3.9.5—Complex numbers

Multiplying by r cisθ scales a modulus by r and adds θ to the argument. Dividing subtracts arguments and divides moduli; powers use De Moivre’s theorem.

Reduce angles modulo 2π, keep all roots when solving zⁿ=w, and convert back to rectangular coordinates only at the end if needed.

The cube roots of 8 are 2 cis(2kπ/3) for k=0,1,2, giving three equally spaced points on a circle.

Taking only the principal root misses the other n−1 roots; complex roots are distributed by equal angular steps.