Transcript geometric representation of complex numbers

11.2a Geometric
Representation of a
Complex Number
Write complex numbers in polar form.
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
2

a
5
So,
z  r cos  (r sin  )i
z  r(cos  i sin  )
We abbreviate this as “cis”
z  rcis
Complex Numbers
Rectangular Form:
Polar Form: z
z  a  bi
 rcis
Convert to Rectangular
•
Formulas:
a  r cos 
b  r sin 
Convert to Polar Form
•
Absolute Value of a Complex
Nubmer ( Magnitude)
• Same as a radius!!!!!
z  3  2i
2
z  4 cis
3
Homework
• 11.2 P. 406 1-12 all
11.2b Complex
Numbers
Expressing Products and Quotient’s of Complex Numbers in
Polar and Rectangular Form.
Operations with complex
numbers
•
Do you want to go thru that every time?
 rcis   tcis   r  t  cis    
Multiply:
 4cis 25  6cis35 
How do you think we can
work with division?
3cis165
4cis 45
SUMMARY
To convert a+bi to polar:
Formulas:
r   a2  b2
b
tan    
a
To convert
rcis
to rectangular:
Formulas:
a  r cos
b  r sin 
Homework
• Pg. 406, 17-21odd, 23, 24