Definition:Complex Number/Polar Form

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Definition

For any complex number $z = x + i y \ne 0$, let:

\(\displaystyle r\) \(=\) \(\displaystyle \cmod z = \sqrt {x^2 + y^2}\) $\quad$ the modulus of $z$, and $\quad$
\(\displaystyle \theta\) \(=\) \(\displaystyle \arg z\) $\quad$ the argument of $z$ (the angle which $z$ yields with the real line) $\quad$

where $x, y \in \R$.

From the definition of $\arg z$:

$(1): \quad \dfrac x r = \cos \theta$
$(2): \quad \dfrac y r = \sin \theta$

which implies that:

$x = r \cos \theta$
$y = r \sin \theta$

which in turn means that any number $z = x + i y \ne 0$ can be written as:

$z = x + i y = r \paren {\cos \theta + i \sin \theta}$

The pair $\polar {r, \theta}$ is called the polar form of the complex number $z \ne 0$.


The number $z = 0 + 0i$ is defined as $\polar {0, 0}$.


Exponential Form

From Euler's Formula:

$e^{i \theta} = \cos \theta + i \sin \theta$

so $z$ can also be written in the form:

$z = r e^{i \theta}$


Also known as

Some sources refer to polar form as trigonometric form.

As $\cos \theta + i \sin \theta$ appears so often in complex analysis, the abbreviation $\operatorname{cis} \theta$ is frequently seen.


Examples

Example: $i$

The imaginary unit $i$ can be expressed in polar form as $\polar {1, \dfrac \pi 2}$.


Example: $-1$

The real number $-1$ can be expressed as a complex number in polar form as $\left\langle{1, \pi}\right)$.


Sources