What Are Complex Conjugates?

There is an operation that can be performed on complex numbers called conjugation, also known as complex conjugation. The conjugate of a complex number $z$ is denoted by $\overset{}{z}$ —pronounced $z$ bar—or $z^{*}$ .

Theory

For a complex number

z = a + b i

, the conjugate is given by

\overset{}{z} = a - b i .

When you conjugate a number $z$ , you change the sign of the imaginary part of $z$ . In polar form, this corresponds to changing the sign of the argument of $z$ . In the complex plane you can think of conjugation of $z$ as a reflection of $z$ across the real axis.

Visualization of conjugation in the complex plane

The conjugate of $z$ has the same norm and the same real part as $z$ . What this means is that real numbers are not affected by conjugation. Real numbers are therefore called fixed points for conjugation. Conjugation is also an involution, meaning the result of a conjugation is the inverse of what you started with. This means that if you conjugate a complex number twice, you return to the starting point. The conjugate of the conjugate of $z$ is therefore $z$ :

\overset{}{\overset{}{z}} = z .

Example 1

Write $\overset{}{z}$ for $z = - 1 + \sqrt{3} i$ in both Cartesian form and in polar form.

In Cartesian form, you find the conjugate by changing the sign of the imaginary part of $z$ . So in Cartesian form, the conjugate of $z$ is therefore

\overset{}{z} = - 1 - \sqrt{3} i .

To write $\overset{}{z}$ in polar form, you first have to find the norm and the argument of $z$ . Here you have the norm $r = 2$ and the argument $𝜃 = \frac{2 π}{3}$ . In polar form, $z$ is therefore $z = 2 e^{i \frac{2 π}{3}}$ . Since you conjugate $z$ by changing the sign of the argument of $z$ , the conjugate $\overset{}{z}$ has norm $r = 2$ and argument $𝜃 = - \frac{2 π}{3}$ . In polar form, the conjugate of $z$ is therefore

\overset{}{z} = 2 e^{- i \frac{2 π}{3}} .

An important property of conjugation is that the product of a complex number $z$ with its own conjugate $\overset{}{z}$ is the square of the norm of $z$ , and therefore a real number:

Formula

For every complex number $z = a + b i$ , you have the following:

\begin{array}{l} z \cdot \overset{}{z} & = (a + b i) (a - b i) \\ = a^{2} + b^{2} \\ = {| z |}^{2} . \end{array}

z \cdot \overset{}{z} = (a + b i) (a - b i) = a^{2} + b^{2} = {| z |}^{2} .

Multiplication of complex numbers can be thought of as a combination of a scaling and a rotation in the complex plane. The arguments of $z$ and $\overset{}{z}$ are opposite: $𝜃$ and $- 𝜃$ . The product of $z$ and $\overset{}{z}$ is therefore a rotation of $z$ from $𝜃$ to $0$ . The product of $z$ and $\overset{}{z}$ is therefore a real number.

The norms of $z$ and $\overset{}{z}$ are equal. Multiplication of $\overset{}{z}$ can therefore be interpreted as a scaling of $z$ from $r$ to $r^{2}$ . The product of $z$ and $\overset{}{z}$ is therefore the square of the norm of $z$ . This property is, among other things, used to carry out division of complex numbers and to find the inverse of complex numbers.

Visualization of the product of a number with its own conjugate.

Rule

Rules for Conjugation

For complex numbers $z$ and $w$ , the following are true:

\begin{array}{l} \overset{}{z} + \overset{}{w} & = \overset{}{z + w}, \\ \overset{}{z} - \overset{}{w} & = \overset{}{z - w}, \\ \overset{}{z} \cdot \overset{}{w} & = \overset{}{z \cdot w}, \\ \frac{\overset{}{z}}{\overset{}{w}} & = \overset{}{(\frac{z}{w})}, w \neq 0 . \end{array}

The rules for conjugation state that the order of conjugation and arithmetic operations doesn’t matter. It is an equivalent process if you first carry out an arithmetic operation, and then conjugate the result, or if you first conjugate, and then carry out an operation.

Example 2

Find the conjugate of $\overset{}{z} \cdot w$

You can use the rules for conjugation to change the order of conjugation and multiplication

\overset{}{\overset{}{z} \cdot w} = \overset{}{\overset{}{z}} \cdot \overset{}{w} .

Then you can use the fact that conjugation is an involution to simplify:

\overset{}{\overset{}{z}} = z .

The conjugate of $\overset{}{z} \cdot w$ is therefore

\overset{}{\overset{}{z} \cdot w} = \overset{}{\overset{}{z}} \cdot \overset{}{w} = z \cdot \overset{}{w} .

Think About This

Can you spot the similarity between the product of a number with its own conjugate and the identity for the difference of squares?

The identity for the difference of squares is used to factorize expressions of the form $a^{2} - b^{2}$ : $\begin{array}{l} a^{2} - b^{2} = (a + b) (a - b) . \end{array}$

By using complex conjugation you can now also factorize expressions of the form $a^{2} + b^{2}$ : $\begin{array}{l} a^{2} + b^{2} = (a + b i) (a - b i) . \end{array}$

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