Download Section 2-4 Complex Numbers

2-4 Complex Numbers
133
Section 2-4 Complex Numbers
Introductory Remarks
The Complex Number System
Complex Numbers and Radicals
Introductory Remarks
The Pythagoreans (c. 500 B.C.) found that the simple quadratic equation
x2 ⫺ 2 ⫽ 0
(1)
had no rational number solution—it is not possible to find the ratio of two integers whose square is 2. If equation (1) were to have a solution, then a new kind
of number had to be invented—an irrational number. The irrational numbers 兹2
and ⫺ 兹2 are both solutions to equation (1). Irrational numbers were not put on
a firm mathematical foundation until the nineteenth century. The rational and irrational numbers together constitute the real number system. Is there any reason to
invent any other kinds of numbers?
Explore/Discuss
1
Graph g(x) ⫽ x2 ⫺ 1 in a standard viewing window and discuss the
relationship between the real zeros of the function and the x intercepts of
its graph. Do the same for f(x) ⫽ x2 ⫹ 1.
Does the simple quadratic equation
x2 ⫹ 1 ⫽ 0
(2)
have a solution? If equation (2) is to have a solution, x2 must be negative. But
the square of a real number is never negative. Thus, equation (2) cannot have any
real number solutions. Once again, a new type of number must be invented—a
number whose square can be negative. These new numbers are among the numbers called complex numbers. The complex numbers evolved over a long period
of time, but, like the real numbers, it was not until the nineteenth century that
they were given a firm mathematical foundation. Table 1 gives a brief history of
the evolution of complex numbers.
T A B L E
1 Brief History of Complex Numbers
Approximate
Date
Person
Event
50
850
1545
1637
1748
1832
Heron of Alexandria
Mahavira of India
Cardano of Italy
Descartes of France
Euler of Switzerland
Gauss of Germany
First recorded encounter of a square root of a negative number.
Said that a negative has no square root, since it is not a square.
Solutions to cubic equations involved square roots of negative numbers.
Introduced the terms real and imaginary.
Used i for 兹苵苵
⫺1.
Introduced the term complex number.
134
2 LINEAR AND QUADRATIC FUNCTIONS
The Complex Number System
We start the development of the complex number system by defining a complex
number and several special types of complex numbers. We then define equality,
addition, and multiplication in this system, and from these definitions the important special properties and operational rules for addition, subtraction, multiplication, and division will follow.
DEFINITION
1
COMPLEX NUMBER
A complex number is a number of the form
a ⴙ bi
Standard Form
where a and b are real numbers and i is called the imaginary unit.
The imaginary unit i introduced in Definition 1 is not a real number. It is a
special symbol used in the representation of the elements in this new complex
number system.
Some examples of complex numbers are
⫹ 5i
3 ⫺ 2i
1
2
0 ⫹ 3i
5 ⫹ 0i
2 ⫺ 13i
0 ⫹ 0i
Particular kinds of complex numbers are given special names as follows:
NAMES FOR PARTICULAR KINDS OF COMPLEX NUMBERS
DEFINITION
2
EXAMPLE
1
Imaginary Unit:
Complex Number:
Imaginary Number:
Pure Imaginary Number:
Real Number:
Zero:
Conjugate of a ⴙ bi:
i
a
a
0
a
0
a
⫹
⫹
⫹
⫹
⫹
⫺
bi a and b real numbers
bi b ⫽ 0
bi ⫽ bi b ⫽ 0
0i ⫽ a
0i ⫽ 0
bi
Special Types of Complex Numbers
Given the list of complex numbers:
3 ⫺ 2i
1
2
0 ⫹ 3i ⫽ 3i
5 ⫹ 0i ⫽ 5
⫹ 5i
2 ⫺ 13i
0 ⫹ 0i ⫽ 0
2-4 Complex Numbers
135
(A) List all the imaginary numbers, pure imaginary numbers, real numbers,
and zero.
(B) Write the conjugate of each.
Solutions
MATCHED PROBLEM
1
(A) Imaginary numbers: 3 ⫺ 2i, 12 ⫹ 5i, 2 ⫺ 13i, 3i
Pure imaginary numbers: 0 ⫹ 3i ⫽ 3i
Real numbers: 5 ⫹ 0i ⫽ 5, 0 ⫺ 0i ⫽ 0
Zero: 0 ⫹ 0i ⫽ 0
1
2 ⫹ 13i
(B) 3 ⫹ 2i
2 ⫺ 5i
0 ⫺ 3i ⫽ ⫺3i
5⫺ 0i ⫽ 5
0 ⫺ 0i ⫽ 0
Given the list of complex numbers:
6 ⫹ 7i
0⫹
2
3i
⫽
2
3i
兹2 ⫺ 13i
0 ⫺ i ⫽ ⫺i
⫺ 兹3 ⫹ 0i ⫽ ⫺ 兹3
0 ⫺ 0i ⫽ 0
(A) List all the imaginary numbers, pure imaginary numbers, real numbers, and
zero.
(B) Write the conjugate of each.
In Definition 2, notice that we identify a complex number of the form a ⫹ 0i
with the real number a, a complex number of the form 0 ⫹ bi, b ⫽ 0, with the
pure imaginary number bi, and the complex number 0 ⫹ 0i with the real number 0. Thus, a real number is also a complex number, just as a rational number
is also a real number. Any complex number that is not a real number is called an
imaginary number. If we combine the set of all real numbers with the set of all
imaginary numbers, we obtain C, the set of complex numbers. The relationship
of the complex number system to the other number systems we have studied is
shown in Figure 1.
FIGURE 1
Natural
numbers
(N )
Zero
Negative
integers
N傺Z傺Q傺R傺C
Integers
(Z )
Noninteger
rational
numbers
Rational
numbers
(Q)
Irrational
numbers
(I )
Real
numbers
(R )
Imaginary
numbers
Complex
numbers
(C )
To use complex numbers, we must know how to add, subtract, multiply, and
divide them. We start by defining equality, addition, and multiplication.
136
2 LINEAR AND QUADRATIC FUNCTIONS
EQUALITY AND BASIC OPERATIONS
DEFINITION
3
1. Equality:
a ⫹ bi ⫽ c ⫹ di if and only if a ⫽ c and b ⫽ d
2. Addition:
(a ⫹ bi) ⫹ (c ⫹ di) ⫽ (a ⫹ c) ⫹ (b ⫹ d)i
3. Multiplication:
(a ⫹ bi)(c ⫹ di) ⫽ (ac ⫺ bd) ⫹ (ad ⫹ bc)i
The basic properties of the real number system are discussed in Appendix A,
Section A-1. Using the definition of addition and subtraction of complex numbers
(Definition 3), it can be shown that the complex number system possesses the
same properties. That is,
BASIC PROPERTIES OF THE COMPLEX NUMBER SYSTEM
1. Addition and multiplication of complex numbers are commutative
and associative operations.
2. There is an additive identity and a multiplicative identity for complex numbers.
3. Every complex number has an additive inverse or negative.
4. Every nonzero complex number has a multiplicative inverse or
reciprocal.
5. Multiplication distributes over addition.
As a consequence of these properties you will not have to memorize the definitions of addition and multiplication of complex numbers in Definition 3.
We can manipulate complex number symbols of the form a ⴙ bi
just like we manipulate real binomials of the form a ⴙ bx, as long
as we remember that i is a special symbol for the imaginary unit,
not for a real number.
The first arithmetic operation we consider is addition.
EXAMPLE
2
Solutions
Addition of Complex Numbers
Carry out each operation and express the answer in standard form:
(A) (2 ⫺ 3i) ⫹ (6 ⫹ 2i)
(B) (⫺5 ⫹ 4i) ⫹ (0 ⫹ 0i)
(A) We could apply the definition of addition directly, but it is easier to use complex number properties.
(2 ⫺ 3i) ⫹ (6 ⫹ 2i) ⫽ 2 ⫺ 3i ⫹ 6 ⫹ 2i
Remove parentheses.
⫽ (2 ⫹ 6) ⫹ (⫺3 ⫹ 2)i Combine like terms.
⫽8⫺i
2-4 Complex Numbers
137
(B) (⫺5 ⫹ 4i) ⫹ (0 ⫹ 0i) ⫽ ⫺5 ⫹ 4i ⫹ 0 ⫹ 0i
⫽ ⫺5 ⫹ 4i
MATCHED PROBLEM
2
Carry out each operation and express the answer in standard form:
(A) (3 ⫹ 2i) ⫹ (6 ⫺ 4i)
(B) (0 ⫹ 0i) ⫹ (7 ⫺ 5i)
Example 2, part B, and Matched Problem 2, part B, illustrate the following
general result: For any complex number a ⫹ bi,
(a ⫹ bi) ⫹ (0 ⫹ 0i) ⫽ (0 ⫹ 0i) ⫹ (a ⫹ bi) ⫽ a ⫹ bi
Thus, 0 ⫹ 0i is the additive identity or zero for the complex numbers. We anticipated this result in Definition 1 when we identified the complex number 0 ⫹ 0i
with the real number 0.
Your graphing utility may be able to perform complex number arithmetic
(check your manual). Figure 2(a) shows the solution of Example 2 on a TI-83
using the standard a ⫹ bi notation. Figure 2(b) shows the same operations on a
TI-86.
FIGURE 2
Complex number arithmetic on
some typical graphing utilities.
(a) TI-83
(b) TI-86
Note that the TI-86 does not support the a ⫹ bi notation for complex numbers. Instead, the complex number a ⫹ bi is entered as the ordered pair (a, b).
For now, if your graphing utility uses ordered pair notation for complex numbers,
just translate each a ⫹ bi form into an ordered pair, perform the operation, and
translate back to a ⫹ bi form.
We now turn to negatives and subtraction, which can be defined in terms of
the additive inverse of a complex number. However, because of the already stated
properties of complex numbers, we can manipulate a ⫹ bi in the same way we
manipulate the real binomial form a ⫹ bx.
EXAMPLE
3
Solutions
Negation and Subtraction
Carry out each operation and express the answer in standard form:
(A) ⫺(4 ⫺ 5i)
(B) (7 ⫺ 3i) ⫺ (6 ⫹ 2i)
(C) (⫺2 ⫹ 7i) ⫹ (2 ⫺ 7i)
(A) ⫺(4 ⫺ 5i) ⫽ (⫺1)(4 ⫺ 5i) ⫽ ⫺4 ⫹ 5i
(B) (7 ⫺ 3i) ⫺ (6 ⫹ 2i) ⫽ 7 ⫺ 3i ⫺ 6 ⫺ 2i
⫽ 1 ⫺ 5i
(C) (⫺2 ⫹ 7i) ⫹ (2 ⫺ 7i) ⫽ ⫺2 ⫹ 7i ⫹ 2 ⫺ 7i ⫽ 0
138
2 LINEAR AND QUADRATIC FUNCTIONS
MATCHED PROBLEM
3
Carry out each operation and express the answer in standard form:
(A) ⫺(⫺3 ⫹ 2i)
(B) (3 ⫺ 5i) ⫺ (1 ⫺ 3i)
(C) (⫺4 ⫹ 9i) ⫹ (4 ⫺ 9i)
Figure 3 shows a check of the calculations in Example 3 on two different graphing utilities.
FIGURE 3
(a) TI-83
(b) TI-86
In general, the additive inverse or negative of a ⫹ bi is ⫺a ⫺ bi since
(a ⫹ bi) ⫹ (⫺a ⫺ bi) ⫽ (⫺a ⫺ bi) ⫹ (a ⫹ bi) ⫽ 0
(see Example 3, part C, and Matched Problem 3, part C).
Now we turn our attention to multiplication. First, we use the definition of
multiplication to see what happens to the complex unit i when it is squared:
(a ⴙ bi)(c ⴙ di)
i 2 ⫽ (0 ⫹ 1i)(0 ⫹ 1i)
(ac ⴚ bd) ⴙ (ad ⴙ bc)i
⫽ (0 ⴢ 0 ⫺ 1 ⴢ 1) ⫹ (0 ⴢ 1 ⫹ 1 ⴢ 0)i
⫽ ⫺1 ⫹ 0i
⫽ ⫺1
Thus, we have proved that
i 2 ⴝ ⴚ1
That is, the square of i is a negative real number and i is a solution to x2 ⫹ 1 ⫽
0. Since i2 ⫽ ⫺1, we define 兹⫺1 to be the imaginary unit i. Thus,
i ⴝ 兹ⴚ1
and
ⴚi ⴝ ⴚ兹ⴚ1
Just as in the case of addition and subtraction, multiplication of complex
numbers can be carried out using the properties of complex numbers stated
above. That is, we can manipulate a ⫹ bi in the same way we manipulate the
real binomial form a ⫹ bx. The key difference is that we replace i2 with ⫺1 each
time it occurs.
EXAMPLE
4
Multiplying Complex Numbers
Carry out each operation and express the answer in standard form:
(A) (2 ⫺ 3i)(6 ⫹ 2i)
(B) 1(3 ⫺ 5i)
(C) i(1 ⫹ i)
(D) (3 ⫹ 4i)(3 ⫺ 4i)
2-4 Complex Numbers
Solutions
139
(A) (2 ⫺ 3i)(6 ⫹ 2i) ⫽ 2(6 ⫹ 2i) ⫺ 3i(6 ⫹ 2i)
⫽ 12 ⫹ 4i ⫺ 18i ⫺ 6i2
⫽ 12 ⫺ 14i ⫺ 6(⫺1)
Replace i2 with ⫺1.
⫽ 18 ⫺ 14i
(B) 1(3 ⫺ 5i) ⫽ 1 ⴢ 3 ⫺ 1 ⴢ 5i ⫽ 3 ⫺ 5i
(C) i(1 ⫹ i) ⫽ i ⫹ i2 ⫽ i ⫺ 1 ⫽ ⫺1 ⫹ i
(D) (3 ⫹ 4i)(3 ⫺ 4i) ⫽ 9 ⫺ 12i ⫹ 12i ⫺ 16i2
⫽ 9 ⫹ 16 ⫽ 25
MATCHED PROBLEM
4
Carry out each operation and express the answer in standard form:
(A) (5 ⫹ 2i)(4 ⫺ 3i) (B) 3(⫺2 ⫹ 6i) (C) i(2 ⫺ 3i) (D) (2 ⫹ 3i)(2 ⫺ 3i)
A graphing utility check of Example 4 is shown in Figure 4.
FIGURE 4
(a) TI-83
(b) TI-86
For any complex number a ⫹ bi,
1(a ⫹ bi) ⫽ (a ⫹ bi)1 ⫽ a ⫹ bi
(see Example 4, part B). Thus, 1 is the multiplicative identity for complex numbers, just as it is for real numbers.
Earlier we stated that every nonzero complex number has a multiplicative
inverse or reciprocal. We will denote this as a fraction, just as we do with real
numbers. Thus,
1
a ⫹ bi
is the reciprocal of
a ⫹ bi
a ⫹ bi ⫽ 0
The following important property of the conjugate of a complex number is
used to express reciprocals and quotients in standard form.
THEOREM
1
PRODUCT OF A COMPLEX NUMBER AND ITS CONJUGATE
(a ⫹ bi)(a ⫺ bi) ⫽ a2 ⫹ b2
A real number
140
2 LINEAR AND QUADRATIC FUNCTIONS
We now turn to the fourth arithmetic operation, division of complex numbers. Division can be performed by making direct use of Theorem 1. As before,
we can manipulate a ⫹ bi in the same way we manipulate the real binomial form
a ⫹ bx, except we replace i2 with ⫺1 each time it occurs. Example 5 should make
the process clear.
EXAMPLE
Reciprocals and Quotients
5
Solutions
Carry out each operation and express the answer in standard form:
1
7 ⫺ 3i
(A)
(B)
2 ⫹ 3i
1⫹i
(A) Multiply numerator and denominator by the conjugate of the denominator:
1
2 ⴚ 3i
1
⫽
ⴢ
2 ⫹ 3i 2 ⫹ 3i 2 ⴚ 3i
⫽
⫽
2 ⫺ 3i
2 ⫺ 3i
⫽
2
4 ⫺ 9i
4⫹9
2 ⫺ 3i
2
3
⫽
⫺ i
13
13 13
To check this answer, we multiply the divisor by the quotient:
冢132 ⫺ 133 i冣 ⫽ 134 ⫺ 136 i ⫹ 136 i ⫺ 139 i
(2 ⫹ 3i)
Check
2
⫽
(B)
7 ⫺ 3i 7 ⫺ 3i 1 ⴚ i
⫽
ⴢ
1⫹i
1⫹i 1ⴚi
⫽
Check
MATCHED PROBLEM
5
4
9
⫹
⫽1
13 13
⫽
7 ⫺ 7i ⫺ 3i ⫹ 3i2
1 ⫺ i2
4 ⫺ 10i
⫽ 2 ⫺ 5i
2
(1 ⫹ i)(2 ⫺ 5i) ⫽ 2 ⫺ 5i ⫹ 2i ⫺ 5i2 ⫽ 7 ⫺ 3i
Carry out each operation and express the answer in standard form:
1
6 ⫹ 7i
(A)
(B)
4 ⫹ 2i
2⫺i
Figure 5 shows the result of carrying out the operations in Example 5 on two
graphing utilities. In the answer to Example 5, part A, note that we used a builtin routine to convert the decimal forms to fraction forms.
FIGURE 5
(a) TI-83
(b) TI-86
2-4 Complex Numbers
EXAMPLE
6
Solutions
141
Combined Operations
Carry out the indicated operations and write each answer in standard form:
2 ⫺ 3i
(B)
(A) (3 ⫺ 2i)2 ⫺ 6(3 ⫺ 2i) ⫹ 13
2i
(A) (3 ⫺ 2i)2 ⫺ 6(3 ⫺ 2i) ⫹ 13 ⫽ 9 ⫺ 12i ⫹ 4i2 ⫺ 18 ⫹ 12i ⫹ 13
⫽ 9 ⫺ 12i ⫺ 4 ⫺ 18 ⫹ 12i ⫹ 13
⫽0
(B) If a complex number is divided by a pure imaginary number, we can make
the denominator real by multiplying numerator and denominator by i.
3
2 ⫺ 3i i 2i ⫺ 3i2 2i ⫹ 3
ⴢ ⫽
⫽⫺ ⫺i
⫽
2
2i
i
2i
⫺2
2
MATCHED PROBLEM
6
Explore/Discuss
2
Carry out the indicated operations and write each answer in standard form:
4⫺i
(B)
(A) (3 ⫹ 2i)2 ⫺ 6(3 ⫹ 2i) ⫹ 13
3i
Natural number powers of i take on particularly simple forms:
i
i 5 ⫽ i 4 ⴢ i ⫽ (1)i ⫽ i
i2 ⫽ ⫺1
i 6 ⫽ i 4 ⴢ i 2 ⫽ 1(⫺1) ⫽ ⫺1
i 3 ⫽ i 2 ⴢ i ⫽ (⫺1)i ⫽ ⫺i
i 7 ⫽ i 4 ⴢ i 3 ⫽ 1(⫺i) ⫽ ⫺i
i 4 ⫽ i 2 ⴢ i 2 ⫽ (⫺1)(⫺1) ⫽ 1
i8 ⫽ i4 ⴢ i4 ⫽ 1 ⴢ 1 ⫽ 1
In general, what are the possible values for in, n a natural number?
Explain how you could easily evaluate in for any natural number n. Then
evaluate each of the following:
(A) i17
(B) i24
(C) i38
(D) i47
If your graphing utility can perform complex arithmetic, use it to check
your calculations in parts A–D.
Complex Numbers and Radicals
Recall that we say that a is a square root of b if a2 ⫽ b. If x is a positive real
number, then x has two square roots, the principal square root, denoted by 兹x,
and its negative, ⫺ 兹x (Section A-7). If x is a negative real number, then x still
has two square roots, but now these square roots are imaginary numbers.
142
2 LINEAR AND QUADRATIC FUNCTIONS
DEFINITION
4
PRINCIPAL SQUARE ROOT OF A NEGATIVE REAL NUMBER
The principal square root of a negative real number, denoted by
兹⫺a, where a is positive, is defined by
兹⫺a ⫽ i兹a
兹⫺3 ⫽ i兹3
兹⫺9 ⫽ i兹9 ⫽ 3i
The other square root of ⫺a, a ⬎ 0, is ⫺ 兹⫺a ⫽ ⫺i兹a.
Note in Definition 4 that we wrote i兹a and i兹3 in place of the standard
forms 兹ai and 兹3i. We follow this convention whenever it appears that i might
accidentally slip under a radical sign (兹ai ⫽ 兹ai, but 兹ai ⫽ i兹a). Definition
4 is motivated by the fact that
(i兹a)2 ⫽ i2a ⫽ ⫺a
EXAMPLE
7
Complex Numbers and Radicals
Write in standard form:
(A) 兹⫺4
Solutions
MATCHED PROBLEM
7
Explore/Discuss
3
(B) 4 ⫹ 兹⫺5
(C)
⫺3 ⫺ 兹⫺5
2
(D)
1
1 ⫺ 兹⫺9
(A) 兹⫺4 ⫽ i兹4 ⫽ 2i
(B) 4 ⫹ 兹⫺5 ⫽ 4 ⫹ i兹5
⫺3 ⫺ 兹⫺5 ⫺3 ⫺ i兹5
3 兹5
⫽
⫽⫺ ⫺
i
(C)
2
2
2
2
1
1
1 ⴢ (1 ⴙ 3i)
⫽
⫽
(D)
1 ⫺ 兹⫺9 1 ⫺ 3i (1 ⫺ 3i) ⴢ (1 ⴙ 3i)
1 ⫹ 3i
1 ⫹ 3i
1
3
⫽
⫽
⫽
⫹ i
2
1 ⫺ 9i
10
10 10
Write in standard form:
(A) 兹⫺16
(B) 5 ⫹ 兹⫺7
(C)
⫺5 ⫺ 兹⫺2
2
(D)
1
3 ⫺ 兹⫺4
From Theorem 1 in Section A-7, we know that if a and b are positive
real numbers, then
兹a兹b ⫽ 兹ab
(3)
Thus, we can evaluate expressions like 兹9 兹4 two ways:
兹9 兹4 ⫽ 兹(9)(4) ⫽ 兹36 ⫽ 6
and
兹9 兹4 ⫽ (3)(2) ⫽ 6
2-4 Complex Numbers
Explore/Discuss
3
continued
CAUTION
143
Evaluate each of the following two ways. Is equation (3) a valid property
to use in all cases?
(A) 兹9 兹⫺4
(B) 兹⫺9 兹4
(C) 兹⫺9 兹⫺4
If your graphing utility can perform complex arithmetic, use it to check
your calculations in parts A–C. If entering 兹⫺4 produces an error
message, try entering 兹⫺4 ⫹ 0i.
Note that in Example 7, part D, we wrote 1 ⫺ 兹⫺9 ⫽ 1 ⫺ 3i before
proceeding with the simplification. This is a necessary step because some
of the properties of radicals that are true for real numbers turn out not to
be true for complex numbers. In particular, for positive real numbers a
and b,
兹a 兹b ⫽ 兹ab
but
兹⫺a 兹⫺b ⫽ 兹(⫺a)(⫺b)
(See Explore/Discuss 3.)
Early resistance to these new numbers is suggested by the words used to name
them: complex and imaginary. In spite of this early resistance, complex numbers
have come into widespread use in both pure and applied mathematics. They are
used extensively, for example, in electrical engineering, physics, chemistry, statistics, and aeronautical engineering. Our first use of them will be in connection with
solutions of second-degree equations in the next section.
Answers to Matched Problems
1. (A) Imaginary numbers: 6 ⫹ 7i, 兹2 ⫺ 13i, 0 ⫺ i ⫽ ⫺i, 0 ⫹ 23i ⫽ 23i
Pure imaginary numbers: 0 ⫺ i ⫽ ⫺i, 0 ⫹ 23i ⫽ 23i
Real numbers: ⫺兹3 ⫹ 0i ⫽ ⫺兹3, 0 ⫺ 0i ⫽ 0
Zero: 0 ⫺ 0i ⫽ 0
(B) 6 ⫺ 7i, 兹2 ⫹ 13i, 0 ⫹ i ⫽ i, 0 ⫺ 23i ⫽ ⫺ 23i, ⫺兹3 ⫺ 0i ⫽ ⫺兹3, 0 ⫹ 0i ⫽ 0
2. (A) 9 ⫺ 2i
(B) 7 ⫺ 5i
3. (A) 3 ⫺ 2i
(B) 2 ⫺ 2i
(C) 0
4. (A) 26 ⫺ 7i
(B) ⫺6 ⫹ 18i
(C) 3 ⫹ 2i
(D) 13
1
5. (A) 15 ⫺ 10
(B) 1 ⫹ 4i
i
6. (A) 0
(B) ⫺ 13 ⫺ 43i
3
2
7. (A) 4i
(B) 5 ⫹ i兹7
(C) ⫺ 52 ⫺ (兹2/2)i
(D) 13
⫹ 13
i
144
2 LINEAR AND QUADRATIC FUNCTIONS
41. (2 ⫺ 3i)2 ⫺ 2(2 ⫺ 3i) ⫹ 9
EXERCISE 2-4
42. (2 ⫺ i)2 ⫹ 3(2 ⫺ i) ⫺ 5
A
43. Let f(x) ⫽ x2 ⫺ 2x ⫹ 2.
(A) Show that the conjugate complex numbers 1 ⫹ i and
1 ⫺ i are both zeros of f.
In Problems 1–26, perform the indicated operations and write
each answer in standard form.
1. (2 ⫹ 4i) ⫹ (5 ⫹ i)
2. (3 ⫹ i) ⫹ (4 ⫹ 2i)
3. (⫺2 ⫹ 6i) ⫹ (7 ⫺ 3i)
4. (6 ⫺ 2i) ⫹ (8 ⫺ 3i)
5. (6 ⫹ 7i) ⫺ (4 ⫹ 3i)
6. (9 ⫹ 8i) ⫺ (5 ⫹ 6i)
7. (3 ⫹ 5i) ⫺ (⫺2 ⫺ 4i)
8. (8 ⫺ 4i) ⫺ (11 ⫺ 2i)
9. (4 ⫺ 5i) ⫹ 2i
10. 6 ⫹ (3 ⫺ 4i)
(B) Does f have any real zeros? Any x intercepts? Explain.
44. Let g(x) ⫽ ⫺x2 ⫹ 4x ⫺ 5.
(A) Show that the conjugate complex numbers 2 ⫹ i and
2 ⫺ i are both zeros of g.
(B) Does g have any real zeros? Any x intercepts?
Explain.
45. Simplify: i18, i32, and i67.
11. (4i)(6i)
12. (3i)(8i)
13. ⫺3i(2 ⫺ 4i)
14. ⫺2i(5 ⫺ 3i)
15. (3 ⫹ 3i)(2 ⫺ 3i)
16. (⫺2 ⫺ 3i)(3 ⫺ 5i)
In Problems 47–50, solve for x and y.
17. (2 ⫺ 3i)(7 ⫺ 6i)
18. (3 ⫹ 2i)(2 ⫺ i)
47. (2x ⫺ 1) ⫹ (3y ⫹ 2)i ⫽ 5 ⫺ 4i
19. (7 ⫹ 4i)(7 ⫺ 4i)
20. (5 ⫹ 3i)(5 ⫺ 3i)
48. 3x ⫹ (y ⫺ 2)i ⫽ (5 ⫺ 2x) ⫹ (3y ⫺ 8)i
46. Simplify: i21, i43, and i52.
21.
1
2⫹i
22.
1
3⫺i
23.
3⫹i
2 ⫺ 3i
49.
(1 ⫹ x) ⫹ ( y ⫺ 2)i
⫽2⫺i
1⫹i
24.
2⫺i
3 ⫹ 2i
25.
13 ⫹ i
2⫺i
26.
15 ⫺ 3i
2 ⫺ 3i
50.
(2 ⫹ x) ⫹ ( y ⫹ 3)i
⫽ ⫺3 ⫹ i
1⫺i
B
In Problems 51–54, solve for z. Express answers in standard
form.
In Problems 27–36, convert imaginary numbers to standard
form, perform the indicated operations, and express answers
in standard form.
51. (2 ⫹ i)z ⫹ i ⫽ 4i
27. (2 ⫺ 兹⫺4) ⫹ (5 ⫺ 兹⫺9)
53. 3iz ⫹ (2 ⫺ 4i) ⫽ (1 ⫹ 2i)z ⫺ 3i
28. (3 ⫺ 兹⫺4) ⫹ (⫺8 ⫹ 兹⫺25)
54. (2 ⫺ i)z ⫹ (1 ⫺ 4i) ⫽ (⫺1 ⫹ 3i)z ⫹ (4 ⫹ 2i)
29. (9 ⫺ 兹⫺9) ⫺ (12 ⫺ 兹⫺25)
55. Explain what is wrong with the following “proof ” that
⫺1 ⫽ 1:
30. (⫺2 ⫺ 兹⫺36) ⫺ (4 ⫹ 兹⫺49)
31. (3 ⫺ 兹⫺4)(⫺2 ⫹ 兹⫺49)
52. (3 ⫺ i)z ⫹ 2 ⫽ i
⫺1 ⫽ i2 ⫽ 兹⫺1 兹⫺1 ⫽ 兹(⫺1)(⫺1) ⫽ 兹1 ⫽ 1
56. Explain what is wrong with the following “proof” that
1/i ⫽ i. What is the correct value of 1/i?
32. (2 ⫺ 兹⫺1)(5 ⫹ 兹⫺9)
33.
5 ⫺ 兹⫺4
7
34.
6 ⫺ 兹⫺64
2
35.
1
2 ⫺ 兹⫺9
36.
1
3 ⫺ 兹⫺16
Write Problems 37–42 in standard form.
1
1
兹1
⫽
⫽
⫽
i
兹⫺1 兹⫺1
兹
1
⫽ 兹⫺1 ⫽ i
⫺1
C
37.
2
5i
38.
1
3i
In Problems 57–62, perform the indicated operations, and
write each answer in standard form.
1 ⫹ 3i
2i
58. (a ⫹ bi) ⫺ (c ⫹ di)
40.
2⫺i
3i
57. (a ⫹ bi) ⫹ (c ⫹ di)
39.
59. (a ⫹ bi)(a ⫺ bi)
60. (u ⫺ vi)(u ⫹ vi)
2-5 Quadratic Equations and Inequalities
61. (a ⫹ bi)(c ⫹ di)
62.
a ⫹ bi
c ⫹ di
63. Show that i4k ⫽ 1, k a natural number.
64. Show that i4k⫹1 ⫽ i, k a natural number.
65. Show that 2 ⫺ i and ⫺2 ⫹ i are square roots of 3 ⫺ 4i.
66. Show that ⫺3 ⫹ 2i and 3 ⫺ 2i are square roots of 5 ⫺ 12i.
67. Describe how you could find the square roots of 8 ⫺ 6i
without using a graphing utility. What are the square roots
of 8 ⫺ 6i?
68. Describe how you could find the square roots of 2i without
using a graphing utility. What are the square roots of 2i?
69. Let Sn ⫽ i ⫹ i2 ⫹ i3 ⫹ ⭈ ⭈ ⭈ ⫹ in, n ⱖ 1. Describe the possible values of Sn.
70. Let Tn ⫽ i2 ⫹ i4 ⫹ i6 ⫹ ⭈ ⭈ ⭈ ⫹ i2n, n ⱖ 1. Describe the
possible values of Tn.
Supply the reasons in the proofs for the theorems stated in
Problems 71 and 72.
71. Theorem: The complex numbers are commutative under
addition.
Proof:
Let a ⫹ bi and c ⫹ di be two arbitrary complex
numbers; then,
Statement
1. (a ⫹ bi) ⫹ (c ⫹ di) ⫽ (a ⫹ c) ⫹ (b ⫹ d)i
2.
⫽ (c ⫹ a) ⫹ (d ⫹ b)i
3.
⫽ (c ⫹ di) ⫹ (a ⫹ bi)
145
Reason
1.
2.
3.
72. Theorem: The complex numbers are commutative under
multiplication.
Proof:
Let a ⫹ bi and c ⫹ di be two arbitrary complex
numbers; then,
Statement
1. (a ⫹ bi) ⴢ (c ⫹ di) ⫽ (ac ⫺ bd) ⫹ (ad ⫹ bc)i
2.
⫽ (ca ⫺ db) ⫹ (da ⫹ cb)i
3.
⫽ (c ⫹ di)(a ⫹ bi)
Reason
1.
2.
3.
Letters z and w are often used as complex variables, where
z ⫽ x ⫹ yi, w ⫽ u ⫹ vi, and x, y, u, and v are real numbers.
The conjugates of z and w, denoted by z and w, respectively,
are given by z ⫽ x ⫺ yi and w ⫽ u ⫺ vi. In Problems 73–80,
express each property of conjugates verbally and then prove
the property.
73. zz is a real number.
74. z ⫹ z is a real number.
75. z ⫽ z if and only if z is real. 76. z ⫽ z
77. z ⫹ w ⫽ z ⫹ w
78. z ⫺ w ⫽ z ⫺ w
79. zw ⫽ z ⴢ w
80. z/w ⫽ z/ w
Section 2-5 Quadratic Equations and Inequalities
Introduction
Solution by Factoring
Solution by Completing the Square
Solution by Quadratic Formula
Solving Quadratic Inequalities
Introduction
In this book we are primarily interested in functions with real number domains
and ranges. However, if we want to fully understand the nature of the zeros of a
function or the roots of an equation, it is necessary to extend some of the definitions in Section 1-4 to include complex numbers. A complex number r is a zero