Polar Form

Polar form exploits the magnitude and direction of complex numbers, making multiplication and powering trivial compared to Cartesian form.

Standard Representation

• $r=|z|$ (Modulus)
• $\theta =\arg (z)$ (Argument), usually in $(-\pi ,\pi ]$ or $[0,2\pi )$.

Multiplication and Division

If $z_1=r_1\text{cis }{\theta }_1$ and $z_2=r_2\text{cis }{\theta }_2$:

1. Product: $z_1{z}_2=r_1{r}_2\text{cis }({\theta }_1+{\theta }_2)$

• Multiply magnitudes, add angles.

2. Quotient: $\frac{z_1}{z_2}=\frac{r_1}{r_2}\text{cis }({\theta }_1-{\theta }_2)$

• Divide magnitudes, subtract angles.

Roots of Unity (Advanced)

The $n$-th roots of unity are the solutions to $z^n=1$.

Properties of $n$-th Roots

Let $\zeta =\text{cis }(2\pi /n)$ be a primitive $n$-th root.

1. The roots are $1,\zeta ,{\zeta }^2,\dots ,{\zeta }^{n-1}$.
2. Sum of roots: $1+\zeta +{\zeta }^2+\cdots +{\zeta }^{n-1}=0$ (for $n>1$).
3. Product of roots: $(-1{)}^{n-1}$.
4. They form the vertices of a regular $n$-gon inscribed in the unit circle.

Practice Problems