3.4 Complex Numbers Essential 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 The imaginary unit i is defined as i = 1. Natural Numbers Classifying Numbers Work with a partner. Determine which subsets of the set of complex numbers contain each number. ATTENDING TO PRECISION To be proficient in math, you need to use clear definitions in your reasoning and discussions with others. a. 9 b. 0 c. 4 d. 4 9 e. f. 1 Complex Solutions of Quadratic Equations Work with a partner. Use the definition of the imaginary unit i to match each quadratic equation with its complex solution. Justify your answers. a. x 4 = 0 b. x + 1 = 0 c. x 1 = 0 d. x + 4 = 0 e. x 9 = 0 f. x + 9 = 0 A. i B. 3i C. 3 D. i E. 1 F. Communicate 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. Section 3.4 Complex Numbers 15
3.4 Lesson What You Will Learn Core Vocabulary imaginary unit i, p. 16 complex number, p. 17 imaginary number, p. 17 pure imaginary number, p. 17 complex conjugates, p. 19 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 = 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, defined as i = 1. Note that i = 1. The imaginary unit i can be used to write the square root of any negative number. Core Concept The Square Root of a Negative Number Property Example 1. If r is a positive real number, then r = i r. 3 = i 3. By the first property, it follows that ( i r ) = r. ( i 3 ) = i 3 = 3 Finding Square Roots of Negative Numbers Evaluate each expression. a. 5 b. 7 c. 5 9 a. 5 = 5 1 = 5i b. 7 = 7 1 = 36 i = 6 i = 6i c. 5 9 = 5 9 1 = 5 3 i = 15i Monitoring Progress Evaluate the expression. 1. 4. 1 3. 36 4. 54 Remember i = 1. So, i 3 = i i = 1 i = i and i 4 = i 3 i = i i = i = 1. Continuing on with the powers of i shows a repeating pattern. Power of i i 0 i 1 i i 3 i 4 i 5 i 6 i 7 i 8 i 9 i 10 i 11 Value 1 i 1 i 1 i 1 i 1 i 1 i Because i 4 = 1, the value of i n is determined by the remainder of n 4. 16 Chapter 3 Quadratic Equations and Complex Numbers
Core Concept Evaluating a Power of i To evaluate a power of i, divide the exponent by four. When the remainder is 0, the power equals i 0 = 1. When the remainder is 1, the power equals i 1 = i. When the remainder is, the power equals i = 1. When the remainder is 3, the power equals i 3 = i. Evaluate i 7. Evaluating a Power of i STUDY TIP When the exponent has more than two digits, you only need to divide the last two digits by four to find the remainder. Divide the exponent by four and find the remainder. 7 4 = 6 R 3 The remainder is 3, so i 7 = i. Monitoring Progress Evaluate the power of i. 5. i 34 6. i 47 7. i 65 8. i 1040 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. Complex Numbers (a + bi) Real Numbers (a + 0i) 1 5 3 π Imaginary Numbers (a + bi, b 0) + 3i 9 5i Pure Imaginary Numbers (0 + bi, b 0) 4i 6i Operations with Complex Numbers Core Concept Sums and Differences of Complex Numbers To add (or subtract) two complex numbers, add (or subtract) their real parts and their imaginary parts separately. Sum of complex numbers: Difference of complex numbers: (a + bi) + (c + di) = (a + c) + (b + d)i (a + bi) (c + di) = (a c) + (b d)i Section 3.4 Complex Numbers 17
Adding and Subtracting Complex Numbers Add or subtract. Write the answer in standard form. a. (8 i ) + (5 + 4i ) b. (7 6i) (3 6i) c. 13 ( + 7i) + 5i a. (8 i ) + (5 + 4i ) = (8 + 5) + ( 1 + 4)i Definition of complex addition = 13 + 3i Write in standard form. b. (7 6i ) (3 6i ) = (7 3) + ( 6 + 6)i Definition of complex subtraction = 4 + 0i Simplify. = 4 Write in standard form. c. 13 ( + 7i ) + 5i = [(13 ) 7i] + 5i Definition of complex subtraction = (11 7i ) + 5i Simplify. = 11 + ( 7 + 5)i Definition of complex addition = 11 i Write in standard form. 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 STUDY TIP When simplifying an expression that involves complex numbers, be sure to simplify i as 1. Multiply. Write the answer in standard form. a. 4i( 6 + i ) b. (9 i )( 4 + 7i ) a. 4i( 6 + i ) = 4i + 4i Distributive Property = 4i + 4( 1) Use i = 1. = 4 4i Write in standard form. b. (9 i )( 4 + 7i ) = 36 + 63i + 8i 14i Multiply using FOIL. = 36 + 71i 14( 1) Simplify and use i = 1. = 36 + 71i + 14 Simplify. = + 71i Write in standard form. Monitoring Progress Perform the operation. Write the answer in standard form. 9. (9 i ) + ( 6 + 7i ) 10. (3 + 7i ) (8 i ) 11. 4 (1 + i ) (5 + 9i ) 1. ( 3i)(10i) 13. i(8 i ) 14. (3 + i )(5 i ) 18 Chapter 3 Quadratic Equations and Complex Numbers
Two complex numbers of the form a + bi and a bi where b 0, are called complex conjugates. The product of complex conjugates is always a real number. For example, (3 + i)(3 i) = 9 6i + 6i + 4 = 13. You can use this fact to write the quotient of two complex numbers in standard form by multiplying the numerator and denominator by the conjugate of the denominator. Write each quotient in standard form. Dividing Complex Numbers a. 4 + 3i 1 i b. 1 4i + 3i c. + 5i 4i ANOTHER WAY When the denominator does not contain a real term, multiply the numerator and denominator by i. So, + 5i 4i = + 5i 4i i i i +5i = 4i i + 5( 1) = 4( 1) 5 + i = 4 = 5 4 1 i. a. 4 + 3i 1 i = 4 + 3i 1 i 1 + i 1 + i 4 + 8i + 3i + 6i = Multiply using FOIL. 1 + i i 4i 4 + 11i + 6( 1) = 1 4( 1) Multiply numerator and denominator by 1 + i, the complex conjugate of 1 i. Simplify and use i = 1. + 11i = 5 Simplify. = 5 + 11 i 5 Write in standard form. b. 1 4i + 3i = 1 4i + 3i 3i 3i c. + 5i 4i 3i 8i + 1i = Multiply using FOIL. 4 6i + 6i 9i 11i + 1( 1) = 4 9( 1) Multiply numerator and denominator by 3i, the complex conjugate of + 3i. Simplify and use i = 1. 10 11i = 13 Simplify. = 10 13 11 i 13 Write in standard form. = + 5i 4i = 4i 4i 8i 0i 16i 8i 0( 1) = 16( 1) 0 8i = 16 Multiply numerator and denominator by 4i, the complex conjugate of 4i. Multiply using FOIL. Simplify and use i = 1. Simplify. = 5 4 1 i Write in standard form. Monitoring Progress Write the quotient in standard form. 15. 3i 4 + i 16. 7 4 + 6i 17. 6 + 5i 3i Section 3.4 Complex Numbers 19
Complex Solutions and Zeros Solving Quadratic Equations Solve (a) x + 4 = 0 and (b) x 11 = 47. LOOKING FOR STRUCTURE Notice that you can use the solutions in Example 6(a) to factor x + 4 as (x + i )(x i ). a. x + 4 = 0 Write original equation. x = 4 Subtract 4 from each side. x = ± 4 Take square root of each side. x = ±i Write in terms of i. The solutions are i and i. b. x 11 = 47 Write original equation. x = 36 Add 11 to each side. x = 18 Divide each side by. x = ± 18 Take square root of each side. x = ±i 18 Write in terms of i. x = ±3i Simplify radical. The solutions are 3i and 3i. Finding Zeros of a Quadratic Function FINDING AN ENTRY POINT 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. 40 30 10 y Find the zeros of f (x) = 4x + 0. 4x + 0 = 0 Set f (x) equal to 0. 4x = 0 Subtract 0 from each side. x = 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 5. Check f ( i 5 ) = 4 ( i 5 ) + 0 = 4 5i + 0 = 4( 5) + 0 = 0 f ( i 5 ) = 4 ( i 5 ) + 0 = 4 5i + 0 = 4( 5) + 0 = 0 4 4 x Monitoring Progress Solve the equation. 18. x = 13 19. x = 38 0. x + 11 = 3 1. x 8 = 36. 3x 7 = 31 3. 5x + 33 = 3 Find the zeros of the function. 4. f (x) = x + 7 5. f (x) = x 4 6. f (x) = 9x + 1 130 Chapter 3 Quadratic Equations and Complex Numbers
3.4 Exercises Dynamic Solutions available at BigIdeasMath.com Vocabulary and Core Concept Check 1. VOCABULARY What is the imaginary unit i defined as and how can you use i?. COMPLETE THE SENTENCE For the complex number 5 + i the imaginary part is and the real part is. 3. WRITING Describe how to add complex numbers. 4. WHICH ONE DOESN T BELONG? Which number does not belong with the other three? Explain your reasoning. 3 + 0i + 5i 3 + 6i 0 7i Monitoring Progress and Modeling with Mathematics In Exercises 5 18, evaluate the expression. (See Example 1.) 5. 36 6. 64 7. 4 8. 16 9. 5 10. 13 11. 18 1. 4 13. 16 14. 3 49 15. 3 7 16. 4 11 17. 4 3 18. 6 63 In Exercises 19 30, evaluate the power of i. (See Example.) 19. i 44 0. i 94 1. i 57. i 115 3. i 79 4. i 340 5. i 1013 6. i 871 7. i 4000 8. i 8000 In Exercises 31 40, add or subtract. Write the answer in standard form. (See Example 3.) 31. (6 i ) + (7 + 3i ) 3. (9 + 5i ) + (11 + i ) 33. (1 + 4i ) (3 7i ) 34. ( 15i ) (4 + 5i ) 35. (1 3i ) + (7 + 3i ) 36. (16 9i ) ( 9i ) 37. 7 (3 + 4i ) + 6i 38. 16 ( 3i ) i 39. 10 + (6 5i ) 9i 40. 3 + (8 + i ) + 7i 41. USING STRUCTURE Write each expression as a complex number in standard form. a. 9 + 4 16 b. 5 9 64 c. 16 + 8 + 36 9. i 4001 30. i 800 Section 3.4 Complex Numbers 131
4. REASONING The additive inverse of a complex number z is a complex number z a such that z + z a = 0. Find the additive inverse of each complex number. a. z = 1 + i b. z = 3 i c. z = + 8i In Exercises 43 50, multiply. Write the answer in standard form. (See Example 4.) 43. 3i( 5 + i ) 44. i(7 i ) 45. (3 i )(4 + i ) 46. (7 + 5i )(8 6i ) 47. (4 i )(4 + i ) 48. (9 + 5i )(9 5i ) 49. (3 6i ) 50. (8 + 3i ) JUSTIFYING STEPS In Exercises 51 and 5, justify each step in performing the operation. 51. 11 (4 + 3i ) + 5i = [(11 4) 3i ] + 5i = (7 3i ) + 5i = 7 + ( 3 + 5)i = 7 + i 5. (3 + i )(7 4i ) = 1 1i + 14i 8i = 1 + i 8( 1) = 1 + i + 8 = 9 + i REASONING In Exercises 53 56, place the tiles in the expression to make a true statement. 53. ( i ) ( i ) = 4i 7 4 3 6 54. ( i ) ( 3 + 4i ) = i 1 9 7 11 55. i( + i ) = 18 10i 5 9 56. i( 5i ) = 4i 4 0 6 In Exercises 57 6, write the complex conjugate of the number. 57. 1 + i 58. 1 + i 59. 5 i 60. 6 i 61. 10 + 10i 6. 6 6i In Exercises 63 70, write the quotient in standard form. (See Example 5.) 5i + 3i 63. 64. 3 4i 7 + 4i 65. 67. 69. 1 + 7i 9 i 9 8i 1 + i 1 + 3i i 66. 68. 70. 5 + i 5i + i 3 + 4i + 3i 4i In Exercises 71 76, solve the equation. Check your solution(s). (See Example 6.) 71. x + 9 = 0 7. x + 49 = 0 73. x 4 = 11 74. x 9 = 15 75. x + 6 = 34 76. x + 7 = 47 13 Chapter 3 Quadratic Equations and Complex Numbers
In Exercises 77 84, find the zeros of the function. (See Example 7.) 77. f (x) = 3x + 6 78. g(x) = 7x + 1 79. h(x) = x + 7 80. k(x) = 5x 15 81. m(x) = x 7 8. p(x) = x + 98 1 83. r (x) = x 4 84. f (x) = 5 x 10 ERROR ANALYSIS In Exercises 85 and 86, describe and correct the error in performing the operation and writing the answer in standard form. 85. 86. 87. NUMBER SENSE Simplify each expression. Then classify your results in the table below. a. ( 4 + 7i ) + ( 4 7i ) b. ( 6i ) ( 10 + 4i ) c. (5 + 15i ) (5 6i ) d. (5 + i )(8 i ) e. (17 3i ) + ( 17 6i ) f. ( 1 + i )(11 i ) g. (7 + 5i ) + (7 5i ) h. ( 3 + 6i ) ( 3 8i ) Real numbers (3 + i )(5 i ) = 15 3i + 10i i Imaginary numbers = 15 + 7i i = i + 7i + 15 (4 + 6i ) = (4) + (6i ) 1 = 16 + 36i = 16 + (36)( 1) = 0 Pure imaginary numbers 88. OPEN-ENDED Find two imaginary numbers whose sum and product are real numbers. How are the imaginary numbers related? In Exercises 89 94, describe the real-number values of x and y for which the sum or difference is (a) a real number, and (b) a pure imaginary number. 89. (x + yi) + (4 + 6i) 90. ( 7x + xi) ( y 4i) 91. ( x 8i) (5 + yi) 9. ( x + yi) + ( 1) 93. x ( yi 3) 94. xi + (i y) 95. REASONING Without performing the calculations, determine whether the product is a real number. Explain your reasoning. (111 3i)( + 46i) 96. HOW DO YOU SEE IT? The graphs of three functions are shown. Which function(s) has real zeros? imaginary zeros? Explain your reasoning. 4 4 4 y h f g 4 x 97. REASONING For what integers a and b is the product a positive real number? Explain. (a bi)(a + bi) 98. MAKING AN ARGUMENT The Product Property of Square Roots states a b = ab. Your friend concludes 4 9 = 36 = 6. Is your friend correct? Explain. Section 3.4 Complex Numbers 133
In Exercises 99 104, write the expression as a complex number in standard form. 99. (3 + 4i ) (7 5i ) + i(9 + 1i ) 100. 3i( + 5i ) + (6 7i ) (9 + i ) 101. (3 + 5i )( 7i 4 ) 10. i 3 (5 1i ) 103. ( + 4i 5 ) + (1 9i 6 ) ( 3 + i 7 ) 104. (8 i 4 ) + (3 7i 8 ) (4 + i 9 ) 105. COMPARING METHODS Describe the two different methods shown for writing the complex expression in standard form. Which method do you prefer? Explain. Method 1 4i ( 3i ) + 4i (1 i ) = 8i 1i + 4i 8i = 8i 1( 1) + 4i 8( 1) = 0 + 1i 106. THOUGHT PROVOKING Opposition to the flow of current in an electrical circuit is called resistance for resistors and reactance for inductors and capacitors. Each of these quantities is measured in ohms (Ω). The table shows the relationship between a component s resistance or reactance and its contribution to impedance. The impedance for a series circuit is the sum of the impedances for the individual components. Component and symbol Resistance or reactance (ohms) Resistor Inductor Capacitor R L C Impedance (ohms) R Li Ci A series circuit is shown with the resistance or reactance of each component labeled. Find the impedance of the circuit. Then draw a circuit that has an impedance of 14 3i ohms. 3Ω 5Ω 4Ω Method 4i( 3i ) + 4i (1 i ) = 4i [( 3i ) + (1 i )] = 4i [3 5i ] = 1i 0i = 1i 0( 1) = 0 + 1i Maintaining Mathematical Proficiency Determine whether the given value of x is a solution of the equation. Alternating current source 107. 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. 108. 3(x ) + 4x 1 = x 1; x = 1 109. x 3 6 = x + 9 3x; x = 5 110. x + 4x = 19 3 x ; x = 4 Write an equation of the parabola in vertex form. Reviewing what you learned in previous grades and lessons 3 111. 6 y (0, 3) 11. ( 1, 5) 4 y 113. y 4 6 x (3, ) (1, ) 6 1 x 4 (, 1) 4 x ( 3, 3) 134 Chapter 3 Quadratic Equations and Complex Numbers