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In mathematics, de Moivre's formula (also known as de Moivre's theorem and de Moivre's identity) states that for any real number x and integer n it is the case that ( + ) = + , where i is the imaginary unit (i 2 = −1).
The theorem appeared in the second edition of The Doctrine of Chances by Abraham de Moivre, published in 1738. Although de Moivre did not use the term "Bernoulli trials", he wrote about the probability distribution of the number of times "heads" appears when a coin is tossed 3600 times.
As Φ 5 (x) = x 4 + x 3 + x 2 + x + 1, the four primitive fifth roots of unity are the roots of this quartic polynomial, which may be explicitly solved in terms of radicals, giving the roots +, where may take the two values 1 and −1 (the same value in the two occurrences). As Φ 6 (x) = x 2 − x + 1, there are two primitive sixth roots of ...
Published in 1738 by Woodfall and running for 258 pages, the second edition of de Moivre's book introduced the concept of normal distributions as approximations to binomial distributions. In effect de Moivre proved a special case of the central limit theorem. Sometimes his result is called the theorem of de Moivre–Laplace.
The real part of the other side is a polynomial in cos x and sin x, in which all powers of sin x are even and thus replaceable through the identity cos 2 x + sin 2 x = 1. By the same reasoning, sin nx is the imaginary part of the polynomial, in which all powers of sin x are odd and thus, if one factor of sin x is factored out, the remaining ...
Theorem of de Moivre–Laplace, a central limit theorem Topics referred to by the same term This disambiguation page lists articles associated with the title De Moivre's theorem .
Maximum power theorem (electrical circuits) Maxwell's theorem (probability theory) May's theorem (game theory) Mazur–Ulam theorem (normed spaces) Mazur's torsion theorem (algebraic geometry) Mean value theorem ; Measurable Riemann mapping theorem (conformal mapping) Mellin inversion theorem (complex analysis) Menelaus's theorem
Abraham de Moivre was born in Vitry-le-François in Champagne on 26 May 1667. His father, Daniel de Moivre, was a surgeon who believed in the value of education. Though Abraham de Moivre's parents were Protestant, he first attended the Christian Brothers' Catholic school in Vitry, which was unusually tolerant given religious tensions in France at the time.