Polar form of a complex number

The Cartesian form $z = a + bi$ is convenient for addition. The polar form $z = r(\cos\theta + i\sin\theta)$ is convenient for multiplication. Switching between the two is the single most important skill in this chapter.

Definitions

For any $z \ne 0$ in the Argand plane, let $r = |z|, \quad \theta = \arg(z).$ Then $\boxed{z = r(\cos\theta + i\sin\theta).}$

This is the polar (or trigonometric) form of $z$. The pair $(r, \theta)$ are polar coordinates of the point. We sometimes write the right side as $r\,\text{cis}\,\theta$ , a shorthand for "$r$ times $\cos\theta + i\sin\theta$".

The argument $\theta$ is determined up to adding $2\pi$. The principal argument $\text{Arg}(z) \in (-\pi, \pi]$ is the canonical choice.

Conversion

Cartesian to polar. Given $z = a + bi$: $r = \sqrt{a^2 + b^2}, \quad \tan\theta = \frac{b}{a} \text{ (with quadrant adjustment)}.$

Polar to Cartesian. Given $r, \theta$: $a = r\cos\theta, \quad b = r\sin\theta.$

Multiplication and division in polar form

Let $z_1 = r_1(\cos\theta_1 + i\sin\theta_1)$ and $z_2 = r_2(\cos\theta_2 + i\sin\theta_2)$.

Then $z_1 z_2 = r_1 r_2 \big(\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)\big),$ $\frac{z_1}{z_2} = \frac{r_1}{r_2}\big(\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)\big).$

Multiply moduli, add arguments. Divide moduli, subtract arguments. This is the geometric content of complex multiplication, and the deepest reason why complex numbers are useful.

Proof of the multiplication formula

$z_1 z_2 = r_1 r_2 (\cos\theta_1 + i\sin\theta_1)(\cos\theta_2 + i\sin\theta_2).$ Expanding: $= r_1 r_2 \big[(\cos\theta_1\cos\theta_2 - \sin\theta_1\sin\theta_2) + i(\sin\theta_1\cos\theta_2 + \cos\theta_1\sin\theta_2)\big].$ By the addition formulas (Chapter 3): $= r_1 r_2 \big[\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)\big]. \qed$

De Moivre's theorem (preview)

For any positive integer $n$ and any complex number $z = r(\cos\theta + i\sin\theta)$: $z^n = r^n (\cos n\theta + i\sin n\theta).$

This is De Moivre's theorem. Repeated application of the multiplication formula gives the case $n \in \mathbb{N}$; the formula extends to negative integers (and ultimately to all real numbers via $z^n = e^{n \ln z}$ in Class XII).

Roots of unity

The $n$-th roots of unity are the complex solutions of $z^n = 1$. They are $z_k = \cos\frac{2\pi k}{n} + i\sin\frac{2\pi k}{n}, \quad k = 0, 1, \dots, n-1.$

They form a regular $n$-gon inscribed in the unit circle, with one vertex at $1$.

Worked examples

Example 1. Express $z = 1 + i$ in polar form.

$r = \sqrt{2}$, $\arg z = \pi/4$. So $z = \sqrt{2}\,(\cos\tfrac{\pi}{4} + i\sin\tfrac{\pi}{4})$.

Example 2. Express $z = -\sqrt{3} + i$ in polar form.

$r = \sqrt{3 + 1} = 2$. Point $(-\sqrt{3}, 1)$ is in quadrant II, so $\theta = \pi - \pi/6 = 5\pi/6$. So $z = 2(\cos\tfrac{5\pi}{6} + i\sin\tfrac{5\pi}{6})$.

Example 3. Compute $(1 + i)^{10}$ using polar form.

$1 + i = \sqrt{2}\,\text{cis}\,\tfrac{\pi}{4}$. So $(1 + i)^{10} = (\sqrt{2})^{10}\,\text{cis}\,\tfrac{10\pi}{4} = 32\,\text{cis}\,\tfrac{5\pi}{2} = 32\,\text{cis}\,\tfrac{\pi}{2} = 32 \cdot i = 32i$.

Example 4. Find the three cube roots of unity.

$z^3 = 1$, so $z_k = \text{cis}\,\tfrac{2\pi k}{3}$ for $k = 0, 1, 2$:

• $z_0 = 1$.
• $z_1 = \cos\tfrac{2\pi}{3} + i\sin\tfrac{2\pi}{3} = -\dfrac{1}{2} + \dfrac{\sqrt{3}}{2} i$ (often called $\omega$).
• $z_2 = \cos\tfrac{4\pi}{3} + i\sin\tfrac{4\pi}{3} = -\dfrac{1}{2} - \dfrac{\sqrt{3}}{2} i = \omega^2$.

Note: $1 + \omega + \omega^2 = 0$ and $\omega^3 = 1$.

Example 5 (harder). Find all complex solutions of $z^4 = 16$.

$16 = 16\,\text{cis}\,0$. Fourth roots: $z_k = 16^{1/4}\,\text{cis}\,\tfrac{2\pi k}{4} = 2\,\text{cis}\,\tfrac{\pi k}{2}$ for $k = 0, 1, 2, 3$:

• $z_0 = 2$.
• $z_1 = 2i$.
• $z_2 = -2$.
• $z_3 = -2i$.

Try it yourself

1. Write $z = -1 - i$ in polar form (principal argument).
2. Write $z = 4\,\text{cis}\,\tfrac{\pi}{3}$ in Cartesian form.
3. Compute $(1 - i)^6$ using polar form.
4. Find the modulus and argument of $\dfrac{1 + i}{1 - i}$.
5. Solve $z^3 = -8$.
6. Find all sixth roots of unity.
7. Compute $(\sqrt{3} + i)^{12}$.
8. Prove $(\cos\theta + i\sin\theta)^n + (\cos\theta - i\sin\theta)^n = 2\cos n\theta$.
9. Find $z$ with $|z| = 2$ and $\arg z = -\tfrac{\pi}{4}$.
10. Show that $\omega = e^{2\pi i/3}$ satisfies $\omega^2 + \omega + 1 = 0$.
11. Find all solutions of $z^2 = i$.
12. Express $\cos 5\theta$ as a polynomial in $\cos\theta$ using De Moivre.

Pitfalls / Tricks

• The argument is not unique. Always specify whether you want the principal value or "any" argument.
• For $z$ in quadrant II or III, $\arctan(b/a)$ gives the wrong sign , add or subtract $\pi$.
• Polar form makes powers and roots easy; Cartesian form makes addition easy. Switch as needed.
• Insight. Multiplication by $\text{cis}\,\theta$ is rotation by angle $\theta$. Multiplication by $r$ is scaling by $r$. Every complex multiplication is a rotation-and-scaling.