CK-12-Pre-Calculus Concepts

http://www.ck12.org Chapter 11. Complex Numbers

11.4 De Moivre’s Theorem and nth Roots

Here you will learn about De Moivre’s Theorem, which will help you to raise complex numbers to powers and find
roots of complex numbers.
You know how to multiply two complex numbers together and you’ve seen the advantages of using trigonometric
polar form, especially when multiplying more than two complex numbers at the same time. Because raising a
number to a whole number power is repeated multiplication, you also know how to raise a complex number to a
whole number power.
What is a geometric interpretation of squaring a complex number?

Watch This

MEDIA

Click image to the left for use the URL below.

URL: http://www.ck12.org/flx/render/embeddedobject/62129

http://www.youtube.com/watch?v=Sf9gEzcVZkU James Sousa: De Moivre’s Theorem: Powers of Complex Num-
bers in Trig Form

Guidance

Recall that ifz 1 =r 1 ·cisθ 1 andz 2 =r 2 ·cisθ 2 withr 26 =0, thenz 1 ·z 2 =r 1 ·r 2 ·cis(θ 1 +θ 2 ).
Ifz 1 =z 2 =z=rcisθthen you can determinez^2 andz^3 :

z^2 =r·r·cis(θ+θ) =r^2 cis( 2 ·θ)

z^3 =r^3 cis( 3 ·θ)

De Moivre’s Theorem simply generalizes this pattern to the power of any positive integer.
zn=rn·cis(n·θ)
In addition to raising a complex number to a power, you can also take square roots, cube roots andnthroots of
complex numbers. Suppose you have complex numberz=rcisθand you want to take thenthroot ofz. In other
words, you want to find a numberv=s·cisβsuch thatvn=z. Do some substitution and manipulation:

vn=z

(s·cisβ)n=r·cisθ

sn·cis(n·β) =r·cisθ