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,
cos x + i sin x
n = cos nx + i sin nx .
How does that help us solve our problem? We also know that for any integer k,
1 = cos 2kπ + i sin 2kπ .
So by de Moivre's theorem we get
1
1/n = cos 2kπ + i sin 2kπ
1/n = cos
2kπ
n
+ i sin
2kπ
n
.
Calculating the nth Roots of 1
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We can use this last equation to find the nth roots of 1. For example, for any value of n, we can use
the formula above with values of k = 0…n − 1. We can use this MATLAB code to experiment with
different values of n:
n =
;
roots = zeros(1, n);
for k = 0:n-1
roots(k+1) = cos(2*k*pi/n) + 1i*sin(2*k*pi/n);
% Calculate the roots
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