Modulus and Argument of Complex Numbers

COMPLEX NUMBER REPRESENTATION

An imaginary (complex) number $z$ can be written as:

$$\begin{aligned} z &= x + yi \\ z &= [ \;\; \underset{\begin{subarray}{c} \uparrow \\ \text{modulus} \end{subarray}}{|z|} , \quad \underset{\begin{subarray}{c} \uparrow \\ \text{argument} \end{subarray}}{\text{arg}(z)} \;\; ] \end{aligned}$$

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Other forms

$z = x + iy$ can be rewritten as:

$$\begin{aligned} z &= |z|\cos(\theta) + i|z|\sin(\theta) \\ z &= |z|(\cos(\theta) + i\sin(\theta)) \\ \therefore z &= |z|\operatorname{cis}(\theta) && \text{where } \operatorname{cis}(\theta) = \cos(\theta) + i\sin(\theta) \end{aligned}$$

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Degrees and Radians

To switch between degrees and radians, use these constants:

$$\boxed{1^\circ = \frac{\pi}{180}} \quad \boxed{1 \text{ rad} = \frac{180}{\pi}}$$

Degrees	Radians
$360^\circ$	$2\pi$
$270^\circ$	$\frac{3}{2}\pi$
$180^\circ$	$\pi$
$135^\circ$	$\frac{3}{4}\pi$
$90^\circ$	$\frac{1}{2}\pi$
$60^\circ$	$\frac{1}{3}\pi$
$45^\circ$	$\frac{1}{4}\pi$
$30^\circ$	$\frac{1}{6}\pi$

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Finding the Modulus and Argument

Finding the modulus of a complex number is very simple, we just use Pythagoreas' Theorem.

$$\begin{aligned} z &= x + yi \\ |z| &= \sqrt{x^2+y^2} \end{aligned}$$

The argument of a complex number $z = x + iy$ is the angle $\theta$ made with the positive real axis ($\text{cis}(\theta)$). Because $\tan(\theta) = \frac{y}{x}$, we use the signs of $x$ and $y$ to determine the correct quadrant.

First, find the reference angle: $\alpha = \tan^{-1} \left| \frac{y}{x} \right|$.

Quadrant	Location	Signs	Calculation	Range of $\theta$
I	Upper Right	$x > 0, y > 0$	$\theta = \alpha$	$0 < \theta < \frac{\pi}{2}$
II	Upper Left	$x < 0, y > 0$	$\theta = \pi - \alpha$	$\frac{\pi}{2} < \theta < \pi$
III	Lower Left	$x < 0, y < 0$	$\theta = -(\pi - \alpha)$	$-\pi < \theta < -\frac{\pi}{2}$
IV	Lower Right	$x > 0, y < 0$	$\theta = -\alpha$	$-\frac{\pi}{2} < \theta < 0$