Introduction

Extending an Angle-Addition Idea

The previous chapter introduced the polar (or $cis$) form of a complex number and showed that multiplying two complex numbers in this form is easy on the angles: $cis\,\theta_1 \cdot cis\,\theta_2 = cis(\theta_1+\theta_2)$. As an immediate consequence, squaring gives $(cis\,\theta)^2 = cis\,2\theta$.

A natural question follows: does this pattern continue for any whole-number power, not just $n=2$? That is, does $(\cos\theta+i\sin\theta)^n$ always simplify to $\cos n\theta + i\sin n\theta$ for every integer $n$? The answer is yes, and the result is known as De Moivre's Theorem, after the French mathematician Abraham de Moivre (1667–1754), who is also remembered for foundational work in probability theory.

What This Chapter Covers

This chapter proves De Moivre's theorem first for integral indices (positive, zero, and negative integers each need their own short argument) and then extends it to rational indices, where some care is needed because a fractional power of a complex number is generally multi-valued. The theorem then becomes the key tool for the chapter's second big idea: finding the $n$th roots of any complex number, and in particular the $n$th roots of unity — the $n$ solutions of $z^n=1$ — along with their elegant geometric picture as equally spaced points on a circle in the Argand plane.