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# Complex conjugates

Complex conjugates are pairs of complex numbers ${\displaystyle z=a+bi}$ and ${\displaystyle z=a-bi}$. The complex conjugate ${\displaystyle {\overline {z}}}$ of a complex number ${\displaystyle z=a+bi=re^{i\theta }}$ is defined as

${\displaystyle {\overline {z}}:=a-bi=re^{-i\theta },\,}$

where ${\displaystyle r}$ is the complex norm and ${\displaystyle \theta }$ is the complex argument.

For example, the complex conjugate of ${\displaystyle 2+i}$ is ${\displaystyle 2-i}$.

Such pairs often occur as roots of cubic equations.

## Properties

${\displaystyle {\frac {z+{\overline {z}}}{2}}={\frac {(a+bi)+(a-bi)}{2}}=a=r\left({\frac {e^{i\theta }+e^{-i\theta }}{2}}\right)=r\cos \theta ,\,}$
${\displaystyle {\frac {z-{\overline {z}}}{2i}}={\frac {(a+bi)-(a-bi)}{2i}}=b=r\left({\frac {e^{i\theta }-e^{-i\theta }}{2i}}\right)=r\sin \theta ,\,}$

and

${\displaystyle z{\overline {z}}=(a+bi)(a-bi)=a^{2}+b^{2}=re^{i\theta }re^{-i\theta }=r^{2}.\,}$

## Complex norm

The complex norm of a complex number ${\displaystyle z}$ is defined as

${\displaystyle |z|:={\sqrt {z{\overline {z}}}}={\sqrt {(a+bi)(a-bi)}}={\sqrt {a^{2}+b^{2}}}={\sqrt {re^{i\theta }re^{-i\theta }}}=r.\,}$