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Complex conjugates

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

${\overline {z}}:=a-bi=re^{-i\theta },\,$ where $r$ is the complex norm and $\theta$ is the complex argument.

For example, the complex conjugate of $2+i$ is $2-i$ .

Such pairs often occur as roots of cubic equations.

Properties

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

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

The complex norm of a complex number $z$ is defined as

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