The Polar Representation of Complex Numbers
Let $z = a + bi \in \mathbb{C}$. One way to represent this number is through its polar representation in the complex plane. Recall from The Absolute Value of Complex Numbers page that the absolute value of $z$ is defined as:
(1)This represents the distance of $z$ from the origin in the complex plane or equivalently, the length of the position vector representing $z$. This position vector will make an angle with the positive real-axis which we define below.
Definition: Let $z = a + bi \in \mathbb{C}$. The Argument of $z$ denoted $\arg (z)$ is the angle $\theta$ made by the position vector of $z$ with the positive real-axis. |
For example, let $z = 1 + i$. Then the angle formed by $z$ with the positive real-axis will be $\displaystyle{\arg (z) = \frac{\pi}{4}}$, or of course more generally, $\displaystyle{\arg (z) = \frac{\pi}{4} + 2k\pi}$ for all $k \in \mathbb{N}$. Often times we will be restricting ourselves to a particular range of arguments so in context it will be clear when there is a unique argument for a complex number.
More precisely, the notation "$\mathrm{Arg} (z)$" will be used to denote that the argument of $z$ is restricted to $0 \leq \mathrm{Arg} (z) < 2 \pi$ called the Principal Branch of the Argument Function.
We are now ready to define the polar representation of a complex number.
Definition: Let $z = a + bi \in \mathbb{C}$, $z \neq 0$. The Polar Representation of $z$ is defined to be $z = r (\cos \theta + i \sin \theta)$ where $r = \mid z \mid$ and $\theta = \arg (z)$ where we simply write $(r, \theta)$. |
We will define the polar representation for only nonzero complex numbers since the complex number $0$ doesn't have an an unambiguous argument.
To justify the formula for the polar representation of a complex number, let $z = a + bi$. Then $z$ is entirely determined by the argument of $z$ and the absolute value of $z$ and moreover, if $r = \mid z \mid$ and $\theta = \arg (z)$ then basic trigonometry gives us that $a = r \cos \theta$ and $b = r \sin \theta$:
Hence:
(2)For example, if $z = 1 + i$ then $r = \sqrt{1 + 1} = \sqrt{2}$, and so the polar representation of $z$ is:
(3)Or rather, $\displaystyle{(r, \theta) = \left ( \sqrt{2}, \frac{\pi}{4} \right )}$.