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Mathematics B30
Module 1
Lesson 1
Mathematics B30
The Complex Number System
1
Lesson 1
Mathematics B30
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Lesson 1
The Complex Number System
Introduction
In this lesson you will learn about a new number called the imaginary number. You will
learn how to simplify expressions which involve imaginary numbers. Imaginary numbers
then, in turn, will be used to define complex numbers.
Complex numbers allows us to be able to solve more mathematical questions.
Mathematics B30
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Lesson 1
Mathematics B30
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Lesson 1
Objectives
After completing this lesson, you will be able to
•
define and illustrate complex numbers.
•
express complex numbers in the form a + bi.
•
add and subtract complex numbers.
•
multiply and divide complex numbers.
•
divide complex numbers using conjugates.
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Lesson 1
Mathematics B30
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Lesson 1
1.1 Number Systems
Number Systems have been a developmental process over hundreds of years. During the
development of Mathematics, number systems were expanded based on the need.
Number Systems
1.
Natural Numbers
1, 2, 3, ...
2.
Whole Numbers
0, 1, 2, ...
3.
Integers
...,  1, 0, 1, ...
4.
Rational Numbers
5.
Irrational Numbers
{ x x is a real number and x cannot be
shown as a quotient of integers.}
2
Examples: 
6.
Real Numbers
{ x x is a number that may be shown as a
p

p and q are integers, and q  0 

q

3
1
 0 .75
 0 .33333 ...
Examples:
4
3
decimal.}
•
The Natural Numbers were the first system of numbers used. That system was used
until the need for the zero was addressed. The Whole Numbers were the system that
resulted from that need.
•
From Whole Numbers, Integers were introduced for the use of the negative numbers.
Up until the Integers, solutions to 7  9 = ? could not be solved.
•
Using only Integers, the problem was that division did not always lead to another
Integer. Example: 1  4  ? The result was the introduction of the Rational
Numbers.
•
The Rational Numbers were satisfactory until the problem arose that not all numbers
could be expressed as the quotient of integers. The Irrational Numbers were then
developed and this meant that equations such as x 2  2  0 could have a solution.
•
The system of Irrational Numbers together with the rational numbers is called the
Real Number system.
Mathematics B30
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Lesson 1
Here, in diagram form, are the number systems you have seen and worked with so far on
your mathematical journey.
Irrational
Numbers
Zero
Real
Numbers
Integers
Natural
Numbers
Rational
Numbers
Non-Integers
You are in a similar situation where you have solved problems that have given you an
impossible solution, or an undefined solution. Here are two such cases.
1.
x 2  6
x  6
•
2.
Solutions with a negative sign under the square root were not allowed
by early mathematicians. Imaginary is what situations like this were
called.
If the graph of a parabola (quadratic equation) has no intersection with the x-axis, we
say that the equation has no real solution. Even though real roots (x-intercepts) do
not exist, the equation and its graph do exist.
y  x2 1
Graph:
Table of Values
•
y
x
y
2
1
0
1
2
5
2
1
2
5
x
The desire by mathematicians to be able to describe every quadratic equation
by some sort of roots was not possible with the real number system.
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Lesson 1
Gradually, uses were found that needed the application of the imaginary numbers, making
it necessary to enlarge the set of real numbers to form the set of complex numbers. By doing
so, the set of complex numbers provides a solution for just about any equation that can be
written.
The definition of a complex number is motivated by the equation x 2  1  0 .
x2 1  0
x 2  1
x   1
No real solution.
To provide a solution for the equation x 2  1  0 , a new number i is defined.
Definition:
1  i
The square root of any negative number can now be defined in terms of i.
 r  1 r
Definition:
 1 r
i r
This definition emphasizes that roots of negative numbers must be dealt with in only one
way. Convert to i times the root of a positive number.
For any real number x, ix is called an imaginary number.
Example 1
Convert
Mathematics B30
 9 into the imaginary number form, ix .
9
Lesson 1
Solution:
 9  19
 1 9
 i3
A common error is mistaking
reason we write
3 i for
3 i , with the i under the radical. For this
3 i as i 3 .
Using the equation x 2  1  0 along with the definition
original equation to obtain another useful equation.
1  i we can substitute into the
x2 1  0
x 2  1
x   1
Definition:
1  i
x  i
Substitute.
x2 1  0
 i 2  1  0
 i 2  1
i 2  1
When finding the product of  3   1 , the product rule for radicals can’t be
used. The rule applies only when both radicals represent real numbers.
Therefore, always change  r (where r  0 ) to the form i r before multiplying
or dividing.
Mathematics B30
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Lesson 1
Right
Multiply:
 3  1
Wrong
Multiply:
 3  1
Solution:
 3  1
Solution:
 3  1
 3  1 
 i 3 i 1

 ii 3  1
 3
i  3
2
  1  3
* Wrong answer due to
incorrect method.
 3
Example 2
Simplify:
a.
b.
c.
1 1
2 6
 4  9  16
Solution:
a.
 1  1  ii
 i2
 1
2 6  i 2 i 6
b.
 i2 2 6
  1  12
 2 3
c.
 4  9  16  i 4 i 9 i 16
 i2i 2  3  4
  1  i 24
 i 24
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Lesson 1
Use the following keystroke pattern to find the product of

MODE
(six times)
1 1 .
 (once) to have a + bi highlighted.
ENTER
2nd QUIT
(2nd x2 )
–
1
)
negative sign (not the subtraction key)
–
1
)
ENTER
negative sign
Answer displayed:  1
Find the product of
Mathematics B30
 2  6 and the product of
12
 4  9  16 using the calculator.
Lesson 1
Activity 1.1
A pattern develops when reviewing the various powers of i. Complete the chart.
Powers of i
Working
Room
Value
i1
i
i2
1
i3
By definition
i 3  i  i 
2
1
i4
i5
i6
i7
i8
i9
i 10
i 11
i 12
i 13
i 14
i 15
i 16
Conclusion
The pattern of the powers of i is _____, _____, _____, _____.
Using the fact that i 4  1 , larger powers of i can be simplified.
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Lesson 1
Example 3
Find each power of i.
a)
b)
i 32
i 103
Solution:
a)
i
32
8
4 32
 
 i
4 8
 1 
8
1
b)
i 103  i 100  i 3
 
25
 i4
25 R3
4 103
 i3
 1   i 3
25
 
 1 i2 i
 i
There are many combinations that can be used to solve questions like Example 3.
Alternate Solution:
a)
i 32  i 2  16
b)
 
 i2
16
 i i 2  51
 
  1 
 i i2
16
51
 i  1 
 i  1
 i
1
Mathematics B30
i 103  i  i 102
51
14
Lesson 1
Example 4
2  3  4 .
Simplify
Solution:
2  3  4  i 2 i 3 i 4

 i 2  3 2

Exercise 1.1
1.
2.
Convert each of the following to the imaginary number form ix.
a.
1 6
d.
 2  3  6 
b.
3
e.
2
c.
 100
f.

6
1 7
Simplify:
a.
b.
4 2  4 2 8 
1  2
h.

i.
i7
j.
i8

1
3
 1 2i
c.
3.

3
4
d.
2i i i 2
k.
i 39
e.
i4
l.
i 93
f.
i 17
m.
i 46
g.
i4 i9
n.
i 84
Simplify:
a.
1  2 1   4
b.
 4  6i  7i 3
c.
 5  7   9
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Lesson 1
1.2 Complex Numbers
The complex number system entails the use of the imaginary numbers, ix.
A complex number is an expression of the form
a  bi , where a and b are real numbers.
2  3i
Example:
real part
•
imaginary
part
Every real number may be considered as a complex number with imaginary part
equal to zero.
Example:
Solution:
•
Express 2 as a complex number.
2  2  0i
If the real part is zero, the remaining part bi is called a pure imaginary number.
Example:
0  7i
Our diagram on the number system we know now looks like the following.
Complex
Numbers
a + bi
a and b are
reals
Imaginary
numbers
a + bi,
b0
Irrational
Numbers
Real
Numbers
a + bi
b0
Zero
Integers
Natural
Numbers
Rational
Numbers
Non-Integers
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Lesson 1
Complex numbers can be represented in a coordinate plane by letting x  yi correspond to
the point x, y  in the following graph.
y
4i
–3 + 3i
2 + 2i
3
x
1 – 3i
–4 – 3i
When the coordinate plane is used for complex numbers
•
the plane is called the complex plane.
•
the horizontal axis (x-axis) is called the real axis.
•
the vertical axis is called the imaginary axis.
The absolute value of a complex number can also be found.
y
c = x + yi
The absolute value of a complex number, c , is the
distance from the point c to the origin 0.
x
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Lesson 1
The distance can be determined using the Pythagorean
Theorem.
hypotenuse 2  side 2  side 2
c  x2  y2
Absolute Value of a complex number.
a  bi  a 2  b 2
Example 1
Plot 3  4 i on a complex plane and find 3  4 i .
Solution:
y
3 + 4i
3  4i  x 2  y2

3 2  4 2
5
3
x
Use the following keystroke pattern to find the absolute value of 3  4 i
MATH
 twice (to highlight CPX)
5 ( to select abs( )
3 + 4 2nd . ) ENTER
Answer displayed: 5
Mathematics B30
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Lesson 1
Example 2
Simplify 4 2  5i  .
Solution:
4
(
2
+
5
i
)
=
8
+
2
0
i
* Only the real numbers are multiplied.
Exercise 1.2
1.
2.
3.
Write as a complex number.
a.
4  2
b.
6 5
c.
7
Plot each of the following on a complex plane. Find a  bi for each as well.
a.
5 i
b.
 3  2i
c.
7  4i
d.
6  8i
Simplify.
a.
65  3i 
b.
6 4  3
c.
8 i 3  3i 5

Mathematics B30

19
Lesson 1
1.3 Addition and Subtraction of Complex Numbers
Addition of complex numbers is performed by adding real parts to real parts and
imaginary parts to imaginary parts. Subtraction is similar.
a  ib   c  id   a  c  i b  d 
a  ib   c  id   a  c  i b  d 
Example 1
Complete the indicated operation.
2  3i    1  4 i   7  i 
Solution:
2  3i    1  4 i   7  i   2  1  7   i 3  4  1
 6  0 i
 6
Use the following keystroke pattern
CLEAR
(
2
+
3
i
[ 2nd
. ]
(
–
1
–
4
i
–
)
)
+
use negative sign button
(
7
–
i
)
ENTER
Addition and subtraction of two complex numbers can be shown graphically by using the
origin as a starting point.
Example 2
Illustrate 4  3i   3  5i  on a graph.
Mathematics B30
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Lesson 1
Solution:
y
(3 + 5i )
Plot the two complex points.
(4 + 3 i)
x
y
(3 + 5 i )
5i
3
Create a parallelogram.
(4 + 3 i )
x
y
(7 + 8 i )
4  3 i   3  5 i   7  8 i 
3  5 i   4  3 i   7  8 i 
3i
(3 + 5i )
4
(4 + 3 i )
x
Mathematics B30
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Lesson 1
Example 3
Illustrate 4  3i    2  4 i  on a graph.
Solution:
 2  4 i 
4  3i 
Plot the two complex points.
y
(–2 + 4i )
Plot the opposite of  2  4 i   2  4 i  .
(4 + 3 i )
x
(2 – 4 i )
Mathematics B30
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Lesson 1
y
Create a parallelogram with 4  3i  and 2  4 i  .
(4 + 3 i )
 4  3i    2  4 i   6  i 
x
(6 – i )
(2 – 4 i )
Exercise 1.3
1.
Complete the indicated operation.
a.
b.
c.
d.
e.
f.
2.
4  7i   3  3i 
6  3i    9  2i 
 8  5i   6  3i 
2  5i    3  4i 
32  3i   4 6  i 
24  3i   35  2i 
Illustrate the following operations on a graph. State your solution.
a.
b.
c.
d.
3  2i   4  3i 
3  4 i   2  2i 
 2  3i   3  2i 
23  i 
Mathematics B30
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Lesson 1
1.4
Multiplication and Division of Complex
Numbers
Multiplication of complex numbers is performed by using the distributive law just like
multiplication of binomials.
(a + bi )(c + di )
= ac + adi + bci + bdi 2
= ac + (ad + bc)i + bd (–1)
= (ac – bd ) + (ad + bc)i
Example 1
Multiply 2  3i  1  4 i  .
Solution:
2  3 i  1  4 i   2  1   2  4 i   3 i  1   3 i  4 i 
2
 2  8 i  3 i  12 i 
 2  11 i  12  1 
 10  11 i
Use the following keystroke pattern.
CLEAR
( 2 + 3 i ) ×
( – 1 – 4
negative
Mathematics B30
i
) ENTER
subtraction
24
Lesson 1
Example 2
Solve 5  2 i  .
2
Solution:
5  2 i 2  5  2 i 5  2 i 
 25  20 i  4 i 2
 25  20 i  4
 21  20 i
Example 3
Multiply 1  6i 1  6i  .
Solution:
1  6 i 1  6 i   1  6 i  6 i  36 i 2
 1  36  1 
 37
The numbers 1  6 i and 1  6 i are called conjugates. Generally, the conjugate of a  bi is
a  bi . As shown in Example 3, the product of a complex number and its conjugate is a
real number a 2  b 2 . The use of conjugates is used to simplify the quotient of two
imaginary numbers.
Use the following keystroke pattern to determine the conjugate of
 1  2i  .
CLEAR
MATH
 (twice)
1 (to select conj( )
1
+
2
i
)
ENTER
negative
Answer displayed:  1  2 i
Mathematics B30
25
Lesson 1
Division of complex numbers involves multiplying the dividend and divisor by the complex
conjugate of the divisor.
(2–3
i) ÷ (–
1+2
i)
D
iv
id
e
n
d
D
iv
is
o
r
Example 4
Calculate 2  3i    1  2i  .
Solution:
2  3 i  1  2 i 
2  3i

 1  2 i  1  2 i  1  2 i 
* The complex conjugate of  1  2i  is  1  2i  .
 2  4 i  3 i  6 i 
2

1  2 i  2 i  4 i 
 2  i  4  3   6  1 

1  4  1 
8 i
8 1

  i
5
5 5
2
Different Sign
Example 5
Calculate 5  7 i   2i .
Solution:
5  7 i 5  7 i  2 i 

2i
2 i  2 i 
* The complex conjugate of 0  2 i is 0  2 i .
 10 i  14 i 2

 4i 2
 10 i  14

4
7 5
   i.
2 2
Mathematics B30
26
Lesson 1
Never leave an imaginary number, bi, in the denominator of a fraction!
Equality of Complex Numbers
Two complex numbers are equal if and only if the real parts are equal and the imaginary
parts are equal.
a  bi  c  di if and only if, a  c and b  d .
Example 6
For what values of x and y does equality hold?
2 x  5 i  2  5  7 yi
Solution:
Equate the real parts and solve.
2x  2  5
2x  7
x
Equate the imaginary parts and solve.
5 i  7 yi
5  7 y
y
Check.
Mathematics B30
7
2
5
7
2 x  5 i  2  5  7 yi
7 
 5
2   5 i  2  5  7    i
2
 7
5  5 i  5  5 i ()
27
Lesson 1
Exercise 1.4
1.
Write each in the form a  bi .
a.
b.
c.
d.
e.
2.
3  2i 3  2i 
i.
j.
3  i   1  i 
 5  i    5  i 
2  1  2i 
2i  2  i 
1  3  2i 
1  i 1
3  2i   2i
1  i    3i 
7  3i 
 12  4 i 
f.
g.
h.
i.
j.
Write the reciprocal of each and then write it in the form a  bi .
a.
b.
c.
d.
e.
4.
i 2  3i 
 1  5i  1  5i 
i  i 
a  bi a  bi 
2  5 i 2
1  3 i 3
f.
g.
h.
Perform each of the following division and write answer in the form a  bi .
a.
b.
c.
d.
e.
3.
63  2i 
5i4  3i 
1  2i 2  3i 
3i
8
2  7i
 5  3i
a  bi
The reciprocal of x is
1
.
x
Solve for x and y in each of the following equations.
a.
b.
c.
d.
e.
3  7 i  x  yi
3  i   2 x  yi   2i  6i
3 x  2 yi  2i  1  6i 
ii  1  2 y  xi
i(2  i )  3 x  iy  2 xi  4 y
Mathematics B30
28
Lesson 1
Self Evaluation
X
10 5
–9 5
4
F
7
4 –2 –2
6i
–4 –9 –3 –9 –4 8i 10 –i –2
–9 5
3
2
4
8 –6
4
–4 –9 9
–i
4 –2
–2 2 i
–9 – i
8 10 –6 10 3
–i
4 –4
–9 6 i 5
F
6i
4
–9 4i
8 10 –9 –6 –2
F
3
2
4
3i
8 i 10 – i –9 3
5i
6i
10
3
2
–9 5
F
–4 4
–6 –9 –3 10 3 –9 –3 4
6i
4i
4 10 3
Evaluate the following equations for clues to the sentence above. Once the blanks are filled,
see if you can shorten the phrase into a well-known proverb!
1.
3  i 3  i   A
11.
N  3  4i
2.
5  4 i 2
12.
3  2i 5i  2  0  5i  16 
3.
3i2  i   2i3  2i   C
13.
4 i  2i   P
4.
1  i 3 1  i i  D
14.
i2  i 3  4 i   Q  i7  5i 
5.
1  i 2 1  i 2
15.
2i
4i
R


2  i 1  2i  5
6.
2i1  3i   2i  2   H
16.
i
1i S


1i
2i
2
7.
1  2i 2  3i   2  5i 1  3i   I
17.
8.
8i1  i   24  3i   K
x 2  6 x  73  0
x  T U
18.
z 2  6 z  34  0
z  V Y
 B  40 i
E
9.
i 63  L
10.
3  2i  5  7i   7i  29   M
Mathematics B30
29
Lesson 1
Summary – Lesson 1
•
Create a summary of this lesson to assist you come examination time.
•
Each summary is to be sent in with the assignment to be evaluated.
•
Items to include in a summary:
•
definitions
•
formulas
•
calculator “shortcuts”
Mathematics B30
30
Lesson 1
Answers to Exercises
Exercise 1.1 1.
a.
b.
c.
d.
e.
f.
6i
i 3
10 i
6i
 6i
 7i
2.
a.
b.
c.
d.
 2
i
 i4
2
h.
i.
j.
k.
i 48  4 i 3
i
1
i
e.
f.
g.
1
l.
m.
n.
i 93  i 4
1
1
a.
b.
c.
5i
15 i
i 5  7 3
Exercise 1.2 1.
a.
b.
c.
4 i 2
5i 6
0i 7
2.
a.
3.
i
i

 
23
i  i

 5  i  x2  y2

 5 2  1 2
 26 units
Mathematics B30
31
Lesson 1
b.
 3  2i  13 units
c.
y
7 + 4i
d.
x
7  4 i  65 units
x
6  8 i  10 units
y
6 – 8i
3.
Mathematics B30
a.
b.
30  18 i
 24  6  3  24  6i 3
c.
 24 i 3 5  24 i 8  24 i 4  24 1   24
* remember the laws of exponents.
 
2
32
2
Lesson 1
Exercise 1.3 1.
2.
a.
7  10 i
b.
3 i
c.
 2  2i
d.
5  9i
e.
6  9i   24  4i   30  5i  56  i 
f.
7
y
a.
(7 + 5 i )
x
0
b.
y
x
0
(5 – 2 i )
Mathematics B30
33
Lesson 1
c.
d.
Exercise 1.4 1.
Mathematics B30
a.
18  12 i
b.
15  20 i
c.
8 i
d.
3  2i
e.
13
f.
26
g.
1
h.
i.
a 2  b2
 21  20 i
j.
 26  18 i
34
Lesson 1
2.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
3.
a.
b.
c.
d.
e.
4.
a.
b.
c.
d.
e.
Mathematics B30
2i
12  5 i
13
2  4i
5
 2  4i
5
3  2i
13
1i
2
2  3i
2
1  i
3
7
0 i
3
0  3i
1  3i
 3i
 3i  1
1




i  0 i
2
3i  3i  9 i
9
3
3
1
1
 i 8  2i 2  i 2
i 2




 0
2
4
4
 8 i 8  i 8  i 8 
1
2  7i
2  7i
2  7i
2
7





i
2
2  7 i 2  7 i 4  49 i
53
53 53
1
 5  3i  5  3i  5  3i
5 3





i
2
 5  3 i  5  3 i 25  9 i
34
34 34
1
a  bi
a  bi
a  bi
a
b

 2
 2
 2
 2
i
2 2
2
2
a  bi a  bi a  b i
a b
a b
a  b2
3, 7
3
 ,5
2
1
,4
3
1
1, 
2
Solve two equations in two unknowns.
7 8 
2  y  2 x , 1  3 x  4 y. Solution  ,  .
 11 11 
35
Lesson 1
Answers to Self Evaluation
A
N
I
N
I
N
T
O F
E
T
H
D
E
D
E
I
E
R I
X
V
H
C
I
D
E
E
U
P
S
A
R
S
L
E
D
I
B
L
E
S
Q
U
A
L
I
T
V
A
T
I
V
S
P
E
F
O
S
A
R
I
Y
A
M
O
O
K
P
I
T
I
A
F
F
L
L
O
I
A
M E
E
D
N
R
S
T
A
H
I
N
T
“Too many cooks spoil the broth.”
17.
x 2  6 x  73  0
a 1
1
Use the ‘Quadratic Formula’.
b  6
c  73
 b  b 2  4 ac
2a
6   256
x
2
x  3  8i
x=
x  T U
Mathematics B30
36
2
Lesson 1
Mathematics B30
Module 1
Assignment 1
Mathematics B30
37
Lesson 1
Mathematics B30
38
Lesson 1
Optional insert: Assignment #1 frontal sheet here.
Mathematics B30
39
Lesson 1
Mathematics B30
40
Lesson 1
Assignment 1
Values
(20)
A.
Multiple Choice: Select the correct answer to complete each of the following
statements and place a check () beside it.
1.
 49 converted to imaginary form, ix, is ***.
____
____
____
____
2.
4
4
i 2 16
16
a.
b.
c.
d.

1

12
is ***.
1
i
i
1
The absolute value of 9  11i is ***.
____
____
____
____
Mathematics B30
a.
b.
c.
d.
The simplest form of the expression
____
____
____
____
4.
undefined
i7
i  7 7 
i49
 2   8 simplified is ***.
____
____
____
____
3.
a.
b.
c.
d.
a.
b.
c.
d.
202
 40
i 202
i  40
41
Lesson 1
5.
2  3i   7  5i  in complex number form is ***.
____
____
____
____
6.
Mathematics B30
a.
b.
c.
d.
30  21 i  3i 2
30  9 i  3i 2
27  21 i
27  9 i

 
 

When simplified,  2 20  3 i  5 2 45  i  3 80  3 i is ***.
____
____
____
____
8.
9  2i
 5  2i
9  8i
 5  8i
When in complex number form, 5  i 6  3i  is ***.
____
____
____
____
7.
a.
b.
c.
d.
a.
b.
c.
d.
14
14
38
38
5  20 i
5  2i
5  20 i
5  2i
 1  i   2  3i  in complex number form is ***.
____
a.
____
b.
____
c.
____
d.
2 2
 i
5 5
5
5


i
13 13
2  i 1
1
5

i
13 13

42
Lesson 1
9.
10.
Mathematics B30
The reciprocal of 2  3 i in complex number form is ***.
____
a.
____
b.
____
c.
____
d.
1 1

2 3i
2
3

i
13 13
1
2  3i
2 3
  i
5 5
The values for x and y, respectively, in the equation
2  3i   x  3 yi    4  6i i are ***.
____
a.
____
b.
____
c.
____
d.
7
3
7
8, 
3
7
, 8
3
7
 ,8
3
 8,
43
Lesson 1
Mathematics B30
44
Lesson 1
Answer Part B and Part C in the space provided. Evaluation of your solution
to each problem will be based on the following.
(15)
B.
•
A correct mathematical method for solving the problem is shown.
•
The final answer is accurate and a check of the answer is shown where
asked for by the question.
•
The solution is written in a style that is clear, logical, well-organized,
uses proper terms, and states a conclusion.
Simplify each to the complex form a  bi .
1.
2i  i 2
2.
 1  3 i 2
3.
4.
Mathematics B30
2 i  3 1  i 
1
1
i
45
Lesson 1
5.
2i  1  i 
6.
2  3i 2  3i 
7.
i 16
8.
i 21
9.
Mathematics B30
(Note: an imaginary number i can be written as 0  i in complex
form)
 48   3
2
46
Lesson 1

 
 
10.
 6 2  i 8  4  i 18  4 3  3 i 50
11.
5i
9i
12.
6 i 9  11 i 42  i 103  2i 121
13.
5  5i  3  7i   2  3i  5  i 
Mathematics B30
47

Lesson 1
(2)
C.
14.
2 x  3 y  5 x  2 yi  13  4 i .
15.
2  3i x  yi   1 .
1.
Show that i,  i , 1 and  1 are each solutions of the equation x 4  1  0 .
Mathematics B30
Solve for x and y.
Solve for x and y.
48
Lesson 1
2.
Illustrate the following operations on a graph. Label all points clearly.
(4)
a.
4  9i   5  2i 
(4)
b.
(12  4 i)  (7  4 i)
(5)
3.
On a separate piece of paper, write a summary of Lesson 1 that would
be suitable as a review for an examination.
50 
Mathematics B30
49
Lesson 1
Assignment 1
Checklist:
Have you attached the requested forms to
the back of your assignment?

Mathematics B30
Summary for Lesson 1
50
Lesson 1