Math 2.0

Number theory Theorem

The nth roots of unity

Builds on Root of unityde Moivre's formulaPolar form

Statement

Let nn be a positive integer. A complex number zz is an nnth root of unity if and only if z=e2πik/n=cos⁡2πkn+isin⁡2πknz = e^{2 \pi i k / n} = \cos \frac{2 \pi k}{n} + i \sin \frac{2 \pi k}{n} for some integer kk with 0≤k<n0 \le k < n.

Remarks

Proof: by de Moivre's formula, each such zz satisfies zn=e2πik=1z^n = e^{2 \pi i k} = 1. Conversely, zn=1z^n = 1 forces ∣z∣=1|z| = 1 and narg⁡z∈2πℤn \arg z \in 2 \pi \mathbb{Z}. For n=3n = 3, the roots are 11 and −12±32i-\frac{1}{2} \pm \frac{\sqrt{3}}{2} i.

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Complex numbers, object, no evidence levelComplex numbersobjectReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionEuler's formula, statement, no evidence levelEuler's formulastatementProduct of exponentials, statement, no evidence levelProduct of exponentia…statementReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionRoot of unity, definition, no evidence levelRoot of unitydefinitionSupremum, definition, no evidence levelSupremumdefinitionde Moivre's formula, statement, no evidence levelde Moivre's formulastatementCompleteness axiom, statement, no evidence levelCompleteness axiomstatementExistence of square roots of nonnegative reals, statement, no evidence levelExistence of square r…statementSquare root, definition, no evidence levelSquare rootdefinitionModulus of a complex number, definition, no evidence levelModulus of a complex …definitionArgument of a complex number, definition, no evidence levelArgument of a complex…definitionPrincipal value of the argument, statement, no evidence levelPrincipal value of th…statementPolar form, statement, no evidence levelPolar formstatementThe nth roots of unity, statement, no evidence levelThe nth roots of unitystatement
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  1. Complex numbers object
  2. Real numbers object
  3. Set bounded above definition
  4. Euler's formula statement
  5. Product of exponentials statement
  6. Real and imaginary parts definition
  7. Root of unity definition
  8. Supremum definition
  9. de Moivre's formula statement
  10. Completeness axiom statement
  11. Existence of square roots of nonnegative reals statement
  12. Square root definition
  13. Modulus of a complex number definition
  14. Argument of a complex number definition
  15. Principal value of the argument statement
  16. Polar form statement
  17. The nth roots of unity statement

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Root_of_unity, Trigonometric expression
About this page
Tag057BH3JC9S
URIm2://057BH3JC9S
Kindstatement, theorem
WritingW0 stub
MSC11R18
StatusThis copy of the site has no build data. Run m2 status 057BH3JC9S.
UsesRoot of unity, de Moivre's formula, Polar form

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