Editing Exponentiation (section)

==={{mvar|n}}th roots of a complex number===
Every nonzero complex number {{mvar|z}} may be written in [[polar form]] as
:<math>z=\rho e^{i\theta}=\rho(\cos \theta +i \sin \theta),</math>
where <math>\rho</math> is the [[absolute value]] of {{mvar|z}}, and <math>\theta</math> is its [[argument (complex analysis)|argument]]. The argument is defined [[up to]] an integer multiple of {{math|2{{pi}}}}; this means that, if <math>\theta</math> is the argument of a complex number, then <math>\theta +2k\pi</math> is also an argument of the same complex number for every integer <math>k</math>.

The polar form of the product of two complex numbers is obtained by multiplying the absolute values and adding the arguments. It follows that the polar form of an {{mvar|n}}th root of a complex number can be obtained by taking the {{mvar|n}}th root of the absolute value and dividing its argument by {{mvar|n}}:
: <math>\left(\rho e^{i\theta}\right)^\frac 1n=\sqrt[n]\rho \,e^\frac{i\theta}n.</math>

If <math>2\pi</math> is added to <math>\theta</math>, the complex number is not changed, but this adds <math>2i\pi/n</math> to the argument of the {{mvar|n}}th root, and provides a new {{mvar|n}}th root. This can be done {{mvar|n}} times (<math>k=0,1,...,n-1</math>), and provides the {{mvar|n}} {{mvar|n}}th roots of the complex number:
: <math>\left(\rho e^{i(\theta+2k\pi)}\right)^\frac 1n=\sqrt[n]\rho \,e^\frac{i(\theta+2k\pi)}n.</math>

It is usual to choose one of the {{mvar|n}} {{mvar|n}}th root as the [[principal root]]. The common choice is to choose the {{mvar|n}}th root for which <math>-\pi<\theta\le \pi,</math> that is, the {{mvar|n}}th root that has the largest real part, and, if there are two, the one with positive imaginary part. This makes the principal {{mvar|n}}th root a [[continuous function]] in the whole complex plane, except for negative real values of the [[radicand]]. This function equals the usual {{mvar|n}}th root for positive real radicands. For negative real radicands, and odd exponents, the principal {{mvar|n}}th root is not real, although the usual {{mvar|n}}th root is real. [[Analytic continuation]] shows that the principal {{mvar|n}}th root is the unique [[complex differentiable]] function that extends the usual {{mvar|n}}th root to the complex plane without the nonpositive real numbers.

If the complex number is moved around zero by increasing its argument, after an increment of <math>2\pi,</math> the complex number comes back to its initial position, and its {{mvar|n}}th roots are [[circular permutation|permuted circularly]] (they are multiplied by <math DISPLAY=textstyle>e^{2i\pi/n}</math>). This shows that it is not possible to define a {{mvar|n}}th root function that is continuous in the whole complex plane.

====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>