8+4i

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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