# 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 ${\displaystyle 6+3i}$ and ${\displaystyle 6-3i}$ are conjugates.

## Notation and terms

For any complex number ${\displaystyle a+bi}$:

• ${\displaystyle a-bi}$ is the complex conjugate ${\displaystyle a+bi}$
• ${\displaystyle a+bi}$ is the complex conjugate ${\displaystyle a-bi}$
• ${\displaystyle a+bi}$ and ${\displaystyle a-bi}$ are called a complex conjugate pair

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

## Argand diagram

In the Argand diagram below:

${\displaystyle z_{1}=6+3i}$

${\displaystyle z_{2}=z_{1}^{*}=6-3i}$

All conjugate pairs are symmetrical about the x (real) axis.

## Special properties

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

${\displaystyle z+z^{*}=(a+bi)+(a-bi)=2a}$

The product of any complex number ${\displaystyle z}$, and its complex conjugate ${\displaystyle z^{*}}$ is always real:

${\displaystyle z.z^{*}=(a+bi)(a-bi)=a^{2}+a.bi-a.bi-b^{2}i^{2}}$

The two terms in ${\displaystyle a.bi}$ cancel out, and since ${\displaystyle i^{2}}$ is -1, the final result is:

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