Importance of Complex Conjugates

Table of Contents

Complex Conjugates are one of the most important aspects of the complex number because they are important in quotient computation and for turning a complex number into a real one. The complex conjugate is very easy to identify and does not require any serious mathematics. However, students must know when to use them and when not to know.

There are two ways the complex number can be expressed. The first is the standard rectangular form, while the other is the polar form. The conjugates of a complex number work better for the standard complex number expression and are only usually employed when dividing two complex numbers.

This article will quickly consider what Complex Conjugates are and how they help in finding the division of complex numbers.

Complex Conjugates Definition

The Complex conjugates of a complex number z = a + bi is simply the same complex number but with an opposite sign: Z = a – bi.

The multiplication of a complex number with its conjugate will lead to a real number. The main importance of the complex conjugates is in the area of carrying out the division. Without the conjugate division, operation in complex numbers will not be possible.

The Complex Conjugate Theorem

The Complex Conjugate theorem states that for every polynomial P with a root given in the form of a complex number a + bi where a and b are real numbers, the conjugate root is a – bi.

How To Solve Division With Complex Conjugates

Step 1: arrange the complex relevant complex numbers in numerator/ denominator form

Step 2: Multiply the numerator and denominator with the complex conjugates of the denominator.

Step 3: A real number will be obtained as the denominator, which will be used to divide the numerator to get the required solution.

Complex Conjugates Examples

Example 1

Let Z = 3 + 5I be a complex number. Find its coinjuagate


Z = 3 + 5i

Clearly, from the definition of complex conjugates; the conjugate of the above complex number will be

Z = 3 – 5i

Example 2

Let Z1 = 7 + 4i and Z2 = 6 + 9i. Divide Z1 by Z2


However, there is no way to perform the division. So we use the complex conjugates pattern

Step 1: arrange the complex relevant complex numbers in numerator/ denominator form

Step 2: Multiply the numerator and denominator with the complex conjugates of the denominator.

Clearly, the Conjugates of the denominator is 6 – 9i. as such, we will have


The complex conjugates are a very important component of the complex plane. While its major involvement is for the standard rectangular complex number expression, it is also indirectly influential for the polar form of complex number expressions. Several examples were considered in this article to outline exactly how complex conjugates work and what students need to know about it. Each step was explicitly explained so that every information could be fully processed and understood. The aim of this explanation is to simply make the other type of complex conjugates calculations easy and straight to the point.

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