Abraham de Moivre was born in Vitry in Champagne on May 26, 1667. He was a
contemporary and friend of Isaac Newton, Edmund Halley, and James Stirling. https://
en.wikipedia.org/wiki/Abraham_de_Moivre
He is best known for de Moivre's theorem that links complex numbers and trigonometry,
and for his work on the normal distribution and probability theory. De Moivre wrote a
book on probability theory, The Doctrine of Chances, said to have been prized by
gamblers. De Moivre first discovered Binet's formula, the closed-form expression for
Fibonacci numbers linking the nth power of the golden ratio φ to the nth Fibonacci
number. He was also the first to postulate the Central Limit Theorem, a cornerstone of
probability theory.
de Moivre's theorem states that for any real x and any integer n,
cosx+isinxn= cosnx +isinnx.How does that help us solve our problem? We also know that for any integer k,
1 = cos 2 kπ +isin 2 kπ.So by de Moivre's theorem we get
1 1/n= cos 2 kπ +isin 2 kπ
1/n
= cos
2 kπ
n +isin2 kπ
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