3.2 · Complex numbers, polar form, powers, and roots

The complex plane separates real and imaginary parts

The number $\displaystyle z=a+bi$ corresponds to point $\displaystyle (a,b)$, with modulus $\displaystyle r=\sqrt{a^2+b^2}$. Choose an argument in the correct quadrant, then write $\displaystyle z=r(\cos\theta+i\sin\theta)$.

Arguments differing by $\displaystyle 2\pi$ describe the same nonzero number. The argument of zero is not uniquely defined.

Multiply moduli and add arguments

Polar multiplication follows the angle-addition formulas: multiply radii and add angles. Division divides moduli and subtracts arguments when the divisor is nonzero.

De Moivre’s Theorem gives $\displaystyle [r(\cos\theta+i\sin\theta)]^n=r^n(\cos n\theta+i\sin n\theta)$ for integers $\displaystyle n$. Convert back to rectangular form when asked.

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

Roots distribute evenly around a circle

The nth roots of a nonzero number of modulus $\displaystyle r$ and argument $\displaystyle \theta$ have modulus $\displaystyle r^{1/n}$ and arguments $\displaystyle (\theta+2\pi k)/n$ for $\displaystyle k=0,\ldots,n-1$.

Listing only the principal argument gives one root instead of all of them. The roots are separated by angle $\displaystyle 2\pi/n$, providing a geometric check on the list.