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Complex Numbers

This is a very terse review of their most important properties. An arbitrary complex number $z$ can be written as:
$\displaystyle z$ $\textstyle =$ $\displaystyle x + i y$ (42)
  $\textstyle =$ $\displaystyle \vert z\vert \cos(\theta) + i \vert z\vert \sin(\theta)$ (43)
  $\textstyle =$ $\displaystyle \vert z\vert e^{i\theta}$ (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.

Robert G. Brown 2004-04-12