Complex Numbers in Polar Form

While complex numbers are usually introduced in rectangular form as $z = a + bi$, they can also be written in polar form. Polar form is incredibly useful because it makes multiplying, dividing, and finding powers of complex numbers much simpler.

Understanding Polar Form

In the complex plane, a complex number $z = a + bi$ can be represented by its distance from the origin and its angle from the positive real axis.

• Modulus ($r$): The distance from the origin to the point. $r = \sqrt{a^2 + b^2}$
• Argument ($\theta$): The angle formed with the positive real axis. $\tan \theta = \frac{b}{a}$

Using trigonometry, we can express the complex number as: $z = r(\cos \theta + i\sin \theta)$ This is often abbreviated using the shorthand notation cis: $z = r\,\text{cis}(\theta)$

Converting Rectangular to Polar Form

Let's convert the complex number $z = 3 - 3i$ into polar form.

1. Find the modulus ($r$): $r = \sqrt{3^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}$
2. Find the argument ($\theta$): $\tan \theta = \frac{-3}{3} = -1$ Since $a$ is positive and $b$ is negative, the point lies in the 4th quadrant. The angle with a tangent of $-1$ in the 4th quadrant is $\theta = \frac{7\pi}{4}$ (or $-\frac{\pi}{4}$).
3. Write in polar form: $z = 3\sqrt{2}\left(\cos\left(\frac{7\pi}{4}\right) + i\sin\left(\frac{7\pi}{4}\right)\right)$ Or simply, $z = 3\sqrt{2}\,\text{cis}\left(\frac{7\pi}{4}\right)$.

Multiplication and Division in Polar Form

Polar form makes multiplication and division incredibly straightforward.

• To multiply: Multiply the moduli and add the arguments. $z_1 z_2 = r_1 r_2 \,\text{cis}(\theta_1 + \theta_2)$
• To divide: Divide the moduli and subtract the arguments. $\frac{z_1}{z_2} = \frac{r_1}{r_2} \,\text{cis}(\theta_1 - \theta_2)$

Example: Multiplication

Multiply $2\,\text{cis}\left(\frac{\pi}{3}\right) \cdot 4\,\text{cis}\left(\frac{\pi}{4}\right)$ and convert the result to rectangular form.

1. Multiply the moduli: $2 \times 4 = 8$
2. Add the arguments: $\frac{\pi}{3} + \frac{\pi}{4} = \frac{4\pi}{12} + \frac{3\pi}{12} = \frac{7\pi}{12}$
3. Polar result: $8\,\text{cis}\left(\frac{7\pi}{12}\right)$

To convert this back to rectangular form $a + bi$, evaluate the sine and cosine of $\frac{7\pi}{12}$:

• $\cos\left(\frac{7\pi}{12}\right) = \frac{\sqrt{2} - \sqrt{6}}{4}$
• $\sin\left(\frac{7\pi}{12}\right) = \frac{\sqrt{6} + \sqrt{2}}{4}$

Now distribute the modulus $8$: $a = 8 \left(\frac{\sqrt{2} - \sqrt{6}}{4}\right) = 2\sqrt{2} - 2\sqrt{6}$ $b = 8 \left(\frac{\sqrt{6} + \sqrt{2}}{4}\right) = 2\sqrt{6} + 2\sqrt{2}$

Rectangular result: $(2\sqrt{2} - 2\sqrt{6}) + i(2\sqrt{6} + 2\sqrt{2})$

De Moivre's Theorem

If you need to raise a complex number to a power, doing it in rectangular form requires tedious binomial expansion. In polar form, De Moivre's Theorem gives us a powerful shortcut:

$z^n = [r(\cos \theta + i\sin \theta)]^n = r^n(\cos(n\theta) + i\sin(n\theta))$

Simply raise the modulus to the power of $n$ and multiply the argument by $n$.