# Complex conjugates

Level: A Level Maths

Complex conjugates are pairs of numbers which have the same real part, and where the imaginary part differs only in sign. For example $6+3i$ and $6-3i$ are conjugates.

## Notation and terms

For any complex number $a+bi$ :

• $a-bi$ is the complex conjugate $a+bi$ • $a+bi$ is the complex conjugate $a-bi$ • $a+bi$ and $a-bi$ are called a complex conjugate pair

For a complex number $z$ , the complex conjugate is written as $z^{*}$ .

## Argand diagram

In the Argand diagram below:

$z_{1}=6+3i$ $z_{2}=z_{1}^{*}=6-3i$ All conjugate pairs are symmetrical about the x (real) axis.

## Special properties

The sum of any complex number $z$ , and its complex conjugate $z^{*}$ is always real. This is because the two imaginary parts cancel out:

$z+z^{*}=(a+bi)+(a-bi)=2a$ The product of any complex number $z$ , and its complex conjugate $z^{*}$ is always real:

$z.z^{*}=(a+bi)(a-bi)=a^{2}+a.bi-a.bi-b^{2}i^{2}$ The two terms in $a.bi$ cancel out, and since $i^{2}$ is -1, the final result is:

$z.z^{*}=a^{2}+b^{2}$ 