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CHAPTER 4.4
CHAPTER 4 COMPLEX NUMBERS
PART 4 –DeMoivre’s Theorem
TRIGONOMETRY MATHEMATICS CONTENT STANDARDS:
 17.0 - Students are familiar with complex numbers. They can represent a
complex number in polar form and know how to multiply complex
numbers in their polar form.
 18.0 - Students know DeMoivre’s theorem and can give nth roots of a complex
number given in polar form
OBJECTIVE(S):
 Students will learn DeMoivre’s Theorem and how to use to find a
power of a complex number.
 Students will learn how to find the nth roots of a complex number.
Powers of Complex Numbers
The trigonometric form of a complex number is used to raise a complex number to a
power. To accomplish this, consider repeated use of the multiplication rule.
z
=
r cos  i sin  
z2
=
r cos  i sin   r cos  i sin  
=
r 2 cos 2  i sin 2 
z3
=
r 2 cos 2  i sin 2  r cos  i sin   =
r 3 cos 3  i sin 3 
z4
=
r 4 cos 4  i sin 4 
z5
=
r 5 cos 5  i sin 5 
This pattern leads to DeMoivre’s Theorem, which is named after the French
mathematician Abraham DeMoivre (1667-1754).
DeMoivre’s Theorem
If z  r cos  i sin  is a complex number and n is a positive integer, then
z n  r cos   i sin  
n
= r n cos n  i sin n 
CHAPTER 4.4
EXAMPLE 1: Finding Powers of a Complex Number


12
Use DeMoivre’s Theorem to find  1  3i .
First convert to trigonometric form using
r
=
r  a 2  b2

=
  arctan
and
b
a
So, the trigonometric form is
 1 3i
=
Then, by DeMoivre’s Theorem, you have
=
=
=
=
=
Roots of Complex Numbers
Recall that a consequence of the Fundamental Theorem of Algebra is that a polynomial
equation of degree n has n solutions in the complex number system. So, the equation
x 6  1 has six solutions, and in this particular case you can find the six solutions by
factoring and using the Quadratic Formula.
x6 1 =
=
Consequently, the solutions are
x = _______, x = ____________________, and x = ____________________
CHAPTER 4.4
Each of these numbers is sixth root of 1. In general, the nth root of a complex number
is defined as follows.
Definition of an nth Root of a Complex Number
The complex number u  a  bi is an nth root of the complex number z
n
if z  u n  a  bi  .
To find a formula for an nth root of a complex number, let u be an nth root of z, where
u=
and
z=
By DeMoivre’s Theorem and the fact that u n  z , you have
s n cos n  i sin n   r cos   i sin  .
Taking the absolute value of each side of this equation, it follows that ____________.
Substituting back into the previous equation and dividing by ____, you get
cos n  i sin n  cos   i sin  .
So, it follows that
=
and
=
Because both sine and cosine have a period of _____, these last two equations have
solutions if and only if the angles differ by a multiple of _____. Consequently, there
must exist an integer k such that
n
=

=
By substituting this value of  into the trigonometric form of u, you get the following:
CHAPTER 4.4
Finding nth Roots of a Complex Number
For a positive integer n, the complex number z  r cos   i sin  has exactly n distinct
nth roots given by
n
  2k
  2k 

r  cos
 i sin

n
n 

where k = 0, 1, 2, …, n - 1
When k exceeds n – 1, the roots begin to repeat.
EXAMPLE 2: Finding the nth Roots of a Real Number
Find all the sixth roots of 1.
First write 1 in the trigonometric form 1 = _______________________________. Then,
by the nth root formula, with n = ____ and r = _____, the roots have the form
6
0  2k
0  2k 

1 cos
 i sin
=
6
6 

So, for k = 0, 1, 2, 3, 4, and 5, the sixth roots are as follows.
k=0
cos 0  i sin 0 =
k=1
cos
k=2
cos
k=3
cos   i sin  =
k=4
cos
4
4
 i sin
=
3
3
k=5
cos
5
5
 i sin
=
3
3
k=6
cos 2  i sin 2 = cos 0  i sin 0 Begins to repeat

3
 i sin

3
=
2
2
 i sin
=
3
3
The n distinct nth roots of 1 are called the nth roots of unity.
CHAPTER 4.4
EXAMPLE 3: Finding the nth Roots of a Complex Number
Find the three cube roots of z  2  2i .
Because z lies in Quadrant _____, the trigonometric form of z is
r cos   i sin 
z  2  2i
r
=
tan  =
=

=
=
z
=
By the formula for nth roots, the cube roots have the form
Finally you obtain the roots
k=0
=
=
k=1
=

k=2
=

DAY