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Complex Numbers
Name______________________
Every system of numbers arises out of a need.
 Why do we need Complex numbers?
 Why aren’t Real numbers enough?
 Do they have any practical applications?
I. Imaginary Numbers
Definition: 1  i which means i  1
so we can write:
2
 4  2i and
1.
 18  3 2 i
Write the following in simplified “I” form:
 24
 100
a)
d)
The numbers
b)
 16
c)
e)
 50
f)
 13
 52
2i and 3 2 i are examples of imaginary numbers. The word imaginary is used to
distinguish them from real numbers like 5,
2 3 . 5  7 , etc.
Definition: A complex number is any number that can be expressed in the form
real numbers.
Examples:
a)
a  bi where a and b are
3   2 is complex since it can be written as 3  2 i
b) 5  1 is a complex number since 5  1  5i
c) 7 is a complex number since 7  7  0i
 0  5i
II. Adding and Subtracting Complex Numbers
Examples:
a) 3i  7i  12i  8i

2.
 
b) 5  2i  6  7i
Perform the indicated operations:
a) 5i  8i 
c)
  11  5i
b)
3  6i   3  6i  
d)
3  7i   4  3i  
2  9i   3  5i  
III. Powers of i
6 0  ____,
 30
 _____ and x 0  _____ . It is also true that i 0  1
i   1 and i 2  1 . To find i 3 we write: i 3  i 2  i  1  i  i
4
2
2
4
If i  i  i , explain why i  1 .
Recall that
Look for the pattern:
i0 
i4 
i8 
i1 
i5 
i9 
i2 
i6 
i 10 
i3 
i7 
i 11 
Write a rule you could use to simplify a power of i:
3.
Use your rule to write each of the following in simplest form:
i 19 
200

d) i
2384

g) i
a)

75
e) i 
1493

h) i
b) i
72
-1-
c) i
98
f) i
39


Complex Numbers
Name______________________
IV. Multiplying Complex Numbers
Examples:

b) 1 
 2   18
Simplify: a)
1 
 2   18

 4 3   16

 4 3   16
 2 i  18 i
 1  2i 3  4i 
 36 i 2
 3  4i  6i  8i
If we multiply before we
change to “i” form we would
get:
 6
 2   18  36  6


c)
e)
g)
i)
k)
3i  2i  
2  i 3  2i  
 9 2  6i  
3  i 3  i  
1  i 1  i 2 
2
2i 1  i 
2

 2i 1  2i  1
2
 3  10i  8
 2i  2i 
 5  10i
 4i 2

 4
Perform the indicated operations:
a)
2
 2i 1  2i  i 2
which is wrong. Always
change to “i” form before
performing any operation.
4.
c) 2i 1  i 


6 3 4 
d) 1  3i 2  4i  
b)
f)
h)


3 i 1 5 3 i  2 
j)
l)
5  3i 2 
72i  3  7i2i  3 
2i 7 3i  4 
7  2i 7  2i  
V. Dividing Complex Numbers
Recall division of irrational numbers:
2 3
Example: To write
4 3
with a rational denominator (rationalize the denominator), we multiply
numerator and denominator by the conjugate of the denominator. The conjugate of
2 3
4 3

4  3 is 4  3 .
2  3 4  3 8  2 3  4 3  3 11  6 3



13
4  3 4  3 16  4 3  4 3  3
The same strategy works for complex numbers. The conjugate of 3  2i is
5. Write the complex conjugate of each of the following:
a) 3  6i
b)  2  7i
c) 4i  2
d) 5i
3  2i .
a  bi is the number a  bi . For any complex
number z , the complex conjugate is written z .
The complex conjugate of a complex number
6.
z  5  2i and w  3  4i , complete each of the following:
a) Evaluate z  z . What kind of number is it? Will this always be true? Why?
If
b) Evaluate z  z . What kind of number is it? Will this always be true? Why?
Evaluate
z  w and z  w . Compare the results.
d) Evaluate
z  w and z  w . Compare the results.
c)
    
e)
Show that z w  zw
f)
Show that
z    z  .
w  w 
g) Show that
z   z .
h) Show that
z   z 
2
1
2
1
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Complex Numbers
Name______________________
This is how we use conjugates to simplify complex fractions.
Examples:
a) Simplify:
2i
5i
b) Simplify:
2  i 5  i 2  i 5  i  10  7i  1 9  7i




5  i 5  i 5  i 5  i 
25  1
26
7.
4i
3  2i
4i

3  2i


10  11i
13
Simplify the following:
4  2i
3i
1  2i
d)
5  6i
24
g) i
1 i
1 i
3  5i
e)
2i
31
h) i
a)
5
 2  6i
5  10i 4  10i

f)
2i
1 i
33
i) 3  i i
b)
c)
VI. Equality Between Complex Numbers
Two complex numbers a  bi and c  di are equal if and only if the real parts are equal and the imaginary
parts are real.
a  bi  c  di if and only if a  c and b  d.
8.
Solve for x and y:
a) 3  xi  y  5i
b)
c)
d)
e)
f)
g)
h)
x  _____ and y  _____
x  _____ and y  _____
x  _____ and y  _____
x  _____ and y  _____
x  _____ and y  _____
x  _____ and y  _____
x  _____ and y  _____
x  _____ and y  _____
2 x  3 yi  10  18i
7 y  18 xi  3  3i
3 x  yi  2  xi
2x  3 y  3x  2 yi  4  i
5x  2 y  2x  yi  3  2i
2x  3y  x  yi  19  12i
3  i   x  yi   4  3i
VII. Complex Numbers as Roots of Quadratic Equations
Page 276: #19b, d, f, h (Answers are in the book)
VIII. Roots of Polynomial Equations
Page 287: Focus D – Complete Steps A to F
Page 288: Answer questions #69d, f, #70b,c, #71
IX. Complex Numbers and Their Graphs
Read page 277: Focus B
Make note of the definitions in the Focus and in the
margin.
Complete Steps A, B, C and D of the Focus
(Use the grid for Steps A and B)
Do page 279: #29, 30
So far we have expressed complex numbers in two forms and graphed them on a Cartesian or rectangular
coordinate system.
 rectangular form (example: 2  3i )
 rectangular coordinates (example: the ordered pair 2,3 )
Another way of graphing complex numbers is by using a Polar Coordinate System.
Read Focus E for a description of polar paper and make note of the important definitions in the margin.
Examine Example 1 on page 290.
Do page 290: #1, 2, 3
 
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Complex Numbers
Name______________________
X. Graphing Equations on Polar Paper
Page 291-293: Work through Investigation 3, Parts 1 and 2 including the Investigation Questions 5-10 and
Check Your Understanding #11 -13.
XI. Trigonometry Review
Since working with complex numbers in polar form involves trigonometry, we shall begin with a brief
review of the trig ratios of special angles:
9. Express in simplest radical form:
a) sin 30 
b) tan 45 
c)
cos120 
d)
sin 315 
e)
cos 585 
f)
sin 810 
10. Find the angle which satisfies the following conditions:
 1 
arcsin  
 , in quadrant III
2

 3

b) arccos 
 2  , in quadrant IV



3

c) arctan  
 3  , in quadrant II


a)
XII. Changing from Rectangular to Polar Coordinates
Suppose we have the point a, b in rectangular
coordinates. Then, since r is the distance from the
origin (or the pole) to the point, we can use


Pythagoras to obtain r  a  b . We can also
tell from the diagram which quadrant is involved
2
r
b
and that
tan  
θ
2
opp b

adj a
a
Example 1: Express
coordinates.
 2,2 in polar
Example 2: Express
coordinates.
3i in polar
 3i  0,3
Step 1: Draw a
sketch and
determine
which quadrant
θ
θ
(0, -3)
Step 2: Find the
value of r .
Step 3: Find the
value of 
r  a2  b2
r  a2  b2
r  44
r
r2 2
b
2
tan   
 1
a 2
In the second quadrant,
Conclusion
r 3
  270
  135
 3i in rectangular coordinates is

3,270 in polar coordinates.
2 2 ,135 in polar coordinates.
11. Change each of the following to polar coordinates:
a)
 2,2
2  2i
g)  4  4i
d)
j)
 2i

b) 1, 3
2
Since the point is on an axis, the angle
should be obvious.
 2,2 in rectangular coordinates is

 3

c)
e) 2


3 , 1
h)
1 3i
2i
i)  2
k)
 3 i
l) 
f)
-4-
2
2

i
2
2
Complex Numbers
Name______________________
XIII. Polar Form of a Complex Number
Consider the complex number a, b which has polar coordinates




r,  .
Draw a perpendicular from a, b to the x-axis. Label the sides of the
right triangle formed.
Then sin   _____ and cos   _____
Since
r
(a,b)
θ
a
b
 cos , then a  r cos and  sin  , then b  r sin 
r
r
The complex number in rectangular form is
a  bi  r cos   r sin  i
a  bi . Using a  r cos and b  r sin  we have
a  bi  r cos   i sin  
r cos  i sin   is called the polar from of a complex number and is often abbreviated as rcis
XIV. Changing from Polar Coordinates to Rectangular Form
Example: Change 2. 210 to rectangular form.


2, 210  2cos 210  i sin 210

2, 210  2 
3
 2
2, 210  
i
 1

2 
3 i
12. Change the following to rectangular form:

d)
3, 135
b)
d)
3, 270
e) 4cis
2 , 315



c)  5,
4
3
f)


2
5cis300
13. In each of the following, one of the ways of writing a complex number is given. Complete the table by
filling in the spaces with the appropriate form.
Rectangular Form
Rectangular Coordinates
Polar Form
Polar Coordinates
2, 240
 1 3i
2  2i
2cis120
 4, 0
1, 90
XV. Using Polar Equations to Simplify Graphing
Page 297: Read focus F and answer Focus Questions #27, 28, 29 (You don’t need to graph these questions)
XVI. The Product of Complex Numbers Working in Polar Form
We know that two complex numbers can be multiplied. If they are in polar form, it will look like this:
r1 cos A  i sin A  r2 cos B  i sin B   r1 r2 cos A  i sin Acos B  i sin B 
r1 cos A  i sin A  r2 cos B  i sin B   r1 r2 cos A cos B  i cos A sin B  i sin A cos B  sin A sin B 
r1 cos A  i sin A  r2 cos B  i sin B   r1 r2 cos A cos B  sin A sin B  i sin A cos B  cos A sin B 
r1 cos A  i sin A  r2 cos B  i sin B   r1 r2 cos A  B   i sin  A  B 
Example:
2cos 45  i sin 45  8cos105  i sin 105  16cos150  i sin 150
14. Find the following products. Leave your answer in polar form.
a) 3 cos120  i sin 120  cos 225  i sin 225



6cis 62  3cis136
2
c) 2cos 90  i sin 90
b)
 2
 1


d) 
 2 cis 30   2 cis 45 



e) 2cis18  3cis 72 (leave your answer in rectangular form)
2
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