9781285057095_AppE.qxp
2/18/13
8:25 AM
Page E1
Appendix E
E
Complex Numbers
E1
Complex Numbers
Use the imaginary unit i to write complex numbers, and add, subtract, and multiply
complex numbers.
Find complex solutions of quadratic equations.
Write the trigonometric forms of complex numbers.
Find powers and nth roots of complex numbers.
Operations with Complex Numbers
Some equations have no real solutions. For instance, the quadratic equation
x2 1 0
Equation with no real solution
has no real solution because there is no real number x that can be squared to produce
1. To overcome this deficiency, mathematicians created an expanded system of
numbers using the imaginary unit i, defined as
i 冪1
Imaginary unit
where i 1. By adding real numbers to real multiples of this imaginary unit,
you obtain the set of complex numbers. Each complex number can be written in
the standard form a bi. The real number a is called the real part of the complex
number a bi, and the number bi (where b is a real number) is called the imaginary
part of the complex number.
2
Definition of a Complex Number
For real numbers a and b, the number
a bi
is a complex number. If b 0, then a bi is called an imaginary number.
A number of the form bi, where b 0, is called a pure imaginary number.
To add (or subtract) two complex numbers, you add (or subtract) the real and
imaginary parts of the numbers separately.
Addition and Subtraction of Complex Numbers
If a bi and c di are two complex numbers written in standard form,
then their sum and difference are defined as follows.
Sum: 共a bi兲 共c di兲 共a c兲 共b d兲i
Difference: 共a bi兲 共c di兲 共a c兲 共b d兲i
9781285057095_AppE.qxp
E2
2/18/13
Appendix E
8:25 AM
Page E2
Complex Numbers
The additive identity in the complex number system is zero (the same as in the
real number system). Furthermore, the additive inverse of the complex number a bi is
共a bi兲 a bi.
Additive inverse
So, you have
共a bi兲 共a bi兲 0 0i 0.
Adding and Subtracting Complex Numbers
a. 共3 i兲 共2 3i兲 3 i 2 3i
3 2 i 3i
共3 2兲 共1 3兲i
5 2i
b. 2i 共4 2i兲 2i 4 2i
4 2i 2i
4
c. 3 共2 3i兲 共5 i兲 3 2 3i 5 i
3 2 5 3i i
0 2i
2i
Remove parentheses.
Group like terms.
Write in standard form.
Remove parentheses.
Group like terms.
Write in standard form.
In Example 1(b), notice that the sum of two complex numbers can be a real
number.
Many of the properties of real numbers are valid for complex numbers as well.
Here are some examples.
Associative Properties of Addition and Multiplication
Commutative Properties of Addition and Multiplication
Distributive Property of Multiplication over Addition
Notice how these properties are used when two complex numbers are multiplied.
REMARK Rather than trying
to memorize the multiplication
rule at the right, you can simply
remember how the Distributive
Property is used to multiply
two complex numbers. The
procedure is similar to
multiplying two polynomials
and combining like terms.
共a bi兲共c di兲 a共c di兲 bi共c di兲
ac 共ad 兲i 共bc兲i 共bd 兲i 2
ac 共ad 兲i 共bc兲i 共bd 兲共1兲
ac bd 共ad 兲i 共bc兲i
共ac bd 兲 共ad bc兲i
Distributive Property
Distributive Property
i 2 1
Commutative Property
Associative Property
The procedure above is similar to multiplying two polynomials and combining like
terms, as in the FOIL method.
9781285057095_AppE.qxp
2/18/13
8:25 AM
Page E3
Appendix E
Complex Numbers
E3
Multiplying Complex Numbers
a. 共3 2i兲共3 2i兲 3共3 2i兲 2i共3 2i兲
9 6i 6i 4i 2
9 6i 6i 4共1兲
94
13
b. 共3 2i兲2 共3 2i兲共3 2i兲
3共3 2i兲 2i共3 2i兲
9 6i 6i 4i 2
9 6i 6i 4共1兲
9 12i 4
5 12i
Distributive Property
Distributive Property
i 2 1
Simplify.
Write in standard form.
Square of a binomial
Distributive Property
Distributive Property
i 2 1
Simplify.
Write in standard form.
In Example 2(a), notice that the product of two complex numbers can be a real
number. This occurs with pairs of complex numbers of the form a bi and a bi,
called complex conjugates.
共a bi兲共a bi兲 a2 abi abi b2i 2
a2 b2共1兲
a2 b2
To write the quotient of a bi and c di in standard form, where c and d are not
both zero, multiply the numerator and denominator by the complex conjugate of the
denominator to obtain
a bi a bi c di
c di
c di c di
共ac bd 兲 共bc ad 兲i .
c2 d 2
冢
冣
Multiply numerator and denominator
by complex conjugate of denominator.
Write in standard form.
Writing Complex Numbers in Standard Form
2 3i 2 3i 4 2i
4 2i 4 2i 4 2i
冢
冣
8 4i 12i 6i 2
16 4i 2
8 6 16i
16 4
2 16i
20
4
1
i
10 5
Multiply numerator and denominator
by complex conjugate of denominator.
Expand.
i 2 1
Simplify.
Write in standard form.
9781285057095_AppE.qxp
E4
2/18/13
Appendix E
8:25 AM
Page E4
Complex Numbers
Complex Solutions of Quadratic Equations
When using the Quadratic Formula to solve a quadratic equation, you often obtain a
result such as 冪3, which you know is not a real number. By factoring out i 冪1,
you can write this number in standard form.
冪3 冪3共1兲 冪3冪1 冪3i
The number 冪3i is called the principal square root of 3.
REMARK The definition of
principal square root uses the
rule
冪ab 冪a冪b
for a > 0 and b < 0. This rule
is not valid when both a and b
are negative.
For example,
冪5冪5 冪5共1兲冪5共1兲
冪5i冪5i
冪25i 2
5i 2
5
whereas
冪共5兲共5兲 冪25 5.
To avoid problems with
multiplying square roots of
negative numbers, be sure to
convert to standard form
before multiplying.
Principal Square Root of a Negative Number
If a is a positive number, then the principal square root of the negative number
a is defined as
冪a 冪ai.
Writing Complex Numbers in Standard Form
a. 冪3冪12 冪3i冪12i
冪36i 2
6共1兲
6
b. 冪48 冪27 冪48i 冪27i
4冪3i 3冪3i
冪3i
2
2
c. 共1 冪3 兲 共1 冪3i 兲
2
共1兲2 2冪3i 共冪3 兲 共i 2兲
1 2冪3i 3共1兲
2 2冪3i
Complex Solutions of a Quadratic Equation
Solve 3x2 2x 5 0.
Solution
共2兲 ± 冪共2兲2 4共3兲共5兲
2共3兲
2 ± 冪56
6
2 ± 2冪14i
6
1 冪14
±
i
3
3
x
Quadratic Formula
Simplify.
Write 冪56 in standard form.
Write in standard form.
9781285057095_AppE.qxp
2/18/13
8:25 AM
Page E5
Appendix E
Just as real numbers can be represented by points on the real number line, you can
represent a complex number
(3, 2)
or
3 + 2i
3
(− 1, 3)
or
2
− 1 + 3i
z a bi
1
−1
E5
Trigonometric Form of a Complex Number
Imaginary
axis
−2
Complex Numbers
1
2
3
Real
axis
(− 2, − 1)
or
−2 − i
as the point 共a, b兲 in a coordinate plane (the complex plane). The horizontal axis is
called the real axis and the vertical axis is called the imaginary axis, as shown in
Figure E.1.
The absolute value of a complex number a bi is defined as the distance between
the origin 共0, 0兲 and the point 共a, b兲.
The Absolute Value of a Complex Number
The absolute value of the complex number z a bi is given by
Figure E.1
ⱍa biⱍ 冪a2 b2.
When the complex number a bi is a real number 共that is, b 0兲, this definition
agrees with that given for the absolute value of a real number.
ⱍa 0iⱍ 冪a2 02 ⱍaⱍ
To work effectively with powers and roots of complex numbers, it is helpful to
write complex numbers in trigonometric form. In Figure E.2, consider the nonzero
complex number a bi. By letting be the angle from the positive real axis (measured
counterclockwise) to the line segment connecting the origin and the point 共a, b兲, you
can write
Imaginary
axis
(a, b)
a r cos r
b
θ
a
and
b r sin where r 冪a2 b2. Consequently, you have
Real
axis
a bi 共r cos 兲 共r sin 兲i
from which you can obtain the trigonometric form of a complex number.
Figure E.2
Trigonometric Form of a Complex Number
The trigonometric form of the complex number z a bi is given by
z r共cos i sin 兲
where a r cos , b r sin , r 冪a2 b2, and tan b兾a. The number
r is the modulus of z, and is called an argument of z.
The trigonometric form of a complex number is also called the polar form.
Because there are infinitely many choices for , the trigonometric form of a complex
number is not unique. Normally, is restricted to the interval 0 < 2, although on
occasion it is convenient to use < 0.
9781285057095_AppE.qxp
E6
2/18/13
Appendix E
8:25 AM
Page E6
Complex Numbers
Trigonometric Form of a Complex Number
Write the complex number z 2 2冪3i in trigonometric form.
Solution
The absolute value of z is
r 2 2冪3i 冪共2兲2 共2冪3 兲2 冪16 4
ⱍ
ⱍ
and the angle is given by
Imaginary
axis
−3
−2
4π
3
1
−1
−2
⏐z⏐ = 4
−3
z = − 2 − 2 3i
Figure E.3
−4
b
a
2冪3
2
冪
3.
tan Real
axis
Because tan共兾3兲 冪3 and because z 2 2冪3i lies in Quadrant III, choose to be 兾3 4兾3. So, the trigonometric form is
z r 共cos i sin 兲
4
4
4 cos
i sin
.
3
3
冢
冣
See Figure E.3.
The trigonometric form adapts nicely to multiplication and division of complex
numbers. Consider the two complex numbers
z1 r1共cos 1 i sin 1 兲 and
z2 r2共cos 2 i sin 2 兲.
The product of z1 and z2 is
z1z2 r1r2共cos 1 i sin 1 兲共cos 2 i sin 2 兲
r1r2关共cos 1 cos 2 sin 1 sin 2 兲 i 共sin 1 cos 2 cos 1 sin 2 兲兴.
Using the sum and difference formulas for cosine and sine, you can rewrite this
equation as
z1z2 r1r2关cos共1 2 兲 i sin共1 2 兲兴.
This establishes the first part of the rule shown below. The second part is left for you to
verify (see Exercise 109).
Product and Quotient of Two Complex Numbers
Let z1 r1共cos 1 i sin 1 兲 and z2 r2共cos 2 i sin 2 兲 be complex
numbers.
z1z2 r1r2关cos共1 2 兲 i sin共1 2 兲兴
z1 r1
关cos共1 2 兲 i sin共1 2 兲兴, z2 0
z2 r2
Product
Quotient
9781285057095_AppE.qxp
2/18/13
8:25 AM
Page E7
Appendix E
Complex Numbers
E7
Note that this rule says that to multiply two complex numbers you multiply moduli
and add arguments, whereas to divide two complex numbers you divide moduli and
subtract arguments.
Multiplying Complex Numbers
Find the product z1z2 of the complex numbers.
冢
z1 2 cos
2
2
i sin
,
3
3
冣
冢
z2 8 cos
11
11
i sin
6
6
冣
Solution
2
2
11
11
i sin
8 cos
i sin
3
3
6
6
2 11
2 11
16 cos
i sin
3
6
3
6
5
5
16 cos
i sin
2
2
16 cos i sin
2
2
16关0 i共1兲兴
16i
冢
冣 冢
z1z2 2 cos
冤 冢
冤
冤
冣
冢
冣冥
冣
Multiply moduli and
add arguments.
冥
冥
5
and are coterminal.
2
2
Check this result by first converting to the standard forms z1 1 冪3i and
z2 4冪3 4i and then multiplying algebraically.
Dividing Complex Numbers
Find the quotient z1兾z2 of the complex numbers.
z1 24共cos 300 i sin 300兲,
z2 8共cos 75 i sin 75兲
Solution
z1 24共cos 300 i sin 300兲
z2
8共cos 75 i sin 75兲
24
关cos共300 75兲 i sin共300 75兲兴
8
3关cos 225 i sin 225兴
冤冢 冣 冢 冣冥
3
冪2
2
i 3冪2 3冪2
i
2
2
冪2
2
Divide moduli and
subtract arguments.
9781285057095_AppE.qxp
E8
2/18/13
Appendix E
8:25 AM
Page E8
Complex Numbers
Powers and Roots of Complex Numbers
To raise a complex number to a power, consider repeated use of the multiplication rule.
z r 共cos i sin 兲
z2 r 2 共cos 2 i sin 2兲
z3 r 3共cos 3 i sin 3兲
⯗
This pattern leads to the next theorem, which is named after the French mathematician
Abraham DeMoivre (1667–1754).
THEOREM E.1 DeMoivre’s Theorem
If z r共cos i sin 兲 is a complex number and n is a positive integer, then
zn 关r 共cos i sin 兲兴 n r n共cos n i sin n兲.
Finding Powers of a Complex Number
Use DeMoivre’s Theorem to find 共1 冪3i 兲 .
12
Solution
First convert the complex number to trigonometric form.
冢
1 冪3i 2 cos
2
2
i sin
3
3
冣
Then, by DeMoivre’s Theorem, you have
共1 冪3i兲12 冤 2冢cos 23 i sin 23 冣冥
12
2
i sin 12
3
4096 共cos 8 i sin 8兲
4096.
冤 冢
212 cos 12
冣
冢
2
3
冣冥
Recall that a consequence of the Fundamental Theorem of Algebra is that a
polynomial equation of degree n has n solutions in the complex number system. Each
solution is an nth root of the equation. The nth root of a complex number is defined
below.
Definition of nth Root of a Complex Number
The complex number u a bi is an nth root of the complex number z when
z un 共a bi兲n.
9781285057095_AppE.qxp
2/18/13
8:25 AM
Page E9
Appendix E
Complex Numbers
E9
To find a formula for an n th root of a complex number, let u be an n th root of z,
where
u s共cos i sin 兲 and
z r共cos i sin 兲.
By DeMoivre’s Theorem and the fact that un z, you have
sn共cos n i sin n兲 r共cos i sin 兲.
Taking the absolute value of each side of this equation, it follows that s n r. Substituting
r for s n in the previous equation and dividing by r, you get
cos n i sin n cos i sin .
So, it follows that
cos n cos and
sin n sin .
Because both sine and cosine have a period of 2, these last two equations have
solutions if and only if the angles differ by a multiple of 2. Consequently, there must
exist an integer k such that
n 2 k
2 k .
n
By substituting this value for into the trigonometric form of u, you get the result stated
in the next theorem.
THEOREM E.2 n th Roots of a Complex Number
For a positive integer n, the complex number z r 共cos i sin 兲 has exactly
n distinct n th roots given by
冢
n r cos
冪
2 k
2 k
i sin
n
n
冣
where k 0, 1, 2, . . . , n 1.
For k > n 1, the roots begin to repeat. For instance, when k n, the angle
2 n 2
n
n
is coterminal with 兾n, which is also obtained when k 0.
The formula for the n th roots of a complex
number z has a nice geometric interpretation,
as shown in Figure E.4. Note that because the
n
nth roots of z all have the same magnitude冪
r,
n
they all lie on a circle of radius冪r with center
at the origin. Furthermore, because successive
n
r
nth roots have arguments that differ by 2兾n,
the n roots are equally spaced along the circle.
Figure E.4
Imaginary
axis
2π
n
2π
n
Real
axis
9781285057095_AppE.qxp
E10
2/18/13
Appendix E
8:25 AM
Page E10
Complex Numbers
Finding the n th Roots of a Complex Number
Find the three cube roots of z 2 2i.
Solution
Because z lies in Quadrant II, the trigonometric form for z is
z 2 2i 冪8 共cos 135 i sin 135兲.
By the formula for n th roots, the cube roots have the form
冢
6 8 cos
冪
135 360k
135 360k
i sin
.
3
3
冣
Finally, for k 0, 1, and 2, you obtain the roots
冪2 共cos 45 i sin 45兲 1 i
冪2 共cos 165 i sin 165兲 ⬇ 1.3660 0.3660i
冪2 共cos 285 i sin 285兲 ⬇ 0.3660 1.3660i.
E Exercises
Performing Operations In Exercises 1–24, perform the
operation and write the result in standard form.
1. 共5 i兲 共6 2i兲
Writing a Complex Conjugate In Exercises 25–32, write
the complex conjugate of the complex number. Then multiply
the number by its complex conjugate.
25. 5 3i
26. 9 12i
3. 共8 i兲 共4 i兲
27. 2 冪5i
28. 4 冪2i
4. 共3 2i兲 共6 13i兲
29. 20i
30. 冪15
31. 冪8
32. 1 冪8
6. 共8 冪18 兲 共4 3冪2i兲
Writing in Standard Form In Exercises 33–42, write the
7. 13i 共14 7i兲
quotient in standard form.
2. 共13 2i兲 共5 6i兲
5. 共2 冪8 兲 共5 冪50 兲
8. 22 共5 8i兲 10i
10
2i
33.
6
i
34. 35.
4
4 5i
36.
3
1i
12. 冪5 冪10
37.
2i
2i
38.
8 7i
1 2i
14. 共冪75 兲2
39.
6 7i
i
40.
8 20i
2i
41.
1
共4 5i兲2
42.
共2 3i兲共5i兲
2 3i
3
5
5
11
9. 共2 2i兲 共3 3 i兲
10. 共1.6 3.2i兲 共5.8 4.3i兲
11. 冪6 冪2
13. 共冪10 兲2
15. 共1 i兲共3 2i兲
16. 共6 2i兲共2 3i兲
17. 6i共5 2i兲
Performing Operations In Exercises 43–46, perform the
18. 8i共9 4i兲
operation and write the result in standard form.
19. 共冪14 冪10i兲共冪14 冪10i兲
20. 共3 冪5 兲共7 冪10 兲
43.
2
3
1i 1i
44.
2i
5
2i 2i
21. 共4 5i兲2
45.
i
2i
3 2i 3 8i
46.
3
1i
i
4i
22. 共2 3i兲
2
23. 共2 3i兲2 共2 3i兲2
24. 共1 2i兲2 共1 2i兲2
9781285057095_AppE.qxp
2/18/13
8:25 AM
Page E11
Appendix E
Complex Numbers
E11
Using the Quadratic Formula In Exercises 47–54, use the
Performing Operations In Exercises 83–86, perform the
Quadratic Formula to solve the quadratic equation.
operation and leave the result in trigonometric form.
47. x2 2x 2 0
48.
x2
6x 10 0
49. 4x2 16x 17 0
50. 9x2 6x 37 0
51. 4x2 16x 15 0
52. 9x2 6x 35 0
53. 16t2 4t 3 0
83.
冤3冢cos 3 i sin 3 冣冥冤4冢cos 6 i sin 6 冣冥
84.
冤 2 冢cos 2 i sin 2 冣冥冤6冢cos 4 i sin 4 冣冥
3
Writing in Standard Form In Exercises 55–62, simplify
关53共cos 140 i sin 140兲兴关23共cos 60 i sin 60兲兴
86.
cos共5兾3兲 i sin共5兾3兲
cos i sin DeMoivre’s Theorem to find the indicated power of the
complex number. Write the result in standard form.
87. 共1 i兲5
55. 6i3 i 2
88. 共2 2i兲6
56. 4i 2 2i 3
89. 共1 i兲10
57. 5i5
90. 共1 i兲12
58. 共i兲3
91. 2共冪3 i兲7
60. 共冪2 兲6
93.
92. 4共1 冪3i兲3
59. 共冪75 兲3
62.
1
共2i兲3
85.
the complex number and write it in standard form.
1
i3
Using DeMoivre’s Theorem In Exercises 87–94, use
54. 5s2 6s 3 0
61.
冢cos 54 i sin 54冣
94. 冤 2冢cos i sin 冣冥
2
2
10
8
Absolute Value of a Complex Number In Exercises
63–68, plot the complex number and find its absolute value.
63. 5i
64. 5
65. 4 4i
66. 5 12i
67. 6 7i
68. 8 3i
Finding n th Roots In Exercises 95–100, (a) use Theorem E.2
to find the indicated roots of the complex number, (b) represent
each of the roots graphically, and (c) write each of the roots in
standard form.
95. Square roots of 5共cos 120 i sin 120兲
96. Square roots of 16共cos 60 i sin 60兲
冢
97. Fourth roots of 16 cos
Writing in Trigonometric Form In Exercises 69–76,
冢
represent the complex number graphically, and find the
trigonometric form of the number.
98. Fifth roots of 32 cos
69. 3 3i
70. 2 2i
99. Cube roots of 71. 冪3 i
72. 1 冪3i
73. 2共1 冪3i兲
74.
75. 6i
76. 4
5
2
共冪3 i兲
4
4
i sin
3
3
5
5
i sin
6
6
冣
冣
125
共1 冪3i兲
2
100. Cube roots of 4冪2共1 i兲
Writing in Standard Form In Exercises 77–82, represent
Solving an Equation In Exercises 101–108, use Theorem E.2
to find all the solutions of the equation and represent the
solutions graphically.
the complex number graphically, and find the standard form of
the number.
101. x 4 i 0
102. x3 1 0
103. x5 243 0
104. x 4 81 0
105. x3 64i 0
106. x6 64i 0
77. 2共cos 150 i sin 150兲
78. 5共cos 135 i sin 135兲
3
79. 2共cos 300 i sin 300兲
3
80. 4共cos 315 i sin 315兲
冢
81. 3.75 cos
冢
82. 8 cos
3
3
i sin
4
4
i sin
12
12
冣
107.
x3
共1 i兲 0
109. Proof
冣
108. x 4 共1 i兲 0
Given two complex numbers
z1 r1共cos 1 i sin 1兲 and z2 r2共cos 2 i sin 2兲
show that
z1 r1
关cos共1 2兲 i sin共1 2兲兴,
z2 r2
z2 0.