# 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. $\overline{r e^{i \phi}} = r e^{-i \phi}$.

> [!note]+ Note that (4) implies that
>   
> $\mid \bar{z} \mid = \mid z \mid$.
> 
> Also write that $z = r e^{i \phi}$ and $z' = r' e^{i \phi'}$.
> 
> Then $zz' = rr'e^{i(\phi + \phi')}$. Then (1) implies that
> 
> $\mid z z' \mid = rr' = \mid z \mid \mid z' \mid$.
> 
> (Nice interplay between complex multiplication with absolute values).