Geometric meaning of De Moivre's Theorem

Since multiplying by \(\omega := \cos \theta + i \sin \theta\) to a complex number \(z\) rotates it through an angle of \(\theta\), it follows that De Moivre’s Theorem

\[\omega^n = (\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta)\]

can be interpreted as \(n\) rotations through \(\theta\) -- amounting to rotating through a total angle of \(n\theta\). This makes the exponential form of a complex number a suitable notation because De Moivre’s Theorem now reads

\[\omega^n = (e^{i\theta})^n = e^{i(n\theta)},\]

which is consistent with the usual exponential law.