Complex conjugates

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

$\bar{z}:=a-bi=r{e}^{-i\theta },$

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

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

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

and

$z\bar{z}=(a+bi)(a-bi)=a^2+b^2=r{e}^{i\theta }r{e}^{-i\theta }=r^2.$

Complex norm

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

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

See also

• Conjugates