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A modest extension of the version of de Moivre's formula given in this article can be used to find the n-th roots of a complex number for a non-zero integer n. (This is equivalent to raising to a power of 1 / n). If z is a complex number, written in polar form as = ( + ),
The 5th roots of unity (blue points) in the complex plane. In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group ...
de Branges's theorem (complex analysis) de Bruijn's theorem (discrete geometry) De Bruijn–Erdős theorem (incidence geometry) De Bruijn–Erdős theorem (graph theory) De Finetti's theorem (probability) De Franchis theorem (Riemann surfaces) De Gua's theorem ; De Moivre's theorem (complex analysis) De Rham's theorem (differential topology)
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 ...
The n-th power of a complex number can be computed using de Moivre's formula, which is obtained by repeatedly applying the above formula for the product: = ⏟ = (( + )) = ( + ). For example, the first few powers of the imaginary unit i are i , i 2 = − 1 , i 3 = − i , i 4 = 1 , i 5 = i , … {\displaystyle i,i^{2}=-1,i^{3}=-i,i ...
However, since all three roots of the cubic are real, this is an instance of casus irreducibilis, and the expression would require taking the cube root of a complex number. [15] [16] Alternatively, by De Moivre's formula:
It follows from the present theorem and the fundamental theorem of algebra that if the degree of a real polynomial is odd, it must have at least one real root. [2] This can be proved as follows. Since non-real complex roots come in conjugate pairs, there are an even number of them; But a polynomial of odd degree has an odd number of roots;
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 ...