Complex numbers.

Let $z=x+iy$ be a complex number.

1. The conjugate $\overline{z}=x-iy$.

   The real part $Re(z) = x = \frac{z+\overline{z}}{2}$.

   The imaginary part $Im(z) = y = \frac{z-\overline{z}}{2i}$.

   The norm $\vert z\vert = \sqrt{z\overline{z}} = \sqrt{x^2+y^2}$.

2. The reciprocal $\frac{1}{z} = \frac{x-iy}{x^2+y^2}$.

3. The direction from $\mathbf{0}$ (for a non zero $z$) $\frac{z}{\vert z\vert}$.

4. The Euler representation. $z=r\exp(i\theta)$ where $r=\vert z\vert$ and $\theta= Arg(z) = \arctan(y/z)$. The angle $\theta$ can be replaced by any angle in the set $arg(z)=\{Arg(z)+2n\pi \vert n \mbox{\rm Z \kern -0.82em Z} \}$.