Math Doubts

Conjugate of a Complex number

Definition

A complex number having an equal real part and an equal imaginary part in magnitude but opposite in sign is called the complex conjugate of a complex number.

Introduction

Let’s learn what the conjugate of a complex number really is in detail. A complex number may have following two similarities with another complex number.

  1. The real part of a complex number is equal to the real part of another complex number.
  2. The imaginary part of a complex number in magnitude is equal to the magnitude of another complex number but they both have opposite signs.

Now, a complex number is called the conjugate of another complex number, and vice-versa.

complex conjugate

Let’s assume that $a$ and $b$ denote two real numbers. A complex number can be written in the following two forms popularly in terms of $a$ and $b$, where $i$ denotes an imaginary unit.

  1. $a+ib$
  2. $a-ib$

The real part of both complex numbers are $a$ and they both are equal in magnitude. Similarly, the imaginary part of both complex numbers are $b$ and they both are equal in magnitude but they have opposite signs.

So, the complex number $a+ib$ is called the conjugate of complex number $a-ib$. Similarly, the complex number $a-ib$ is called the conjugate of complex number $a+ib$.

Notation

Now, let’s learn how to represent the conjugates of complex numbers in mathematics.

A complex number is denoted by $z$ simply in mathematics. So, $z \,=\, a+ib$. The conjugate of a complex number is written as a straight line over $z$, which means $\overline{z}$. Similarly, it is also written by displaying an asterisk symbol at superscript position of $z$, which means $z^\ast$.

  1. $\overline{z} \,=\, a-ib$
  2. $z^\ast \,=\, a-ib$

So, you can use any one of them to represent the complex conjugate of a complex number.

Examples

Find conjugate of $2+3i$

Let’s denote the complex number $2+3i$ by $z$, which means $z \,=\, 2+3i$. Now, follow below two steps to find the complex conjugate of any complex number.

  1. The real part of complex number $z$ is $2$. So, the real part of its conjugate is also $2$.
  2. The magnitude of its imaginary part is $3$. The magnitude of imaginary part of its conjugate should also be $3$ but its sign should be opposite.

Therefore, the complex conjugate of complex number is $2-3i$. It is written as $\overline{z} \,=\, 2-3i$ or $z^\ast \,=\, 2-3i$ mathematically.

Now, let’s learn more about the complex conjugate of a complex number in detail.

Geometric Representation

Learn how to represent the conjugate of a complex number geometrically in an argand diagram.

Properties of Conjugates

The list properties of conjugates of complex numbers with proofs and understandable examples.