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Complex Numbers,
the Complex
Plane &
Demoivre’s
Theorem
Complex Numbers are numbers in the form of
a  bi
where a and b are real numbers and i, the imaginary
unit, is defined as follows:
i  1
2
i  1
And the powers of i are as follows:
i i
2
i  1
3
2
i  i i  1 i  i
4
i 1
1
The value of in, where n is any number can be found by
dividing n by 4 and then dealing only with the remainder.
Why?
Examples:
1)
i ?
18
18  4  4 with a remainder of 2
i  i Then from the chart on the previous slide
18
2
i  i  1
18
2)
2
i  ? 27  4  6 with a remainder of 3
27
3
i  i Then from the chart on the previous slide
27
i  i  i
27
3
In a complex number
a  bi
a is the real part and bi is the imaginary part.
When b=0, the complex number is a real number.
When a0, and b0, as in 5+8i, the complex number is an
imaginary number.
When a=0, and b0, as in 5i, the complex number is a pure
imaginary number.
Lesson Overview 9-5A
Lesson Overview 9-5B
5-Minute Check Lesson 9-6A
The Complex Plane
Imaginary
z  a  bi
Axis
O
Real
Axis
Let
z  a  bi
be a complex number.
The magnitude or modulus of z, denoted by z
As the distance from the origin to the point (x, y).
z  a b
2
2
is defined
Imaginary
Axis
y
z  a  bi
|z|
O
x
Real
Axis
r cos   i sin  
is sometimes abbreviated as
r cis 
z =-3 + 4i
-3
Imaginary
4
Axis
Real Axis
z = -3 + 4i is in Quadrant II
x = -3 and
y=4
z =-3 + 4i
4
r 5
-3
 
Find the reference angle ()
by solving
y
tan  
x
4
tan  
3
1 4
  tan
3
  53.13
z =-3 + 4i
4
r 5
-3
 
Find r:
r
 3   1
2
r2
2
Imaginary
4
Axis
3
-3
2
1
Real Axis
3 i
Find the reference angle () by solving

3
 1
2
y
tan  
x
1
tan 
3
1  1
  tan
3
  30

3
 1
2
  360  30  330
z  r cos   i sin  
z  2cos 330  i sin 330
Find the cosine of 330 and substitute the value.
Find the sine of 330 and substitute the value.
Distribute the r
Write in standard (rectangular) form.
5
5 

2 cos
 i sin

6
6 

5
3
cos

6
2

3 1 
2 
 i 
2
2


5 1
sin

6 2
  3 i
Lesson Overview 9-7A
Product Theorem
Lesson Overview 9-7B
Quotient Theorem
5-Minute Check Lesson 9-8A
5-Minute Check Lesson 9-8B
Powers and Roots of Complex
Numbers
DeMoivre’s Theorem
81 81 3
i
2
2
What if you wanted to perform the
operation below?
6  2 3i 
4
Lesson Overview 9-8A
Lesson Overview 9-8B
Theorem Finding Complex Roots
roots
Find the complex fifth roots of
The five complex roots are:
for k = 0, 1, 2, 3,4 .
32  0i
2
2

32

0
r  a b
2
  tan 1
2
0
32
  tan 1 0
 322
 32
 0
32 cos  0   i sin  0  
5
  0 360k 
 0 360k  
32 cos  
  i sin  

5 
5 
5
 5
2 cos  72k   i sin  72k  
2 cos  72k   i sin  72k  
k 0
2 cos  0   i sin  0    2 1  0i 
k 1
2 cos  72   i sin  72    .62  1.90i
k 2
2 cos 144   i sin 144    1.62 1.18i
k 3
2 cos  216   i sin  216    1.62 1.18i
k 4
2 cos  288  i sin  288    .62 1.90i
2
The roots of a complex number a cyclical in nature.
That is, when the points are plotted on a polar plane or
a complex plane, the points are evenly spaced around
the origin
Complex Plane
4
2
5
-5
-2
-4
6
Polar plane
4
2
5
5
2
4
6
Polar plane
4
2
5
5
2
4
6
44
22
-5
5
55
-2
2
-4
4
To find the principle root, use DeMoivre’s theorem using
rational exponents.
That is, to find the principle pth root of
a  bi
Raise it to the
1
power
p
Example
Find
3
i
First express
You may assume it is the principle root
you are seeking unless specifically stated
otherwise.
i
0i
as a complex number in standard form.
Then change to polar form
b
tan  
a
1
tan  
0
r  a 2  b2
1
  tan
0
1 cos  90   i sin  90  
1
r 1
  90
1 cos  90   i sin  90  
Since we are looking for the cube root, use DeMoivre’s Theorem
and raise it to the 1 power
3
 1

1

1 cos  90   i sin  90  

3

 3
1
3
1 cos  30   i sin  30  
 3
1
1 
i   3  1 i
2
2 2
 2
Example:
Find the 4th root of
 2  i 
2  i
1
4
Change to polar form
5 cos 153.4   i sin 153.4  
Apply DeMoivre’s Theorem
 1

1
5 cos   153.4   i sin   153.4 
4
 4

1
4
1.22 cos  38.4   i sin  38.4  
.96  .76i