## Complex Numbers

A complex number can be written in Cartesian coordinates as $z=x+iy$, where $x$ is the real part and $y$ is the imaginary part. Or it can be written in polar coordinates as $z=re^{i\theta}$, where $r$ is the absolute value and $e^{i\theta}$ is the point on the unit circle corresponding to the angle $\theta$ measured in the usual anti-clockwise direction from the positive $x$-semiaxis. The angle $\theta$ is called the argument, but unfortunately it is only defined up to multiples of $2\pi i$. The unimodular number $e^{i\theta}$ could be referred to as the direction of $z$.

The complex conjugate of $z$ is the complex number $\bar{z}=x-iy=r/e^{i\theta}$.

Then the Cartesian decomposition correspond to the following additive trick:

$\displaystyle 2z=(z+\bar{z})+(z-\bar{z})$

While the polar decomposition corresponds to a multiplicative trick:

$\displaystyle z^2=(z\cdot\bar{z})(z/\bar{z})$

By this I mean that $z+\bar{z}$ is twice the real part of $z$, while $z-\bar{z}$ is twice the imaginary part of $z$ times $i$. And on the other hand, $z\cdot\bar{z}$ is the square of the absolute value of $z$, and $z/\bar{z}$ is the square of the direction of $z$.