Download Complex Numbers

9.5
9.5
In this
section
Complex Numbers
(9-25)
477
COMPLEX NUMBERS
In this chapter we have seen quadratic equations that have no solution in the set of
real numbers. In this section you will learn that the set of real numbers is contained
in the set of complex numbers. Quadratic equations that have no real number
solutions have solutions that are complex numbers.
●
Definition
●
Operations with Complex
Numbers
Definition
●
Square Roots of Negative
Numbers
If we try to solve the equation x 2 1 0 by the square root property, we get
●
Complex Solutions to
Quadratic Equations
x 2 1
or
x 1.
1 has no meaning in the real number system, the equation has no real
Because solution. However, there is an extension of the real number system, called the
complex numbers, in which x 2 1 has two solutions.
The complex numbers are formed by adding the imaginary unit i to the real
number system. We make the definitions that
1
i i 2 1.
and
In the complex number system, x 2 1 has two solutions: i and i.
The set of complex numbers is defined as follows.
helpful
hint
All numbers are ideas. They
exist only in our minds. So in a
sense we “imagine” all numbers. However, the imaginary
numbers are a bit harder to
imagine than the real numbers and so only they are
called imaginary.
Complex Numbers
The set of complex numbers is the set of all numbers of the form
a bi,
1, and i 2 1.
where a and b are real numbers, i In the complex number a bi, a is called the real part and b is called the imaginary part. If b 0, the number a bi is called an imaginary number.
In dealing with complex numbers, we treat a bi as if it were a binomial with
variable i. Thus we would write 2 (3)i as 2 3i. We agree that 2 i3, 3i 2,
and i3 2 are just different ways of writing 2 3i. Some examples of complex
numbers are
2 3i, 2 5i,
0 4i,
9 0i,
and
0 0i.
For simplicity we will write only 4i for 0 4i. The complex number 9 0i is the
real number 9, and 0 0i is the real number 0. Any complex number with b 0,
is a real number. The diagram in Fig. 9.3 shows the relationships between the
complex numbers, the real numbers, and the imaginary numbers.
Complex numbers
Real numbers
–3, π,
5
—
2 ,
––
0, – 9, √ 2
Imaginary numbers
—–
i, 2 + 3i, √–5 , – 3 – 8i
FIGURE 9.3
478
(9-26)
Chapter 9
Quadratic Equations and Quadratic Functions
Operations with Complex Numbers
We define addition and subtraction of complex numbers as follows.
Addition and Subtraction of Complex Numbers
(a bi) (c di) (a c) (b d)i
(a bi) (c di) (a c) (b d)i
According to the definition, we perform these operations as if the complex numbers
were binomials with i a variable.
E X A M P L E
study
1
tip
Stay calm and confident. Take
breaks when you study. Get six
to eight hours of sleep every
night and keep reminding
yourself that working hard
all of the semester will really
pay off.
Adding and subtracting complex numbers
Perform the indicated operations.
a) (2 3i) (4 5i)
b) (2 3i) (1 i)
c) (3 4i) (1 7i)
d) (2 3i) (2 5i)
Solution
a) (2 3i) (4 5i) 6 8i
b) (2 3i) (1 i) 1 4i
c) (3 4i) (1 7i) 3 4i 1 7i
2 3i
d) (2 3i) (2 5i) 2 3i 2 5i
4 2i
■
We define multiplication of complex numbers as follows.
Multiplication of Complex Numbers
(a bi)(c di) (ac bd ) (ad bc)i
There is no need to memorize the definition of multiplication of complex numbers. We multiply complex numbers in the same way we multiply polynomials: We
use the distributive property. We can also use the FOIL method and the fact that
i2 1.
E X A M P L E
2
Multiplying complex numbers
Perform the indicated operations.
a) 2i(1 3i)
c) (5i)2
e) (3 2i)(3 2i)
Solution
a) 2i(1 3i) 2i 6i 2
2i 6(1)
6 2i
b) (2 5i)(6 7i)
d) (5i)2
Distributive property
i 2 1
9.5
Complex Numbers
(9-27)
479
b) Use FOIL to multiply these complex numbers:
calculator
(2 5i)(6 7i) 12 14i 30i 35i 2
12 16i 35(1)
close-up
Many calculators can perform operations with complex numbers.
i 2 1
12 16i 35
47 16i
c) (5i)2 25i 2 25(1) 25
d) (5i)2 (5)2i 2 25(1) 25
e) (3 2i)(3 2i) 9 4i 2
9 4(1)
94
13
■
Notice that the product of the imaginary numbers 3 2i and 3 2i in Example 2(e) is a real number. We call 3 2i and 3 2i complex conjugates of each
other.
Complex Conjugates
The complex numbers a bi and a bi are called complex conjugates of
each other. Their product is the real number a2 b2.
E X A M P L E
helpful
3
hint
The last time we used “conjugate” was to refer to expressions such as 1 3 and
1 3
as conjugates. Note
that we use the rule for the
product of a sum and a difference to multiply the radical
conjugates or the complex
conjugates.
Complex conjugates
Find the product of the given complex number and its conjugate.
a) 4 3i
b) 2 5i
c) i
Solution
a) The complex conjugate of 4 3i is 4 3i:
(4 3i)(4 3i) 16 9i 2 16 9 25
b) The conjugate of 2 5i is 2 5i :
(2 5i)(2 5i ) 4 25i 2 4 25 29
c) The conjugate of i is i:
(i)(i) i 2 (1) 1
■
To divide a complex number by a real number, we divide each part by the real
number. For example,
4 6i
2 3i.
2
We use the idea of complex conjugates to divide by a complex number. The process
is similar to rationalizing the denominator.
480
(9-28)
Chapter 9
Quadratic Equations and Quadratic Functions
Dividing by a Complex Number
To divide by a complex number, multiply the numerator and denominator of
the quotient by the complex conjugate of the denominator.
E X A M P L E
4
Dividing complex numbers
Perform the indicated operations.
2
6
a) b) 3 4i
2i
3 2i
c) i
Solution
a) Multiply the numerator and denominator by 3 4i, the conjugate of 3 4i:
2(3 4i)
2
(3
4i)(3 4i)
3 4i
6 8i
2
9 16i
6 8i
25
6
25
9 16i 2 9 16(1) 25
8
25
i
b) Multiply the numerator and denominator by 2 i, the conjugate of 2 i :
6(2 i)
6
2 i (2 i)(2 i)
12 6i
4 i2
12 6i
5
12 6
i
5
5
4 i 2 4 (1) 5
c) Multiply the numerator and denominator by i, the conjugate of i:
3 2i
3i 2i2 3i 2
(3 2i)(i)
2 3i
i(i)
i
i2
1
■
Square Roots of Negative Numbers
In Example 2 we saw that both
(5i)2 25
and
(5i)2 25.
Because the square of each of these complex numbers is 25, both 5i and 5i are
square roots of 25. When we use the radical notation, we write
25 5i.
The square root of a negative number is not a real number, it is a complex
number.
9.5
Complex Numbers
(9-29)
481
Square Root of a Negative Number
For any positive number b,
b ib.
For example, 9 i9
3i and 7 i7. Note that the expression 7
i
i, where i is under the radical. For
could easily be mistaken for the expression 7
this reason, when the coefficient of i contains a radical, we write i preceding the
radical.
E X A M P L E
5
Square roots of negative numbers
Write each expression in the form a bi, where a and b are real numbers.
12
2 2 18
a) 2 4
b) c) 2
3
Solution
a) 2 4 2 i4
2 2i
12 2 i1
2 2
b) 2
2
2 2i3
2
1 i3
12 4 3 23
8
18 2 i1
2 c) 3
3
2 3i2
3
2
i2
3
18 92 32
■
Complex Solutions to Quadratic Equations
The equation x 2 4 has no real number solutions, but it has two complex solutions, which can be found as follows:
helpful
x 2 4
4 i4 2i
x hint
The fundamental theorem of
algebra says that an nth degree polynomial equation has
exactly n solutions in the
system of complex numbers.
They don’t all have to be
different. For example, a
quadratic equation such as
x 2 6x 9 0 has two solutions, both of which are 3.
Check:
(2i)2 4i 2 4(1) 4
(2i)2 4i 2 4
Both 2i and 2i are solutions to the equation.
Consider the general quadratic equation
ax 2 bx c 0,
where a, b, and c are real numbers. If the discriminant b2 4ac is positive, then the
quadratic equation has two real solutions. If the discriminant is 0, then the equation
482
(9-30)
Chapter 9
Quadratic Equations and Quadratic Functions
has one real solution. If the discriminant is negative, then the equation has two complex solutions. In the complex number system, all quadratic equations have solutions. We can use the quadratic formula to find them.
E X A M P L E
6
calculator
close-up
The answers in Example 6(b)
can be checked with a calculator as shown here.
Quadratics with imaginary solutions
Find the complex solutions to the quadratic equations.
b) 2x 2 3x 5 0
a) x 2 2x 5 0
Solution
a) To solve x 2 2x 5 0, use a 1, b 2, and c 5 in the quadratic
formula:
2
)2
4(1
)(
5)
2 (
x 2(1)
16 2 4i
2 1 2i
2
2
We can use the operations with complex numbers to check these solutions:
(1 2i)2 2(1 2i) 5 1 4i 4i 2 2 4i 5
1 4i 4 2 4i 5 0
You should verify that 1 2i also satisfies the equation. The solutions are
1 2i and 1 2i.
b) To solve 2x 2 3x 5 0, use a 2, b 3, and c 5 in the quadratic
formula:
3 32
4
(2
)(5)
x 2(2)
31 3 i3
1
3 4
4
Check these answers. The solutions are
3 i31
4
3 i31
4
and .
■
The following box summarizes the basic facts about complex numbers.
Complex Numbers
i 1 and i 2 1.
A complex number has the form a bi, where a and b are real numbers.
The complex number a 0i is the real number a.
b ib
.
If b is a positive real number, then The complex conjugate of a bi is a bi.
Add, subtract, and multiply complex numbers as if they were binomials
with variable i.
7. Divide complex numbers by multiplying the numerator and denominator by
the conjugate of the denominator.
8. In the complex number system, all quadratic equations have solutions.
1.
2.
3.
4.
5.
6.
9.5
WARM-UPS
(9-31)
483
True or false? Explain your answer.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
9.5
Complex Numbers
(3 i) (2 4i) 5 3i True
(4 2i)(3 5i) 2 26i True
(4 i)(4 i) 17 True
i 4 1 True
5 5i False
36 6i False
The complex conjugate of 2 3i is 2 3i. False
Zero is the only real number that is also a complex number. False
Both 2i and 2i are solutions to the equation x 2 4. False
Every quadratic equation has at least one complex solution. True
EXERCISES
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. What is a complex number?
A complex number is a number of the form a bi where a
and b are real and i 1.
2. What is the relationship between the complex numbers, the
real numbers, and the imaginary numbers.
The real numbers together with the imaginary numbers
make up the set of complex numbers.
3. How do you add or subtract complex numbers?
Complex numbers are added or subtracted like binomials in
which i is the variable.
4. What is a complex conjugate?
The complex conjugate of a bi is a bi.
5. What is the square root of a negative number in the complex number system?
If b 0 then b ib.
6. How many solutions do quadratic equations have in the
complex number system?
In the complex number system all quadratic equations have
at least one solution.
Perform the indicated operations. See Example 1.
7. (3 5i) (2 4i)
8. (8 3i) (1 2i)
5 9i
9 5i
9. (1 i) (2 i)
10. (2 i) (3 5i)
1
5 4i
11. (4 5i) (2 3i)
12. (3 2i) (7 6i)
2 8i
4 8i
13. (3 5i) (2 i)
14. (4 8i) (2 3i)
1 4i
2 5i
15. (8 3i) (9 3i)
16. (5 6i) (3 6i)
1
8
1
1 1
17. i i
2
4 2
3 1
i
4 2
2
1 1
18. i i
3
4 2
1
5
i
12 2
Perform the indicated operations. See Example 2.
19. 3(2 3i)
20. 4(3 2i)
6 9i
12 8i
21. (6i)2
22. (3i)2
36
9
23. (6i)2
24. (3i)2
36
9
25. (2 3i)(3 5i)
26. (4 i)(3 6i)
21 i
6 27i
27. (5 2i)2
28. (3 4i)2
21 20i
7 24i
29. (4 3i)(4 3i)
30. (3 5i)(3 5i)
25
34
31. (1 i)(1 i)
32. (3 i)(3 i)
2
10
Find the product of the given complex number and its conjugate. See Example 3.
33. 2 5i 29
34. 3 4i 25
35. 4 6i 52
36. 2 7i 53
37. 3 2i 13
38. 4 i 17
39. i 1
40. 2i 4
Perform the indicated operations. See Example 4.
41. (2 6i) 2 1 3i
42. (3 6i) (3) 1 2i
2 8i
43. 1 4i
2
484
(9-32)
Chapter 9
Quadratic Equations and Quadratic Functions
6 9i
44. 2 3i
3
4 6i
45. 3 2i
2i
3 8i
46. 8 3i
i
4i
8
12
47. i
3 2i 13 13
20 25
5
48. i
4 5i 41 41
2i 3 4
49. i
2i 5 5
i5
50. 1
5i
4 12i
51. 4i
3i
71. x 2 4x 5 0
2 i, 2 i
73. y 2 13 6y
3 2i, 3 2i
75. x 2 4x 7 0
2 i3, 2 i3
77. 9y 2 12y 5 0
2i 2i
, 3
3
79. x 2 x 1 0
1 i3
1 i3
, 2
2
81. 4x 2 8x 9 0
2 i5 2 i5
, 2
2
4 10i
15 23
52. i
5i
13 13
Write each expression in the form a bi, where a and b are real
numbers. See Example 5.
53. 5 9
54. 6 16
5 3i
6 4i
55. 3 7
3 i7
56. 2 3
2 i3
2 12
57. 2
1 i3
6 18
58. 3
2 i2
8 20
59. 4
1
2 i5
2
4 28
61. 6
2 1
i7
3 3
6 24
60. 2
2 100
63. 10
1
i
5
3 81
64. 9
1
i
3
3 i6
6 45
62. 6
1
1 i5
2
Find the complex solutions to each quadratic equation. See
Example 6.
65. x 2 81 0
66. x 2 100 0
9i, 9i
10i, 10i
2
67. x 5 0
68. x 2 6 0
i5, i5
i6
, i6
69. 3y 2 2 0
6 6
i, i
3
3
70. 5y 2 3 0
15 15
i, i
5
5
72. x 2 6x 10 0
3 i, 3 i
74. y 2 29 4y
2 5i, 2 5i
76. x 2 10x 27 0
5 i2, 5 i2
78. 2y 2 2y 1 0
1 1 1 1
i, i
2 2 2 2
80. 4x 2 20x 27 0
5 1
5 1
, i2
i2
2 2
2 2
82. 9x 2 12x 10 0
2 1
2 1
, i6
i6
3 3
3 3
Solve each problem.
83. Evaluate (2 3i)2 4(2 3i) 9.
6 24i
84. Evaluate (3 5i)2 2(3 5i) 5.
17 20i
85. What is the value of x 2 8x 17 if x 4 i?
0
86. What is the value of x 2 6x 34 if x 3 5i?
0
87. Find the product [x (6 i)][x (6 i)].
x 2 12x 37
88. Find the product [x (3 7i)][x (3 7i)].
x 2 6x 58
89. Write a quadratic equation that has 3i and 3i as its
solutions.
x2 9 0
90. Write a quadratic equation that has 2 5i and 2 5i as its
solutions.
x 2 4x 29 0
GET TING MORE INVOLVED
91. Discussion. Determine whether each given number is in
each of the following sets: the natural numbers, the integers,
the rational numbers, the irrational numbers, the real numbers, the imaginary numbers, and the complex numbers.
3
a) 54
b) c) 35
d) 6i e) i5
8
Natural: 54, integers: 54, rational: 54, 38, irrational:
35
, real: 54, 38, 35
, imaginary: 6i, complex: all
92. Discussion. Which of the following equations have real
solutions? Imaginary solutions? Complex solutions?
a) 3x 2 2x 9 0
b) 5x 2 2x 10 0
1
d) 7w 2 12 0
c) x 2 x 3 0
2
Real: b, imaginary: d, complex: all