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Independent Study
Name: ___________________________
Unit 5: Complex Numbers
Introduction
Solving the equation x 2  2 x  2  0 using the quadratic formula yields:
SCHEDULE:
 2  4  4 1  2
2
 2 48
x
2
2 4
x
2
What does  4 mean?
It is known that  4  2 because (2) 2  4
x
and
Nov 6: Quiz: A-G
Nov 20: Quiz H-M
Dec 4: Test
 4  2 because (2) 2  4
 4 cannot equal any real number because the square of any number (positive or negative) is
always positive. Since there is no real number whose square is negative, what should be done with  4 ?
In fact
One possibility would be to ignore it, that is, to solve only equations that will involve even roots of positive
numbers. The second possibility would be to develop a new number system that would contain the
solution… the Complex Number System.
I. Imaginary Numbers are based on and derived from the following definition:
1  i which means i 2  1
Using this definition it is possible to express the square root of a negative number using “i” notation:
 4  (4)  (1)  2  1  2i
 18  (9)  (2)  (1)  3 (2)  (1)  3 2i
1. Write the following in “i” notation:
a.
 24 = _____________
d.
 100 = ____________
b.
 16 = _____________
e.
 50 = _____________
c.
 13 = _____________
f.
 52 = _____________
The numbers 2i and 3 2i are examples of imaginary numbers. The term “imaginary” is used to
distinguish them from real numbers, such as 5, 3 2 , 5  7 , etc. Imaginary numbers have real applications
in the study of aerodynamics, map construction, the theory of electrical conduits and quantum mechanics.
II. The Complex Number System
A. Definition: A complex number is any number that can be expressed in the
form, a  bi , where a and b are real numbers.
Examples: 3   2 is a complex number since it can be written as 3  2i
 8 is a complex number since it can be written as 0  2 2i
7 is a complex number since it can be written as 7  0i
B. Adding and Subtracting Complex Numbers
Examples: 3i  7i  12i  8i
(5  2i)  (6  7i)  11  5i
1. Perform the indicated operations:
a. 5i  8i
=_____________
b. (3  7i )  (4  3i ) =_____________
d. (2  9i )  (3  5i )
= _____________
e. (2  8i)  (3  7i )  (2  3i ) = _____________
c. (3  6i )  (3  6i ) = ____________
C. Powers of i
We know that 6 0  ______ and (3) 0  ______ and x 0  ______. It is also true that i 0  1 .
To find i 3 , write i 3  i 2  i  1  i  i
1
Independent Study
Name: ___________________________
Unit 5: Complex Numbers
To find i 4 , write i 4  i 2  i 2  1  1  1
Repeat the above procedure to complete the following table:
i0  1
i1  i
i2 
i3 
i4
i5
i6
i7




i8 
i9 
i 10 
i 11 
i 12
i 13
i 14
i 15




Did you discover a pattern? ________
What is it? _______________________________________________________
________________________________________________________________
Alternative method to find any power of i: Divide the exponent by 4. The remainder (0, 1, 2, or 3)
can become the new exponent.
i 41  i 1  i
( 41  4  10 Remainder 1)
482
2
i  i  1 ( 482  4  120 Remainder 2)
Example:
1. Use the above method to complete the following:
a. i 19 = ___________
e. i 75 = ___________
b. i 72 = ___________
f.
i 37 = ___________
c. i 98 = ___________
g. i 2384 = ___________
d. i 200 = ___________
h. i 1481 = ___________
D. Multiplying Complex Numbers
Examples:
a.
 2   18
 ( 2i )  ( 18i )
Note: If we multiply before we change to “i” notation
we get:
 2   18
 36  i 2
 36
 6
6
This is incorrect. Always change to “i” notation first.
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b. (1   4 )(3   16 )  (1  2i)(3  4i)  3  4i  6i  8i  3  10i  8  5  10i
c. 2i(1  i) 2  2i(1  i)(1  i)  2i(1  2i  i 2 )  2i(1  2i  1)  2i(2i)  4i 2  4
1. Perform the indicated operations:
a. (3i )( 2i )
= _______________________________________
b.
c.
 6( 3   6)
(2  i)(3  2i)
d. (1  3i )( 2  4i )
e.
 9 (2  6i )
= _______________________________________
= _______________________________________
= _______________________________________
= _______________________________________
(5  3i) 2
= _______________________________________
g. (3  i )(3  i )
= _______________________________________
f.
h. 7(2i  3)  7i(2i  3) = _______________________________________
i.
(1  i)(1  i) 2
= _______________________________________
j.
2i 7 (3i  4)
= _______________________________________
k.
(2 3i  1)(5 3i  2) = _______________________________________
2
Independent Study
Name: ___________________________
Unit 5: Complex Numbers
E. Dividing Complex Numbers
Before dividing complex numbers, it is necessary to review rationalizing denominators involving
irrational numbers. The goal is to have no radical sign in the denominator. To rationalize you must
multiply the numerator and denominator by the conjugate of the denominator.
Example 1:
2 3
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
( 4  3 ) is the conjugate of ( 4  3 ) . Complex numbers also have conjugates. For any complex
number ( a  bi ), the conjugate is ( a  bi ). Therefore the conjugate of (3  2i ) is (3  2i ) , and the
conjugate of (2i ) is (2i ) .
Example 2:
2i 2i 5i
10  7i  i 2
10  7i  1 9  7i





2
5  i 5  i 5  i 25  5i  5i  i
25  1
26
Example 3:
3  2i 3  2i 4  i
12  11i  2i 2
12  11i  2 10  11i





2
4i
4  i 4  i 16  4i  4i  i
16  1
17
1. Give the conjugate of each of the following:
a. 3  6i
b.  2  7i
c.
4i  2
d. 4i
e.  5  8i
f.
4
2. Simplify the following:
a)
4  2i
3i
b)
5
 2  6i
c)
1  2i
5  6i
d)
3  5i
2i
e)
5  10i 4  10i

2i
1 i
f)
i 24
g) i 31
h) (3  i )  i 33
F. Equality Between Complex Numbers
Two complex numbers (a  bi ) and (c  di ) are equal if and only if (iff) the real parts are equal and
the imaginary parts are equal.
Rule: (a  bi)  (c  di ) iff a  c and b  d
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Independent Study
Name: ___________________________
Unit 5: Complex Numbers
1. Use this definition to solve the following for x and y :
a. 3  xi  y  5i
x  ______ y  ______
b. 2 x  3 yi  10  18i
x  ______ y  ______
G. Numbers as Roots of Quadratic Equations
Example 1:
Solve: x 2  2 x  2  0
x
 2  4  4  1  2  2   4  2  2i


 1 i
2
2
2
Example 2:
Solve: x 2  49  0
x 2  49
x    49
x  7 i
1. Solve the following:
a. x 2  6 x  13  0
b.
c.
4x 2  4x  9
d. 2  x 
x  64  0
2
2x  1
x
H. Rectangular Coordinates and Graphing Complex Numbers
There are four ways of expressing complex numbers – rectangular form (this is the same as
a  bi ) rectangular coordinates, polar form and polar coordinates. Thus far you have dealt with
complex numbers in rectangular form. In order to graph complex numbers it is necessary to express
them in coordinate form or as an ordered pair. The complex number z  a  bi can be expressed as
the ordered pair (a, b) and thus can be
represented by a point in a plane with Cartesian
coordinates. This plane is called the complex
plane or Argand plane.
A
The complex plane is formed by considering the
horizontal axis as the real axis and the vertical
axis as the imaginary axis.
A. (1  2i )  (1,2)
B. (4i )  (0,4)
B
Real
1. The following numbers are in rectangular
form. Express each in rectangular
coordinates and graph on the axis provided
above.
a. 2  3i
= __________
b.  2  3i
= __________
c.
3 3 3 3i

2
2
Imaginary
= __________
4
Independent Study
Name: ___________________________
Unit 5: Complex Numbers
I. Polar Coordinates
You graphed complex numbers previously, using a Cartesian or rectangular coordinate system.
Another way of graphing is by using a Polar Coordinate System. This involves using a “grid” made
up of concentric circles and trays emanating from the common center of the circles. The centre of the
concentric circles is called the pole. The ordered pair (r ,  ) is called the polar coordinates of point P.
To graph (3,45) for example, first determine the ray which makes a 45 degree angle with the ray
labelled 0 degrees. Then, move 3 units from the pole along this ray.
1. Graph the following polar coordinates on the polar
coordinate plane provided.
a. (2,60)
b. (3,150)
c. (5,330)
d. (1,945)
180o
90o
(3,45)
0o
o
270
Trigonometry Review
Working with complex numbers in polar form involves trigonometry. Review the trigonometric
functions of special angles by completing the following.
1. Express in simplest radical form:
a. sin 30  ____________
d. sin 315  ____________
b. tan 45  ____________
e. cos 585  ____________
c. cos120  ____________
f.
sin 810  ____________
J. Changing form Rectangular to Polar Coordinates
( a, b)
r
b
θ
a
Consider the point (a, b) in rectangular coordinates. r is the distance from the origin (the pole) to the
point.


Use the Pythagorean theorem to obtain r  a 2  b 2
Determine the quadrant from the diagram, and

tan  
opp b

adj a
Example 1: Express (2,2) in polar coordinates:
Step 1: Sketch to determine the quadrant.
Step 2: Determine the length of r.
r  a b
2
r  44
r2 2
2
θ
Step 3: Calculate the value of θ.
b
a
 2 
  tan 1 

2
  tan 1  
  tan 1 (1)
  135
Therefore, (2,2) in rectangular coordinates is (2 2 ,135) in polar coordinates.
5
Independent Study
Name: ___________________________
Unit 5: Complex Numbers
Example 2: Express  3i in polar coordinates
Step 1: Sketch to determine the quadrant.
θ
 3i  (0,3) in rectangular coordinates.
Step 2: Determine the length of r
r  a2  b2
r  09
r 3
Step 3: Calculate the value of θ.
When the point is on an axis, the angle should be obvious. In this case   270 .
Therefore,  3i is (3,270) in polar coordinates.
1. Express the following in polar coordinate form:
a. (2,2)
g.  4  4i
b. (1, 3 )
h. 1 3i
c.
( 3 ,1)
d. 2  2i
e. 2
f.
i.
2
j.
 2i
k.  3  i
2i
l.
 2
2

i
2
2
K. Polar Form of a Complex Number
Consider the diagram of the complex number (a, b) which
( a, b)
r
b
has polar coordinates (r , ) .
θ
a
a
 cos  
therefore a  r cos
r
b
 sin  
therefore b  r sin 
r
The complex number in rectangular form in a  bi , but we can substitute in our equivalent values for
a and b, yielding:
a  bi  r cos   r sin i  r (cos   i sin  )
r (cos   i sin  ) is called the polar form of a complex number and is often abbreviated as rcis .
6
Independent Study
Name: ___________________________
Unit 5: Complex Numbers
L. Converting from Polar Coordinates to Rectangular Form
Example:
Convert ( 2,210) to rectangular form:
( 2,210)  2(cos 210  i sin 210)
 3
1
( 2,210)  2
 i 
2 
 2
( 2,210)   3  i
1. Express in rectangular form.
a. (3,135)
d. (3,270)
b. (2,315)
e. 4cis
 
c.  5, 
 2
f.
4
3
5cis300
2. In each of the following, one of the ways of writing a complex number is given. Complete the
table by filling in the other spaces with the appropriate form.
Rectangular Form
Rectangular Coordinates
Polar Form
Polar Coordinates
( 2,240)
 1 3i
2  2i
2cis120
(4,0)
 
1, 
 2
M. The Product of Complex Numbers Working in Polar Form
Consider 2 complex numbers: r1 (cos A  i sin A) and r2 (cos B  i sin B)
Multiplication yields:
r1 r2 (cos A  i sin A)(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 r2 cos A cos B  sin A sin B  isin A cos B  cos A sin B 
 r1 r2 cos( A  B)  i sin( A  B)
Example:
2(cos 45  i sin 105) 8(cos 105  i sin 105)
 16(cos 150  i sin 150)
7
Independent Study
Name: ___________________________
Unit 5: Complex Numbers
1. Find the following products. Leave your answers in polar form.
a. 3(cos120  i sin 120) (cos 225  i sin 225)
b.
6cis62 (3cis136)
 

 
c. 2 cos  i sin 
2
2 
 
2
 2
   1  
cis 
cis 
d. 
6  2
6
 2
2
e. (2cis18)  (3cis 72)
8