Overview

Polar form turns multiplication into angle addition and scaling. It is the fastest path for high powers.

Core Skills

Convert to Polar Form

Compute $r=|z|$ and $\theta=\arg z$ so $z=r(\cos\theta+i\sin\theta)$.

Multiply and Divide

Multiply magnitudes and add arguments; divide magnitudes and subtract arguments.

Reduce Angles

Always reduce angles modulo $2\pi$ to keep arguments in a standard range.

Key Ideas

• $z = r(\cos\theta + i\sin\theta) = re^{i\theta}$.
• De Moivre: $z^n = r^n(\cos n\theta + i\sin n\theta)$.
• Multiply by adding arguments and multiplying magnitudes.

Worked Example

Compute $(1+i)^{10}$.

$1+i = \sqrt{2}e^{i\pi/4}$. Then $(1+i)^{10} = (\sqrt{2})^{10} e^{i\cdot 10\pi/4} = 2^5 e^{i\pi/2} = 32i$.

More Examples

Example 1: Power

Compute $(2e^{i\pi/3})^4$.

$2^4 e^{i4\pi/3} = 16\left(\cos\frac{4\pi}{3}+i\sin\frac{4\pi}{3}\right)$.

Example 2: Product

If $z_1=3e^{i\pi/6}$ and $z_2=2e^{i\pi/4}$, find $z_1z_2$.

$6e^{i(\pi/6+\pi/4)} = 6e^{i5\pi/12}$.

Example 3: Division

Compute $\frac{4e^{i\pi/2}}{2e^{i\pi/3}}$.

$2e^{i(\pi/2-\pi/3)} = 2e^{i\pi/6}$.

Strategy Checklist

• Convert to polar form before powering.
• Add or subtract arguments correctly.
• Reduce angles to a standard range.

Common Pitfalls

• Using degrees in one step and radians in another.
• Forgetting to reduce angles modulo $2\pi$.
• Mixing up magnitude and argument operations.

Practice Problems