Complex Numbers

This is a very terse review of their most important properties. An arbitrary complex number $z$ can be written as:

$$z = x + i y$$ (42)

$$= \vert z\vert \cos(\theta) + i \vert z\vert \sin(\theta)$$ (43)

$$= \vert z\vert e^{i\theta}$$ (44)

where $x = \vert z\vert\cos(\theta)$, $y = \vert z\vert\sin(\theta)$, and $\vert z\vert = \sqrt{x^2 + y^2}$. All complex numbers can be written as a real amplitude $\vert z\vert$ times a complex exponential form involving a phase angle. Again, it is difficult to convey how incredibly useful this result is without further study, but I commend it to your attention.