The Polar Representation of Complex Numbers

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)
\begin{align} \quad \mid z \mid = \sqrt{a^2 + b^2} \end{align}

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.
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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 $(r, \theta)$.
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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)
\begin{align} \quad z = a + bi = r \cos \theta + i r \sin \theta = r (\cos \theta + i \sin \theta) \end{align}

For example, if $z = 1 + i$ then $r = \sqrt{1 + 1} = \sqrt{2}$, and so the polar representation of $z$ is:

(3)
\begin{align} \quad z = \sqrt{2} \left ( \cos \left ( \frac{\pi}{4} \right ) + i \sin \left ( \frac{\pi}{4} \right ) \right ) \end{align}

Or rather, $\displaystyle{(r, \theta) = \left ( \sqrt{2}, \frac{\pi}{4} \right )}$.

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