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Complex Numbers

We assume that the student has the basics of the complex numbers. The purpose of this chapter is to remind this basics and introduce some more advanced material which is relevant to this course.

Cartesian and polar forms of complex numbers

A complex number in the cartesian form $z$ can be written as $z=a+ib$ where $a, b$ are real numbers. It is possible to show such numbers in the complex plane. Another name for the complex plane is the Argand plane.

—figure here

By using trigonometry, we can compute two new quantities,

\item Magnitude of the complex number $z$ is denoted by $|z|=R=\sqrt{a^2+b^2}$. Note that magnitude of a complex number is always a positive real number, ie, $|z|.> 0$.
\item Angle of a complex number $z$ is denoted by $\angle{z}$ where $\theta = \mathrm{arctan}\left( \frac{b}{a} \right)+2k\pi$, where $k \in \mathbb{I}$. Note that the angle is not unique.
Given $|z|=R$ and  $\angle{z}=\theta$

Euler’s theorem:      $a+ib = Re^{i\theta}$