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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).
de Moivre's illustration of his piecewise linear approximation. De Moivre's law first appeared in his 1725 Annuities upon Lives, the earliest known example of an actuarial textbook. [6] Despite the name now given to it, de Moivre himself did not consider his law (he called it a "hypothesis") to be a true description of the pattern of human ...
de Moivre's theorem may be: de Moivre's formula, a trigonometric identity; Theorem of de Moivre–Laplace, a central limit theorem This page was last edited on 28 ...
According to the de Moivre–Laplace theorem, as n grows large, the shape of the discrete distribution converges to the continuous Gaussian curve of the normal distribution. In probability theory , the de Moivre–Laplace theorem , which is a special case of the central limit theorem , states that the normal distribution may be used as an ...
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 Christian Brothers' Catholic school in Vitry, which was unusually tolerant given religious tensions in France at the time.
On a note more distantly related to combinatorics, the second section also discusses the general formula for sums of integer powers; the free coefficients of this formula are therefore called the Bernoulli numbers, which influenced Abraham de Moivre's work later, [16] and which have proven to have numerous applications in number theory.
In these cases, the formula for the roots is much simpler, as for the solvable de Moivre quintic x 5 + 5 a x 3 + 5 a 2 x + b = 0 , {\displaystyle x^{5}+5ax^{3}+5a^{2}x+b=0\,,} where the auxiliary equation has two zero roots and reduces, by factoring them out, to the quadratic equation
This concept is attributed to Abraham de Moivre (1718), [1] although it first appears in a paper of Daniel da Silva (1854) [2] and later in a paper by J. J. Sylvester (1883). [3] Sometimes the principle is referred to as the formula of Da Silva or Sylvester, due to these publications.