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Section 4.2 Complex Numbers 155
Classifying Numbers
Work with a partner. Determine which subsets of the set of complex numbers contain each number. a. √
— 9 b. √
— 0 c. −√
— 4
d. (^) √
— 4 — 9
e. √
— 2 f. √
— − 1
Complex Solutions of Quadratic Equations
Work with a partner. Use the defi nition of the imaginary unit i to match each quadratic equation with its complex solution. Justify your answers. a. x^2 − 4 = 0 b. x^2 + 1 = 0 c. x^2 − 1 = 0 d. x^2 + 4 = 0 e. x^2 − 9 = 0 f. x^2 + 9 = 0
A. i B. 3 i C. 3 D. 2 i E. 1 F. 2
Communicate Your AnswerCommunicate Your Answer
3. What are the subsets of the set of complex numbers? Give an example of a number in each subset. 4. Is it possible for a number to be both whole and natural? natural and rational? rational and irrational? real and imaginary? Explain your reasoning.
USING PRECISE
MATHEMATICAL
LANGUAGE
To be profi cient in math, you need to use clear defi nitions in your reasoning and discussions with others.
Essential QuestionEssential Question What are the subsets of the set of
complex numbers?
In your study of mathematics, you have probably worked with only real numbers , which can be represented graphically on the real number line. In this lesson, the system of numbers is expanded to include imaginary numbers. The real numbers and imaginary numbers compose the set of complex numbers.
Complex Numbers
Real Numbers Imaginary Numbers
Rational Numbers Irrational Numbers
Integers
Whole Numbers
Natural Numbers
The imaginary uniti is defi ned as i = √
— −1.
4.2 Complex Numbers
2A.4.F 2A.7.A
TEXAS ESSENTIAL
KNOWLEDGE AND SKILLS
156 Chapter 4 Quadratic Equations and Complex Numbers
Lesson
Finding Square Roots of Negative Numbers
Find the square root of each number. a. √
— − 25 b. √
— − 72 c. − 5 √
— − 9
SOLUTION
a. √
— − 25 = √
— (^25) ⋅√
— − 1 = 5 i b. √
— − 72 = √
— (^72) ⋅√
— − 1 = √
— (^36) ⋅√
— (^2) ⋅ i = 6 √
— 2 i = 6 i √
— 2 c. − 5 √
— − 9 = − 5 √
— (^9) ⋅√
— − 1 = − (^5) ⋅ 3 ⋅ i = − 15 i
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Find the square root of the number.
1. √
— − 4 2. √
— − 12 3. −√
— − 36 4. 2 √
— − 54
A complex number written in standard form is a number a + bi where a and b are real numbers. The number a is the real part , and the number bi is the imaginary part. a + bi If b ≠ 0, then a + bi is an imaginary number. If a = 0 and b ≠ 0, then a + bi is a pure imaginary number. The diagram shows how different types of complex numbers are related.
4.2 What You Will LearnWhat You Will Learn
Define and use the imaginary unit i. Add, subtract, and multiply complex numbers. Find complex solutions and zeros.
The Imaginary Unit i Not all quadratic equations have real-number solutions. For example, x^2 = − 3 has no real-number solutions because the square of any real number is never a negative number. To overcome this problem, mathematicians created an expanded system of numbers using the imaginary unit i , defi ned as i = √
— −1. Note that i^2 = −1. The imaginary unit i can be used to write the square root of any negative number.
imaginary unit i , p. 156 complex number, p. 156 imaginary number, p. 156 pure imaginary number, p. 156
Core VocabularyCore Vocabullarry
CoreCore ConceptConcept
The Square Root of a Negative Number
Property Example
1. If r is a positive real number, then √—− r = i √— r. √
— − 3 = i √
— 3
2. By the fi rst property, it follows that (^ i √— r )^2 = − r. (^ i √
— 3 ) 2 = i^2 ⋅ 3 = − 3
Complex Numbers ( a + bi ) Real Numbers ( a + 0 i )
Imaginary Numbers ( a + bi , b ≠ 0)
Pure Imaginary Numbers (0 + bi , b ≠ 0)
− 1
2 + 3 i 9 − 5 i
−4 i 6 i
5 3
π 2
158 Chapter 4 Quadratic Equations and Complex Numbers
Solving a Real-Life Problem
Electrical circuit components, such as resistors, inductors, and capacitors, all oppose the fl ow of current. This opposition is called resistance for resistors and reactance for inductors and capacitors. Each of these quantities is measured in ohms. The symbol used for ohms is Ω, the uppercase Greek letter omega.
Component and symbol
Resistor Inductor Capacitor
Resistance or reactance (in ohms)
R L C
Impedance (in ohms) R Li (^) − Ci
The table shows the relationship between a component’s resistance or reactance and its contribution to impedance. A series circuit is also shown with the resistance or reactance of each component labeled. The impedance for a series circuit is the sum of the impedances for the individual components. Find the impedance of the circuit.
SOLUTION
The resistor has a resistance of 5 ohms, so its impedance is 5 ohms. The inductor has a reactance of 3 ohms, so its impedance is 3 i ohms. The capacitor has a reactance of 4 ohms, so its impedance is − 4 i ohms. Impedance of circuit = 5 + 3 i + (− 4 i ) = 5 − i
The impedance of the circuit is (5 − i ) ohms.
To multiply two complex numbers, use the Distributive Property, or the FOIL method, just as you do when multiplying real numbers or algebraic expressions.
Multiplying Complex Numbers
Multiply. Write the answer in standard form. a. 4 i (− 6 + i ) b. (9 − 2 i )(− 4 + 7 i )
SOLUTION
a. 4 i (− 6 + i ) = − 24 i + 4 i^2 Distributive Property = − 24 i + 4(−1) Use i^2 = −1. = − 4 − 24 i Write in standard form. b. (9 − 2 i )(− 4 + 7 i ) = − 36 + 63 i + 8 i − 14 i^2 Multiply using FOIL. = − 36 + 71 i − 14(−1) Simplify and use i^2 = −1. = − 36 + 71 i + 14 Simplify. = − 22 + 71 i Write in standard form.
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7. WHAT IF? In Example 4, what is the impedance of the circuit when the capacitor is replaced with one having a reactance of 7 ohms? Perform the operation. Write the answer in standard form. 8. (9 − i ) + (− 6 + 7 i ) 9. (3 + 7 i ) − (8 − 2 i ) 10. − 4 − (1 + i ) − (5 + 9 i ) 11. (− 3 i )(10 i ) 12. i (8 − i ) 13. (3 + i )(5 − i )
STUDY TIP
When simplifying an expression that involves complex numbers, be sure to simplify i^2 as −1.
E
t i u
5 Ω
3 Ω 4 Ω
Alternating current source
Section 4.2 Complex Numbers 159
Complex Solutions and Zeros
Solving Quadratic Equations
Solve (a) x^2 + 4 = 0 and (b) 2 x^2 − 11 = −47.
SOLUTION
a. x^2 + 4 = 0 Write original equation. x^2 = − 4 Subtract 4 from each side. x = ± √
— − 4 Take square root of each side. x = ± 2 i Write in terms of i.
The solutions are 2 i and − 2 i. b. 2 x^2 − 11 = − 47 Write original equation. 2 x^2 = − 36 Add 11 to each side. x^2 = − 18 Divide each side by 2. x = ± √
— − 18 Take square root of each side. x = ± i √
— 18 Write in terms of i. x = ± 3 i √
— 2 Simplify radical.
The solutions are 3 i √
— 2 and − 3 i √
—
Finding Zeros of a Quadratic Function
Find the zeros of f ( x ) = 4 x^2 + 20.
SOLUTION
4 x^2 + 20 = 0 Set f ( x ) equal to 0. 4 x^2 = − 20 Subtract 20 from each side. x^2 = − 5 Divide each side by 4. x = ± √
— − 5 Take square root of each side. x = ± i √
— 5 Write in terms of i.
So, the zeros of f are i √
— 5 and − i √
—
Check
f (^ i √
— 5 )^ = 4( i √
— 5 ) 2
- 20 = (^4) ⋅ 5 i^2 + 20 = 4(−5) + 20 = 0 ✓
f (^ − i √
— 5 )^ = 4(^ − i √
— 5 ) 2
- 20 = (^4) ⋅ 5 i^2 + 20 = 4(−5) + 20 = 0 ✓
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Solve the equation.
14. x^2 = − 13 15. x^2 = − 38 16. x^2 + 11 = 3 17. x^2 − 8 = − 36 18. 3 x^2 − 7 = − 31 19. 5 x^2 + 33 = 3
Find the zeros of the function.
20. f ( x ) = x^2 + 7 21. f ( x ) = − x^2 − 4 22. f ( x ) = 9 x^2 + 1
ANALYZING
MATHEMATICAL
RELATIONSHIPS
Notice that you can use the solutions in Example 6(a) to factor x^2 + 4 as ( x + 2 i )( x − 2 i ).
FORMULATING A PLAN The graph of f does not intersect the x -axis, which means f has no real zeros. So, f must have complex zeros, which you can find algebraically.
x
y 30
10
− 4 − 2 2 4
00
f ( x ) = 4 x^2 + 20
Section 4.2 Complex Numbers 161
In Exercises 37–44, multiply. Write the answer in standard form. (See Example 5.)
37. 3 i (− 5 + i ) 38. 2 i (7 − i ) 39. (3 − 2 i )(4 + i ) 40. (7 + 5 i )(8 − 6 i ) 41. (4 − 2 i )(4 + 2 i ) 42. (9 + 5 i )(9 − 5 i ) 43. (3 − 6 i )^2 44. (8 + 3 i )^2
JUSTIFYING STEPS In Exercises 45 and 46, justify each step in performing the operation.
45. 11 − (4 + 3 i ) + 5 i
= [(11 − 4) − 3 i ] + 5 i
= (7 − 3 i ) + 5 i
= 7 + (− 3 + 5) i
= 7 + 2 i
46. (3 + 2 i )(7 − 4 i )
= 21 − 12 i + 14 i − 8 i^2
= 21 + 2 i − 8(−1)
= 21 + 2 i + 8
= 29 + 2 i
REASONING In Exercises 47 and 48, place the tiles in the expression to make a true statement.
47. (____ − ____ i ) – (____ − ____ i ) = 2 − 4 i
48. ____ i (____ + ____ i ) = − 18 − 10 i
In Exercises 49–54, solve the equation. Check your solution(s). (See Example 6.)
49. x^2 + 9 = 0 50. x^2 + 49 = 0 51. x^2 − 4 = − 11 52. x^2 − 9 = − 15 53. 2 x^2 + 6 = − 34 54. x^2 + 7 = − 47
In Exercises 55–62, find the zeros of the function. (See Example 7.)
55. f ( x ) = 3 x^2 + 6 56. g ( x ) = 7 x^2 + 21 57. h ( x ) = 2 x^2 + 72 58. k ( x ) = − 5 x^2 − 125 59. m ( x ) = − x^2 − 27 60. p ( x ) = x^2 + 98 61. r ( x ) = − —^12 x^2 − 24 62. f ( x ) = −^1 — 5 x^2 − 10
ERROR ANALYSIS In Exercises 63 and 64, describe and correct the error in performing the operation and writing the answer in standard form.
63.
65. NUMBER SENSE Simplify each expression. Then classify your results in the table below. a. (− 4 + 7 i ) + (− 4 − 7 i ) b. (2 − 6 i ) − (− 10 + 4 i ) c. (25 + 15 i ) − (25 − 6 i ) d. (5 + i )(8 − i ) e. (17 − 3 i ) + (− 17 − 6 i ) f. (− 1 + 2 i )(11 − i ) g. (7 + 5 i ) + (7 − 5 i ) h. (− 3 + 6 i ) − (− 3 − 8 i )
Real numbers
Imaginary numbers
Pure imaginary numbers
66. MAKING AN ARGUMENT The Product Property
states √— a ⋅√
— b = √
— ab. Your friend concludes √
—
— –9 = √
— 36 = 6. Is your friend correct? Explain.
(3 + 2 i )(5 − i ) = 15 − 3 i + 10 i − 2 i 2 = 15 + 7 i − 2 i 2 = − 2 i 2 + 7 i + 15
(4 + 6 i )^2 = (4) 2 + (6i )^2 = 16 + 36 i 2 = 16 + (36)( − 1) = − 20
162 Chapter 4 Quadratic Equations and Complex Numbers
67. FINDING A PATTERN Make a table that shows the powers of i from i^1 to i^8 in the fi rst row and the simplified forms of these powers in the second row. Describe the pattern you observe in the table. Verify the pattern continues by evaluating the next four powers of i. 68. HOW DO YOU SEE IT? The graphs of three functions are shown. Which function(s) has real zeros? imaginary zeros? Explain your reasoning.
x
h f g
y 4
2
− 4
− 4 4
In Exercises 69–74, write the expression as a complex number in standard form.
69. (3 + 4 i ) − (7 − 5 i ) + 2 i (9 + 12 i ) 70. 3 i (2 + 5 i ) + (6 − 7 i ) − (9 + i ) 71. (3 + 5 i )(2 − 7 i^4 ) 72. 2 i^3 (5 − 12 i )
73. (2 + 4 i^5 ) + (1 − 9 i^6 ) − (^3 + i^7 )
74. (8 − 2 i^4 ) + (3 − 7 i^8 ) − ( 4 + i^9 ) 75. OPEN-ENDED Find two imaginary numbers whose sum and product are real numbers. How are the imaginary numbers related? 76. COMPARING METHODS Describe the two different methods shown for writing the complex expression in standard form. Which method do you prefer? Explain.
Method 1 4 i (2 − 3 i ) + 4 i (1 − 2 i ) = 8 i − 12 i 2 + 4 i − 8 i 2 = 8 i − 12( − 1) + 4 i − 8( − 1) = 20 + 12 i
Method 2 4 i(2 − 3 i ) + 4 i (1 − 2 i ) = 4 i [(2 − 3 i ) + (1 − 2 i )] = 4 i [3 − 5 i ] = 12 i − 20 i 2 = 12 i − 20( − 1) = 20 + 12 i
77. CRITICAL THINKING Determine whether each statement is true or false****. If it is true, give an example. If it is false, give a counterexample. a. The sum of two imaginary numbers is an imaginary number. b. The product of two pure imaginary numbers is a real number. c. A pure imaginary number is an imaginary number. d. A complex number is a real number. 78. THOUGHT PROVOKING Create a circuit that has an impedance of 14 − 3 i.
Maintaining Mathematical ProficiencyMaintaining Mathematical Proficiency
Determine whether the given value of x is a solution to the equation. (Skills Review Handbook)
79. 3( x − 2) + 4 x − 1 = x − 1; x = 1 80. x^3 − 6 = 2 x^2 + 9 − 3 x ; x = − 5 81. − x^2 + 4 x = (^19) — 3 x^2 ; x = − —^34
Write an equation in vertex form of the parabola whose graph is shown. (Section 3.4) 82.
x
y 6
2
2 4 6
(0, 3)
(1, 2)
x
y 4
2
− 2
2
(−1, 5)
(−3, −3)
x
y
− 4
− 2
− 2 4
(2, −1)
(3, −2)
Reviewing what you learned in previous grades and lessons
