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Direct Proof of De Moivre's Theorem
In §2.10, De Moivre's theorem was introduced as a consequence of
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
using mathematical induction
and elementary trigonometric
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
Since it is true by hypothesis that
multiplying both sides by
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
, 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
therefore can be used to determine all of the th roots of unity
(see §3.12 above).
However, no definition of 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).
Previous: Roots of UnityNext: Euler_Identity Problems
About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA)
, teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/