Editing Exponentiation (section)

====Roots of unity====
{{Main|Root of unity}}

[[File:One3Root.svg|thumb|right|The three third roots of {{math|1}}]]
The {{mvar|n}}th roots of unity are the {{mvar|n}} complex numbers such that {{math|1=''w''<sup>''n''</sup> = 1}}, where {{mvar|n}} is a positive integer. They arise in various areas of mathematics, such as in [[discrete Fourier transform]] or algebraic solutions of algebraic equations ([[Lagrange resolvent]]).

The {{mvar|n}} {{mvar|n}}th roots of unity are the {{mvar|n}} first powers of <math>\omega =e^\frac{2\pi i}{n}</math>, that is <math>1=\omega^0=\omega^n, \omega=\omega^1, \omega^2,..., \omega^{n-1}.</math> The {{mvar|n}}th roots of unity that have this generating property are called ''primitive {{mvar|n}}th roots of unity''; they have the form <math>\omega^k=e^\frac{2k\pi i}{n},</math> with {{mvar|k}} [[coprime integers|coprime]] with {{mvar|n}}. The unique primitive square root of unity is <math>-1;</math> the primitive fourth roots of unity are <math>i</math> and <math>-i.</math>

The {{mvar|n}}th roots of unity allow expressing all {{mvar|n}}th roots of a complex number {{mvar|z}} as the {{mvar|n}} products of a given {{mvar|n}}th roots of {{mvar|z}} with a {{mvar|n}}th root of unity.

Geometrically, the {{mvar|n}}th roots of unity lie on the [[unit circle]] of the [[complex plane]] at the vertices of a [[regular polygon|regular {{mvar|n}}-gon]] with one vertex on the real number 1.

As the number <math>e^\frac{2k\pi i}{n}</math> is the primitive {{mvar|n}}th root of unity with the smallest positive [[argument (complex analysis)|argument]], it is called the ''principal primitive {{mvar|n}}th root of unity'', sometimes shortened as ''principal {{mvar|n}}th root of unity'', although this terminology can be confused with the [[principal value]] of <math>1^{1/n}</math>, which is 1.<ref>{{cite book |title=Introduction to Algorithms |edition=second |author-last1=Cormen |author-first1=Thomas H. |author-last2=Leiserson |author-first2=Charles E. |author-last3=Rivest |author-first3=Ronald L. |author-last4=Stein |author-first4=Clifford |publisher=[[MIT Press]] |date=2001 |isbn=978-0-262-03293-3}} [http://highered.mcgraw-hill.com/sites/0070131511/student_view0/chapter30/glossary.html Online resource] {{webarchive|url=https://web.archive.org/web/20070930201902/http://highered.mcgraw-hill.com/sites/0070131511/student_view0/chapter30/glossary.html |date=2007-09-30}}.</ref><ref>{{cite book |title=Difference Equations: From Rabbits to Chaos |title-link= Difference Equations: From Rabbits to Chaos |edition=[[Undergraduate Texts in Mathematics]] |author-last1=Cull |author-first1=Paul |author-last2=Flahive |author-first2=Mary |author-link2=Mary Flahive |author-last3=Robson |author-first3=Robby |date=2005 |publisher=Springer |isbn=978-0-387-23234-8}} Defined on p. 351.</ref><ref>{{MathWorld |title=Principal root of unity |id=PrincipalRootofUnity}}</ref>