In §2.10, De Moivre's theorem was introduced as a consequence of Euler's identity:
To provide some further insight into the ``mechanics'' of Euler's identity, we'll provide here a direct proof of De Moivre's theorem for integer using mathematical induction and elementary trigonometric identities.
Proof: To establish the ``basis'' of our mathematical induction proof, we may simply observe that De Moivre's theorem is trivially true for . Now assume that De Moivre's theorem is true for some positive integer . Then we must show that this implies it is also true for , i.e.,
multiplying both sides by yields
From trigonometry, we have the following sum-of-angle identities:
These identities can be proved using only arguments from classical geometry.3.8Applying these to the right-hand side of Eq. (3.3), with and , gives Eq. (3.2), and so the induction step is proved.
De Moivre's theorem establishes that integer powers of lie on a circle of radius 1 (since , for all ). It therefore can be used to determine all of the th roots of unity (see §3.12 above). However, no definition of e emerges readily from De Moivre's theorem, nor does it establish a definition for imaginary exponents (which we defined using Taylor series expansion in §3.7 above).