Download Notes - 9.3 (4e)

Math 120 Notes 11.3 –The Complex Plane
I.
The complex plane
A.
Define: real axis, imaginary axis
1.
Plot the following complex numbers
a.
3 + 3i
b.
-2 – i
c.
-3
d.
-2i
2.
Let z  x  y i be a complex number
a.
What is z ?
b.
Find the magnitude for the complex numbers
above.
3.
Find a formula to convert z from rectangular to polar
form, then covert the above points into polar form (degrees).
4.
Plot the following polar coordinates.
2 cos 45  i sin 45
a.

4  cos120
b.
5.
B.

 i sin120

Convert the polar coordinates above into rectangular form.
Euler’s Formula
1.
Fill in the following chart:
x
e ix - rectangular form
e ix - polar form
0

2

3
2
2

6

4
2.
What conclusions can you make from the above analysis?
3.
Convert the following into polar form
i
i 3
5e 3
8e 4
a.
b.
4.
Write the following in exponential form
2 cos 45  i sin 45
a.
b.
Math 120 Notes 11.3

4  cos120

 i sin120

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III.
Operations on complex numbers
A.
Addition / subtraction
1.
Rectangular form
a.
2  3i    5  8i 
b.
2.
B.
7  6i    3  4i 


Multiplication
1.
Rectangular form:
2.

Polar form: 2 cos 45  i sin 45  5 cos 30  i sin 30
C.
D.
[HUH?]
3  3i  4  8i 
Polar form
2 cos 45  i sin 45  5 cos 30  i sin30 
a.



b.
So in general, r1  cos 1  i sin 1   r2  cos 2  i sin 2   = ?








Use the formula to find 2 2 cos 35  i sin35   10 cos 70  i sin70 



Try multiplying  3  3i  4  8i  by converting to polar form first.
c.
3.

Division
1.
Rectangular form:
2.
Polar form
4  7i
3  2i
a.
What do you think is the formula for
c.
Use the formula to find

8  cos 20
r1  cos 1  i sin 1 
=?
r2  cos 2  i sin 2 

 i sin 20 
12 cos 75  i sin 75
Power – De Moivre’s Theorem
1.
Using the multiplication rule, if z  r  cos   i sin  , what is z 2 , z 3 , and z n ?
2.
3.
Prove your result for z n using exponential notation.
Use De Moivre’s Theorem to write the following in standard form.

2 cos 20  i sin20 


b.
 

 
 i sin  
 4  cos
10
10

 
Regular
Coordinates
Complex #s
Vectors
Math 120 Notes 11.3

a.
3
5
Rectangular
Polar
r ,  
Exponential
-
x yi
r  cos   i sin 
re i
r 
-
x , y 
x, y  xiy j
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