GEOMETRIC
REPRESENTATION OF
COMPLEX NUMBERS
A Complex Number is in the form: z = a+bi
We can graph complex numbers on the axis shown below:
4
2
-5
5
-2
-4
Imaginary Axis
Real axis
ABSOLUTE VALUE OF A
COMPLEX NUMBER
z  3  4i
4
2
-5
5
-2
•An arrow is drawn from
the origin to represent the
complex number.
-4
•The length of the arrow
is the absolute value of the
complex number.
REPRESENTING COMPLEX
NUMBERS USING RECTANGULAR
VS. POLAR COORDINATES
a  r cos 
8
(a,b)=(r,)
6
b  r sin 
z  a  bi
4
b
So, z  r co s   ( r sin  ) i
z  r (co s   i sin  )
2

a
5
We abbreviate this as “cis”
z  rcis 
Complex Numbers
Rectangular Form: z  a  bi
Polar Form: z
 r cis 
Example:
Convert z  3 cis 5 5

to rectangular form.
Formulas:
a  r cos 
b  r sin 
Example:
Convert z   2  3i to polar form.
Formulas:
r  
2
a b
2
bI
F
tan  
Ha K
Example:
What is the absolute value of the following complex numbers:
z  3  2i
z  4 cis
2
3
Multiply:  3 cis165   4 cis 45 
Do you want to go thru that every time?
 rcis    tcis    r  t  cis  


Multiply:  4 cis 25   6 cis 35 
Divide:
3 cis165
4 cis 45
SUMMARY
To convert a+bi to polar:
Formulas:
r  
ta n 
To convert rcis  to rectangular:
a
2
 b
2
bI
F

H
aK
Formulas:
a  r cos 
b  r sin 