Complex Numbers

Definition

$\mathbb{C}$ is obtained from $\mathbb{R}$ by adjoining the imaginary unit $i$ such that $i^2=-1$.

In particular, a complex number is of the form
$z=x+iy$
with $x$, $y$ being real, we write
$\mathrm{Re}z=x,\mathrm{Im}z=y.$

Absolute Value

Let $z=x+iy$. Its absolute value (or norm) is defined by
$\mid z\mid \sqrt{x^2+y^2}.$