Complex Conjugation

Definition

Let $z=x+iy$. Then its complex conjugate is
$\bar{z}:=x-iy.$

Properties

The following hold:

1. $\mid z{\mid }^2=z\bar{z}$, ($=r^2$)
2. $z+\bar{z}=2\mathrm{Re}z$,
3. $z-\bar{z}=2i\mathrm{Im}z$,
4. $\overline{r{e}^{i\varphi }}=r{e}^{-i\varphi }$.

Note that (4) implies that

$\mid \bar{z}\mid =\mid z\mid$.

Also write that $z=r{e}^{i\varphi }$ and $z^{\prime }=r^{\prime }{e}^{i{\varphi }^{\prime }}$.

Then $z{z}^{\prime }=r{r}^{\prime }{e}^{i(\varphi +{\varphi }^{\prime })}$. Then (1) implies that

$\mid z{z}^{\prime }\mid =r{r}^{\prime }=\mid z\mid \mid z^{\prime }\mid$.

(Nice interplay between complex multiplication with absolute values).