A point in the complex plane can be described by its distance and angle as well as by its coordinates, just like polar coordinates. For complex numbers this polar description makes multiplication remarkably simple, as the next lesson shows.
Key idea
$$z = a + bi = r(\cos\theta + i\sin\theta),$$ where $r = |z| = \sqrt{a^2 + b^2}$ is the modulus and $\theta$ is the argument, the angle from the positive real axis (with $\tan\theta = \frac{b}{a}$ in the right quadrant). Going back: $a = r\cos\theta$, $b = r\sin\theta$. Some books write $r\operatorname{cis}\theta$ for short.
Why it works
$z$ is the point $(a, b)$. Its polar coordinates are $(r, \theta)$ with $a = r\cos\theta$ and $b = r\sin\theta$, so $$a + bi = r\cos\theta + ir\sin\theta = r(\cos\theta + i\sin\theta).$$
Worked examples
Worked example: To polar form
Write $-1 + i\sqrt{3}$ in polar form.
- $r = \sqrt{1 + 3} = 2$.
- The point $(-1, \sqrt{3})$ is in Quadrant II with reference angle $\frac{\pi}{3}$ (since $\tan = \sqrt{3}$), so $\theta = \frac{2\pi}{3}$.
Answer: $2\left(\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3}\right)$
Worked example: Back to $a + bi$
Write $4\left(\cos\frac{7\pi}{4} + i\sin\frac{7\pi}{4}\right)$ in the form $a + bi$.
- $\cos\frac{7\pi}{4} = \frac{\sqrt{2}}{2}$ and $\sin\frac{7\pi}{4} = -\frac{\sqrt{2}}{2}$.
- $4 \cdot \frac{\sqrt{2}}{2} = 2\sqrt{2}$.
Answer: $2\sqrt{2} - 2\sqrt{2}\,i$
Worked example: On an axis
Write $-3i$ in polar form.
- It is the point $(0, -3)$: distance $3$, straight down.
Answer: $3\left(\cos\frac{3\pi}{2} + i\sin\frac{3\pi}{2}\right)$
Common mistake
Reading the argument off a calculator's $\tan^{-1}\frac{b}{a}$ without checking the quadrant, the same trap as with polar coordinates. For $-1 + i\sqrt{3}$ that gives $-\frac{\pi}{3}$, which points into Quadrant IV. Plot the point first.