Root of Unity

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The 👁 n
th roots of unity are roots 👁 e^(2piik/n)
of the cyclotomic equation

which are known as the de Moivre numbers. The notations 👁 zeta_k
, 👁 epsilon_k
, and 👁 epsilon_k
, where the value of 👁 n
is understood by context, are variously used to denote the 👁 k
th 👁 n
th root of unity.

👁 +1
is always an 👁 n
th root of unity, but 👁 -1
is such a root only if 👁 n
is even. In general, the roots of unity form a regular polygon with 👁 n
sides, and each vertex lies on the unit circle.

See also

Cyclotomic Equation, Cyclotomic Polynomial, de Moivre's Identity, de Moivre Number, nth Root, Primitive Root of Unity, Principal Root of Unity, Unit, Unity

Portions of this entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd and Weisstein, Eric W. "Root of Unity." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RootofUnity.html

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