Polar and exponential form

Plot a complex number as the point (a, b) and you can describe it two ways: by its coordinates, or by how far it is from the origin and in which direction. The second pair — modulus r and argument θ — gives the polar form r(cos θ + i sin θ), and Euler's formula compresses that to re^(iθ).

The payoff is multiplication. In a + bi form, multiplying two complex numbers is a four-term expansion. In polar form it is: multiply the moduli, add the arguments. Multiplying by a complex number of modulus 1 is therefore a pure rotation, which is why complex numbers are the natural language for anything periodic — alternating current, oscillations, waves. Raising to a power follows immediately as de Moivre's formula, r^n(cos nθ + i sin nθ), turning a messy repeated multiplication into one step.

Two details catch people out. The argument is not unique: adding 2π to it names the same number, so every complex number has infinitely many arguments and one must pick a principal value when a single answer is needed. And going the other way — taking an nth root — is where the multiplicity becomes visible: dividing the argument by n and stepping round by 2π/n gives exactly n distinct roots, evenly spaced on a circle. It is roots that come in n copies, not powers.