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### Usefulness of De Moivre's Theorem

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Date: 2/10/96 at 10:9:18
From: Anonymous
Subject: Use of DeMoivre's Theorem

What is the usefulness of DeMoivre's theorem?
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Date: 2/10/96 at 12:18:0
From: Doctor Sarah
Subject: Re: Use of DeMoivre's Theorem

Hello!

De Moivre's theorem involves the branch of trigonometry that
deals with imaginary quantities.

Here's some information found by doing a Web search on de Moivre.  It's
from a Web page called The English Mathematicians of the Eighteenth
Century:

_A Short Account of the History of Mathematics_ (4th edition, 1908) by
W. W. Rouse Ball.  The URL is:

http://www.maths.tcd.ie/pub/HistMath/People/18thCentury/RouseBall/RB_Engl18C.html

"He is best known for having, together with Lambert, created
that part of trigonometry which deals with imaginary quantities.
Two theorems on this part of the subject are still connected with
his name, namely, that which asserts that sin nx + i cos nx is one
of the values of (sin x + i cos x)^n, and that which gives the
various quadratic factors of x^(2n) - 2p x^n + 1."

Here's something about de Moivre's life:

"Demoivre (more correctly written as de Moivre) was born at
Vitry on May 26, 1667, and died in London on November 27,
1754. His parents came to England when he was a boy, and his
education and friends were alike English. His interest in the
higher mathematics is said to have originated in his coming by
chance across a copy of Newton's _Principia_. From the eulogy
on him delivered in 1754 before the French Academy it would
seem that his work as a teacher of mathematics had led him to the
house of the Earl of Devonshire at the instant when Newton, who
had asked permission to present a copy of his work to the earl,
was coming out. Taking up the book, and charmed by the far-
reaching conclusions and the apparent simplicity of the
reasoning, Demoivre thought nothing would be easier than to
master the subject, but to his surprise found that to follow the
argument overtaxed his powers. He, however, bought a copy,
and as he had but little leisure he tore out the pages in order to
carry one or two of them loose in his pocket so that he could
study them in the intervals of his work as a teacher.
Subsequently he joined the Royal Society, and became intimately
connected with Newton, Halley, and other mathematicians of the
English school. The manner of his death has a certain interest
for psychologists. Shortly before it he declared that it was
necessary for him to sleep some ten minutes or a quarter of an
hour longer each day than the preceding one. The day after he
had thus reached a total of something over twenty-three hours he
slept up to the limit of twenty-four hours, and then died in his
sleep.

"His chief works, other than numerous papers in the
_Philosophical Transactions_, were _The Doctrine of Chances_,
published in 1718, and the _Miscellanea Analytica_, published in
1730. In the former the theory of recurring series was first
given, and the theory of partial fractions which Cotes's
premature death had left unfinished was completed, while the
rule for finding the probability of a compound event was
enunciated. The latter book, besides the trigonometrical
propositions mentioned above, contains some theorems in
astronomy, but they are treated as problems in analysis."

-Doctor Sarah,  The Math Forum

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Associated Topics:
High School Trigonometry

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