Training Complex Numbers Polar Form & De Moivre's Theorem
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Polar Form & De Moivre's Theorem

26 min Complex Numbers
Every complex number z = a + bi can be written in polar form z = r(cos θ + i sin θ) = r·e^(iθ), where r = |z| is the modulus and θ = arg(z) = arctan(b/a) is the argument. Multiplication in polar form multiplies moduli and adds arguments: z₁z₂ = r₁r₂·e^(i(θ₁+θ₂)). De Moivre's theorem states (r·e^(iθ))ⁿ = rⁿ·e^(inθ), making it easy to compute powers and derive trigonometric identities. Euler's formula e^(iπ) + 1 = 0 elegantly connects the five most important constants in mathematics. Polar form is the natural language for rotations, oscillations, and signal processing.

Polar Form & De Moivre's Theorem

Polar (Trigonometric) Form

A complex number $z = a+bi$ can be written as:

$$z = r(\cos\theta + i\sin\theta) = r\operatorname{cis}\theta$$

where $r = |z| = \sqrt{a^2+b^2}$ and $\theta = \arg(z) = \arctan\frac{b}{a}$ (adjusted for quadrant).

Multiplication in Polar Form

$$z_1 z_2 = r_1 r_2 \operatorname{cis}(\theta_1 + \theta_2)$$

$$\frac{z_1}{z_2} = \frac{r_1}{r_2} \operatorname{cis}(\theta_1 - \theta_2)$$

De Moivre's Theorem

$$[r\operatorname{cis}\theta]^n = r^n \operatorname{cis}(n\theta)$$

Example 1

Convert $z = 1 + i$ to polar form.

  1. $r = \sqrt{1+1} = \sqrt{2}$, $\theta = \arctan(1/1) = \pi/4$.
  2. $$z = \sqrt{2}\operatorname{cis}\frac{\pi}{4}$$
Example 2

Use De Moivre's Theorem to compute $(1+i)^8$.

  1. $(\sqrt{2})^8 \operatorname{cis}(8 \cdot \pi/4) = 16\operatorname{cis}(2\pi) = 16(1+0i) = 16$
Example 3

Convert $z = 3\operatorname{cis}(\pi/3)$ to rectangular form.

  1. $a = 3\cos(\pi/3) = 3/2$, $b = 3\sin(\pi/3) = 3\sqrt{3}/2$.
  2. $z = \frac{3}{2} + \frac{3\sqrt{3}}{2}i$

Practice Problems

1. Convert $z = -1+i$ to polar form.
2. Convert $z = 2\operatorname{cis}(\pi/6)$ to rectangular form.
3. Compute $(1+i)^4$ using De Moivre.
4. Find $r$ and $\theta$ for $z = -3$.
5. Multiply $2\operatorname{cis}(\pi/4) \cdot 3\operatorname{cis}(\pi/3)$.
6. Convert $z = 4i$ to polar form.
7. Compute $(\sqrt{3}+i)^3$ using De Moivre.
8. Divide $6\operatorname{cis}(2\pi/3)$ by $2\operatorname{cis}(\pi/6)$.
9. Find $|z|$ and $\arg(z)$ for $z = -2-2i$.
10. Compute $i^6$ using polar form.
11. Convert $5\operatorname{cis}(\pi)$ to rectangular.
12. What is $\operatorname{cis}(0)$?
Show Answer Key

1. $r = \sqrt{2}$, $\theta = 3\pi/4$; $z = \sqrt{2}\operatorname{cis}(3\pi/4)$

2. $2\cos(\pi/6)+2i\sin(\pi/6) = \sqrt{3}+i$

3. $(\sqrt{2})^4\operatorname{cis}(\pi) = 4(-1) = -4$

4. $r = 3$, $\theta = \pi$

5. $6\operatorname{cis}(7\pi/12)$

6. $r = 4$, $\theta = \pi/2$; $4\operatorname{cis}(\pi/2)$

7. $r = 2$, $\theta = \pi/6$; $2^3\operatorname{cis}(\pi/2) = 8i$

8. $3\operatorname{cis}(\pi/2) = 3i$

9. $|z| = 2\sqrt{2}$, $\arg = 5\pi/4$ (QIII)

10. $i = \operatorname{cis}(\pi/2)$; $i^6 = \operatorname{cis}(3\pi) = -1$

11. $-5$

12. $1$

📍 Rectangular ↔ Polar Converter
r = |z|
θ = arg(z)
zⁿ (De Moivre)
Division & Conjugates Complex Roots & Applications