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# 6.13: DeMoivre's Theorem and nth Roots

Difficulty Level: At Grade Created by: CK-12
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### Vocabulary Language: English

$n^{th}$ roots of unity

$n^{th}$ roots of unity

The $n^{th}$ roots of unity are the $n^{th}$ roots of the number 1.
complex number

complex number

A complex number is the sum of a real number and an imaginary number, written in the form $a + bi$.
complex plane

complex plane

The complex plane is the graphical representation of the set of all complex numbers.
De Moivre's Theorem

De Moivre's Theorem

De Moivre's theorem is the only practical manual method for identifying the powers or roots of complex numbers. The theorem states that if $z= r(\cos \theta + i \sin \theta)$ is a complex number in $r cis \theta$ form and $n$ is a positive integer, then $z^n=r^n (\cos (n\theta ) + i\sin (n\theta ))$.
trigonometric polar form

trigonometric polar form

To write a complex number in trigonometric form means to write it in the form $r\cos\theta+ri\sin\theta$. $rcis\theta$ is shorthand for this expression.

Sep 26, 2012

Feb 26, 2015