Editing Root of unity (section)

==Trigonometric expression==
[[Image:3rd roots of unity correction.svg|thumb|right|The cube roots of unity]]

[[De Moivre's formula]], which is valid for all [[real number|real]] {{mvar|x}} and integers {{mvar|n}}, is

:<math>\left(\cos x + i \sin x\right)^n = \cos nx + i \sin nx.</math>

Setting {{math|1=''x'' = {{sfrac|2π|''n''}}}} gives a primitive {{mvar|n}}th root of unity – one gets

:<math>\left(\cos\frac{2\pi}{n} + i \sin\frac{2\pi}{n}\right)^{\!n} = \cos 2\pi + i \sin 2\pi = 1,</math>
but
:<math>\left(\cos\frac{2\pi}{n} + i \sin\frac{2\pi}{n}\right)^{\!k} = \cos\frac{2k\pi}{n} + i \sin\frac{2k\pi}{n} \neq 1</math>
for {{math|1=''k'' = 1, 2, …, ''n'' − 1}}. In other words, 
:<math>\cos\frac{2\pi}{n} + i \sin\frac{2\pi}{n}</math>
is a primitive {{mvar|n}}th root of unity.

This formula shows that in the [[complex plane]] the {{mvar|n}}th roots of unity are at the vertices of a [[regular polygon|regular {{mvar|n}}-sided polygon]] inscribed in the [[unit circle]], with one vertex at 1 (see the plot for {{math|1=''n'' = 3}} on the right). This geometric fact accounts for the term "cyclotomic" in such phrases as [[cyclotomic field]] and [[cyclotomic polynomial]]; it is from the Greek roots "[[wikt:κύκλος|cyclo]]" (circle) plus "[[wikt:τόμος|tomos]]" (cut, divide).

[[Euler's formula]]

:<math>e^{i x} = \cos x + i \sin x,</math>

which is valid for all real {{mvar|x}}, can be used to put the formula for the {{mvar|n}}th roots of unity into the form

:<math>e^{2\pi i \frac{k}{n}}, \quad 0 \le k < n.</math>

It follows from the discussion in the previous section that this is a primitive {{mvar|n}}th-root if and only if the fraction {{math|{{sfrac|''k''|''n''}}}} is in lowest terms; that is, that {{mvar|k}} and {{mvar|n}} are coprime.  An [[irrational number]] that can be expressed as the [[complex number|real part]] of the root of unity; that is, as <math>\cos(2\pi k/n)</math>, is called a [[trigonometric number]].