Download 2.1 The Complex Number System

2.1 The Complex Number System
The approximate speed of a car prior to an
accident can be found using the length of the
tire marks left by the car after the brakes have
been applied. The formula s = 121d
gives the
speed, s, in kilometres per hour, where d is the
length of the tire marks, in metres. Radical
expressions like 121d
can be simplified.
I NVESTIGATE & I NQUIRE
Copy the table. Complete it by replacing
each ■ with a whole number.
4 × 9 = ■
=■
4
× 9
=■×■=■
9 × 16 = ■
=■
9
× 16
=■×■=■
25 × 4 = ■
=■
25
× 4
=■×■=■
36
■
= = ■
■
4
100
■
= = ■
■
25
144
= ■ = ■
■
9
= ■
=■
4
100
= ■
=■
25
144
= ■
=■
9
36
Compare the two results in each of the
first three rows of the table.
1.
2.
If a and b are whole numbers, describe how ab
is related to a × b.
Use a calculator to test your statement from question 2
for each of the following.
a) 5
× 9
b) 3
× 7
4. Compare the two results in each of the last three rows of the table.
a
is related
5. If a and b are whole numbers, and b ≠ 0, describe how
b
a
to .
b
3. Technology
100 MHR • Chapter 2
Use a calculator to test your statement from question 5
for each of the following.
a) 36
÷ 2
b) 15
÷ 3
6. Technology
7. Write the expression 121d
in the form ■ d, where ■ represents a
whole number.
8. Determine the speed of a car that leaves tire marks of each of the
following lengths. Round the speed to the nearest kilometre per
hour, if necessary.
a) 64 m
b) 100 m
c) 15 m
The following properties are used to simplify radicals.
• ab
= a × b, a ≥ 0, b ≥ 0
•
a a
= , a ≥ 0, b > 0
b b
A radical is in simplest form when
• the radicand has no perfect square factors other than 1
8 = 22
• the radicand does not contain a fraction
4 = 2
• no radical appears in the denominator of a fraction
1
3
=
3
3
1
1
Recall that the radicand is the
expression under the radical sign.
EXAMPLE 1 Simplifying Radicals
Simplify.
a)
75
b)
48
6
c)
2
9
SOLUTION
a) 75
b)
= 25
× 3
= 53
48
48
= 6
6
= 8
= 4
× 2
= 22
2.1 The Complex Number System • MHR 101
c)
2
9 = 9
2
1
2
= or 2
3
3
Note that, in Example 1, numbers like 75
and
9 are called entire radicals.
2
1
Numbers like 53
and 2 are called mixed radicals.
3
EXAMPLE 2 Multiplying Radicals
Simplify.
a) 92
× 47
b)
23 × 56
SOLUTION
92
× 47 = 9 × 4 × 2 × 7
= 3614
b) 23
× 56 = 2 × 5 × 3 × 6
= 1018
= 10 × 9 × 2
= 10 × 3 × 2
= 302
a)
EXAMPLE 3 Simplifying Radical Expressions
6 − 45
Simplify .
3
SOLUTION
6 − 45
6 − 9
× 5
= 3
3
6 − 35
=
3
= 2 − 5
102 MHR • Chapter 2
In mathematics, we can find the square roots of negative numbers as well as
positive numbers. Mathematicians have invented a number defined as the
principal square root of negative one. This number, i, is the imaginary unit,
with the following properties.
i = −1
and i2 = −1
In general, if x is a positive real number, then −x
is a pure imaginary
number, which can be defined as follows.
= −1
× x
−x
= ix
So, −5
= −1
× 5
= i5
−
−
−
−
To ensure that √xi is not read as √xi , we write √xi as i √x.
Despite their name, pure imaginary numbers are just as real as real numbers.
When the radicand is a negative number, there is an extra rule for expressing
a radical in simplest form.
• A radical is in simplest form when the radicand is positive.
Numbers such as i, i6, 2i, and −3i are examples of pure imaginary
numbers in simplest form.
EXAMPLE 4 Simplifying Pure Imaginary Numbers
Simplify.
a)
−25
b)
−12
SOLUTION
−25
= −1
× 25
=i×5
= 5i
b) −12
= −1
× 12
= i × 4 × 3
= i × 2 × 3
= 2i3
a)
2.1 The Complex Number System • MHR 103
When two pure imaginary numbers are multiplied, the result is a real
number.
EXAMPLE 5 Multiplying Pure Imaginary Numbers
Evaluate.
3i × 4i
a)
b)
2i × (−5i)
c)
(3i2 )
2
SOLUTION
3i × 4i = 3 × 4 × i2
= 12 × (−1)
= −12
2
b) 2i × (−5i) = 2 × (−5) × i
= −10 × (−1)
= 10
2
c) (3i2
) = 3i2 × 3i2
= 32 × i2 × (2 )2
= 9 × (−1) × 2
= −18
a)
A complex number is a number in the form a + bi, where a and b are
real numbers and i is the imaginary unit. We call a the real part and
bi the imaginary part of a complex number. Examples of complex
numbers include 5 + 2i and 4 − 3i. Complex numbers are used for
applications of mathematics in engineering, physics, electronics, and
many other areas of science.
If b = 0, then a + bi = a. So, a real number, such as 5, can be thought
of as a complex number, since 5 can be written as 5 + 0i.
If a = 0, then a + bi = bi. Numbers of the form bi, such as 7i, are pure
imaginary numbers. Complex numbers in which neither a = 0 nor b = 0
are referred to as imaginary numbers.
104 MHR • Chapter 2
Complex Numbers
a
+
bi
↑
↑
real
imaginary
part
part
The following diagram summarizes the complex number system.
Complex Numbers
a + bi, where a and b are real numbers and i = −1
.
b=0
a, b ≠ 0
Real Numbers
–2
e.g., 5, 2, − 7, 3.6, 3
Imaginary Numbers
(e.g., 4 + 3i, 3 − 2i)
Rational Numbers
Irrational Numbers
Can be expressed as
Cannot be expressed as
the ratio of two integers. the ratio of two integers.
–– 6
e.g., 3, − 4, 0.27, − (e.g., 7, − 2, π)
7
a=0
Pure Imaginary Numbers
(e.g., 7i, −3i, i2)
Integers
Whole numbers and their opposites
(e.g., 4, −4, 0, 9, −9)
Whole Numbers
Positive integers and zero
(e.g., 0, 3, 7, 11)
Natural Numbers
Positive integers
(e.g., 1, 5, 8, 23)
EXAMPLE 6 Simplifying Complex Numbers
Simplify.
a)
3 − −24
b)
10 + –32
2
SOLUTION
a)
3 − −24
= 3 − −1
× 24
= 3 − i × 4 × 6
= 3 − i × 26
= 3 − 2i6
−
The expression 3 − 2i√6 cannot be simplified further, because the
real part and the imaginary part are unlike terms.
2.1 The Complex Number System • MHR 105
b)
10 + –32
10 + –1
× 32
= 2
2
× 2
10 + i × 16
= 2
10 + i × 42
= 2
10 + 4i2
= 2
= 5 + 2i2
Key
Concepts
• Radicals are simplified using the following properties.
= a × b, a ≥ 0, b ≥ 0
ab
a a
= , a ≥ 0, b > 0
b b
• The number i is the imaginary unit, where i2 = −1 and i = −1
.
• A complex number is a number in the form a + bi, where a and b are real
numbers and i is the imaginary unit.
Communicate
Yo u r
Understanding
Describe the difference between an entire radical and a mixed radical.
Describe how you would simplify
10 + 20
14
a) 60
b) c) 2
2
3. Describe how you would simplify
9 + –54
a) −28
b) 3
2
4. Describe how you would evaluate (−3i) .
1.
2.
Practise
A
1. Simplify.
a) 12
d) 50
200
j) 60
m) 128
g)
20
e) 24
b)
106 MHR • Chapter 2
45
f) 63
c)
32
k) 18
n) 90
h)
44
l) 54
o) 125
i)
Simplify.
14
10
a) b) 7
2
33
7
e) f) 3
4
2715
1275
i) j) 35
43
2.
Simplify.
a) 2
× 10
c) 15
× 5
e) 43
× 7
g) 22
× 36
i) 33
× 415
k) 6
× 3
× 2
60
3
20
g) 9
42
k) 8
c)
40
5
38
h) 2
22
l) 18
d)
3.
3 × 6
d) 7
× 11
f) 36
× 36
h) 25
× 310
j) 47
× 214
l) 27
× 31 × 7
b)
Simplify.
10 + 155
21 − 76
6 + 8
a) b) c) 5
7
2
12 − 27
–10 − 50
–12 + 48
d) e) f) 3
5
4
4.
Simplify.
a) −9
b) −25
e) −13
f) −23
i) −54
j) −−4
5.
−81
d) −5
g) −12
h) −40
k) −−20
l) −−60
c)
6. Evaluate.
a) 5i × 5i
c) (−2i) × (−2i)
e) 2i × (−5i)
b) 2i × 3i
d) (−3i) × (−4i)
f) (−3i) × 6i
7. Simplify.
3
a) i
5
c) i
2
e) 5i
2
g) 3(−2i)
b) i
d) 4i × 5i
7
f) −i
3
h) i(4i)
4
(i2
)2
l) (i6
)(−i6
)
(3i)(−6i)
k) −(i5
)2
j)
(2i3 )
o) (4i5
)(−2i5
)
n)
i)
m)
2
Simplify.
a) 4 + −20
c) 10 + −75
e) −2 − −90
(−5i2 )
2
8.
7 − −18
d) 11 − −63
f) −6 − −52
b)
Simplify.
15 + 20i5
a) 5
10 − –16
c) 2
–8 + –32
e) 4
9.
14 − 28i6
14
12 + –27
d) 3
–21 − –98
f) 7
b)
Apply, Solve, Communicate
Express each of the following as an integer.
a) 5
2
b) (5
)2
c) (−5)
2
d) −5
2
10.
e)
−(5 )2
f)
−(−5)
2
B
11. Communication Classify each of the following numbers into one or
more of these sets: real, rational, irrational, complex, imaginary, pure
imaginary. Explain your reasoning.
a) 5
b) −4
c) 1 + 3
d) 3 + i6
2.1 The Complex Number System • MHR 107
Express the exact area of the triangle in
simplest radical form.
12. Measurement
10
2
6
2
A square has an area of 675 cm . Express the side length
in simplest radical form.
13. Measurement
There are many variations on the game of chess.
Most are played on square boards that consist of a number of small
squares. However, some variations do not use the familiar 64-square
board.
a) If each small square on a Grand Chess board is 2 cm by 2 cm, each
diagonal of the whole board measures 800
cm. How many small
squares are on the board?
We b C o n n e c t i o n
b) A Japanese variation of chess is called Chu
www.school.mcgrawhill.ca/resources/
Shogi. If each small square on a Chu Shogi
To learn more about variations on the game
board measures 3 cm by 3 cm, each diagonal
of chess, visit the above web site. Go to
of the whole board measures 2592
cm. How
Math Resources, then to MATHEMATICS
many small squares are on the board?
11, to find out where to go next. Summarize
14. Application
15. Pattern a)
Simplify i2, i3, i 4, i5, i6, i7, i8, i9,
how a variation that interests you is played.
i10, i11, and i12.
b) Describe the pattern in the values.
n
c) Describe how to simplify i , where n is a whole number.
48 94 85
99
d) Simplify i , i , i , and i .
16. Measurements of lengths and areas
a) Determine the length of the diagonal of
each of the following Ontario
flags. Write each answer in simplest radical form.
3
2
1
2
4
6
Describe the relationship between the length of the diagonal and either
dimension of the flag.
c) Use the relationship from part b) to predict the length of the diagonal of a
150 cm by 75 cm Ontario flag. Leave your answer in simplest radical form.
b)
108 MHR • Chapter 2
Describe the relationship between the length of the diagonal and the area
of the Ontario flag.
e) Use the relationship from part d) to predict the length of the diagonal of
an Ontario flag with an area of 24 200 cm2. Leave your answer in simplest
radical form.
d)
C
17. Communication Check if x = −i3
is a solution to the equation
x2 + 3 = 0. Justify your reasoning.
Simplify each of the following by first expressing it as the product of
two cube roots.
3
3
3
3
a) 16
b) 32
c) 54
d) 81
18.
19. Equations
a)
x2 = 14
Solve. Express each answer in simplest radical form.
x
30
b) 5x = 50
c) = 6
d) = 5
x
3
For the property ab
= a × b, explain the restrictions a ≥ 0, b ≥ 0.
a a
= , the restrictions are a ≥ 0, b > 0.
b) For the property
b b
Why is the second restriction not b ≥ 0?
20. a)
PATTERN
Power
Subtracting 9 from the two-digit positive integer 21 results in the reversal of the
digits to give 12.
1. List all two-digit positive integers for which the digits are reversed when you
subtract 9.
2. Find all the two-digit positive integers for which the digits are reversed when
you subtract
a) 18 b) 27 c) 36
3. Describe the pattern in words.
4. Use the pattern to find all the two-digit positive integers for which the digits
are reversed when you subtract
a) 54 b) 72
2.1 The Complex Number System • MHR 109