Proof By Induction (for Integer n)
The truth of de Moivre's theorem can be established by mathematical induction for natural numbers, and extended to all integers from there. Consider S(n):
For n > 0, we proceed by mathematical induction. S(1) is clearly true. For our hypothesis, we assume S(k) is true for some natural k. That is, we assume
Now, considering S(k+1):
We deduce that S(k) implies S(k+1). By the principle of mathematical induction it follows that the result is true for all natural numbers. Now, S(0) is clearly true since cos (0x) + i sin(0x) = 1 +i 0 = 1. Finally, for the negative integer cases, we consider an exponent of -n for natural n.
The equation (*) is a result of the identity, for z = cos nx + i sin nx. Hence, S(n) holds for all integers n.
Read more about this topic: De Moivre's Formula
Famous quotes containing the words proof and/or induction:
“There are some persons in this world, who, unable to give better proof of being wise, take a strange delight in showing what they think they have sagaciously read in mankind by uncharitable suspicions of them.”
—Herman Melville (18191891)
“They relieve and recommend each other, and the sanity of society is a balance of a thousand insanities. She punishes abstractionists, and will only forgive an induction which is rare and casual.”
—Ralph Waldo Emerson (18031882)