Complex Numbers: Classifying, Adding, and Multiplying, Study notes of Algebra

Download Complex Numbers: Classifying, Adding, and Multiplying and more Study notes Algebra in PDF only on Docsity! 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. x2 − 4 = 0 b. x2 + 1 = 0 c. x2 − 1 = 0 d. x2 + 4 = 0 e. x2 − 9 = 0 f. x2 + 9 = 0 A. i B. 3i C. 3 D. 2i E. 1 F. 2 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. 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 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 Irrational NumbersRational Numbers Integers Whole Numbers Natural Numbers The imaginary unit i is defi ned as i = √ — −1. Complex Numbers4.2 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 = 5i b. √ — −72 = √ — 72 ⋅ √ — −1 = √ — 36 ⋅ √ — 2 ⋅ i = 6 √ — 2 i = 6i √ — 2 c. −5 √ — −9 = −5 √ — 9 ⋅ √ — −1 = −5 ⋅ 3 ⋅ i = −15i Monitoring Progress Help in English and Spanish at BigIdeasMath.com 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 Learn Defi ne 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, x2 = −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 Vocabulary 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 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 + 0i) Imaginary Numbers (a + bi, b ≠ 0) Pure Imaginary Numbers (0 + bi, b ≠ 0) −1 2 + 3i 9 − 5i 6i−4i 5 3 π 2 Section 4.2 Complex Numbers 159 Complex Solutions and Zeros Solving Quadratic Equations Solve (a) x2 + 4 = 0 and (b) 2x2 − 11 = −47. SOLUTION a. x2 + 4 = 0 Write original equation. x2 = −4 Subtract 4 from each side. x = ± √ — −4 Take square root of each side. x = ±2i Write in terms of i. The solutions are 2i and −2i. b. 2x2 − 11 = −47 Write original equation. 2x2 = −36 Add 11 to each side. x2 = −18 Divide each side by 2. x = ± √ — −18 Take square root of each side. x = ±i √ — 18 Write in terms of i. x = ±3i √ — 2 Simplify radical. The solutions are 3i √ — 2 and −3i √ — 2. Finding Zeros of a Quadratic Function Find the zeros of f (x) = 4x2 + 20. SOLUTION 4x2 + 20 = 0 Set f (x) equal to 0. 4x2 = −20 Subtract 20 from each side. x2 = −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 ) 2 + 20 = 4 ⋅ 5i 2 + 20 = 4(−5) + 20 = 0 ✓ f ( −i √ — 5 ) = 4 ( −i √ — 5 ) 2 + 20 = 4 ⋅ 5i 2 + 20 = 4(−5) + 20 = 0 ✓ Monitoring Progress Help in English and Spanish at BigIdeasMath.com Solve the equation. 14. x2 = −13 15. x2 = −38 16. x2 + 11 = 3 17. x2 − 8 = −36 18. 3x2 − 7 = −31 19. 5x2 + 33 = 3 Find the zeros of the function. 20. f (x) = x2 + 7 21. f (x) = −x2 − 4 22. f (x) = 9x2 + 1 ANALYZING MATHEMATICAL RELATIONSHIPS Notice that you can use the solutions in Example 6(a) to factor x2 + 4 as (x + 2i )(x − 2i ). 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 fi nd algebraically. x y 30 10 2 4−4 −2 f(x) = 4x2 + 20 160 Chapter 4 Quadratic Equations and Complex Numbers Exercises Dynamic Solutions available at BigIdeasMath.com4.2 In Exercises 5–12, fi nd the square root of the number. (See Example 1.) 5. √ — −36 6. √ — −64 7. √ — −18 8. √ — −24 9. 2 √ — −16 10. −3 √ — −49 11. −4 √ — −32 12. 6 √ — −63 In Exercises 13–20, fi nd the values of x and y that satisfy the equation. (See Example 2.) 13. 4x + 2i = 8 + yi 14. 3x + 6i = 27 + yi 15. −10x + 12i = 20 + 3yi 16. 9x − 18i = −36 + 6yi 17. 2x − yi = 14 + 12i 18. −12x + yi = 60 − 13i 19. 54 − 1 — 7 yi = 9x − 4i 20. 15 − 3yi = 1 — 2 x + 2i In Exercises 21–30, add or subtract. Write the answer in standard form. (See Example 3.) 21. (6 − i ) + (7 + 3i ) 22. (9 + 5i ) + (11 + 2i ) 23. (12 + 4i ) − (3 − 7i ) 24. (2 − 15i ) − (4 + 5i ) 25. (12 − 3i ) + (7 + 3i ) 26. (16 − 9i ) − (2 − 9i ) 27. 7 − (3 + 4i ) + 6i 28. 16 − (2 − 3i ) − i 29. −10 + (6 − 5i ) − 9i 30. −3 + (8 + 2i ) + 7i 31. USING STRUCTURE Write each expression as a complex number in standard form. a. √ — −9 + √ — −4 − √ — 16 b. √ — −16 + √ — 8 + √ — −36 32. REASONING The additive inverse of a complex number z is a complex number za such that z + za = 0. Find the additive inverse of each complex number. a. z = 1 + i b. z = 3 − i c. z = −2 + 8i In Exercises 33–36, fi nd the impedance of the series circuit. (See Example 4.) 33. 9Ω 12Ω 7Ω 34. 6Ω 4Ω 9Ω 35. 8Ω 3Ω 2Ω 36. 14Ω 7Ω 8Ω Monitoring Progress and Modeling with Mathematics Vocabulary and Core Concept Check 1. VOCABULARY What is the imaginary unit i defi ned as and how can you use i? 2. COMPLETE THE SENTENCE For the complex number 5 + 2i, 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 2 + 5i √ — 3 + 6i 0 − 7i Section 4.2 Complex Numbers 161 In Exercises 37–44, multiply. Write the answer in standard form. (See Example 5.) 37. 3i(−5 + i ) 38. 2i(7 − i ) 39. (3 − 2i )(4 + i ) 40. (7 + 5i )(8 − 6i ) 41. (4 − 2i )(4 + 2i ) 42. (9 + 5i )(9 − 5i ) 43. (3 − 6i )2 44. (8 + 3i )2 JUSTIFYING STEPS In Exercises 45 and 46, justify each step in performing the operation. 45. 11 − (4 + 3i ) + 5i = [(11 − 4) − 3i ] + 5i = (7 − 3i ) + 5i = 7 + (−3 + 5)i = 7 + 2i 46. (3 + 2i )(7 − 4i ) = 21 − 12i + 14i − 8i 2 = 21 + 2i − 8(−1) = 21 + 2i + 8 = 29 + 2i REASONING In Exercises 47 and 48, place the tiles in the expression to make a true statement. 47. (____ − ____i ) – (____ − ____i ) = 2 − 4i 7 4 3 6 48. ____i(____ + ____i ) = −18 − 10i −5 9 2 In Exercises 49–54, solve the equation. Check your solution(s). (See Example 6.) 49. x2 + 9 = 0 50. x2 + 49 = 0 51. x2 − 4 = −11 52. x2 − 9 = −15 53. 2x2 + 6 = −34 54. x2 + 7 = −47 In Exercises 55–62, fi nd the zeros of the function. (See Example 7.) 55. f (x) = 3x2 + 6 56. g(x) = 7x2 + 21 57. h(x) = 2x2 + 72 58. k(x) = −5x2 − 125 59. m(x) = −x2 − 27 60. p(x) = x2 + 98 61. r (x) = − 1 — 2x2 − 24 62. f (x) = − 1 — 5x2 − 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. 64. 65. NUMBER SENSE Simplify each expression. Then classify your results in the table below. a. (−4 + 7i ) + (−4 − 7i ) b. (2 − 6i ) − (−10 + 4i ) c. (25 + 15i ) − (25 − 6i ) d. (5 + i )(8 − i ) e. (17 − 3i ) + (−17 − 6i ) f. (−1 + 2i )(11 − i ) g. (7 + 5i ) + (7 − 5i ) h. (−3 + 6i ) − (−3 − 8i ) Real numbers Imaginary numbers Pure imaginary numbers 66. MAKING AN ARGUMENT The Product Property states √ — a ⋅ √ — b = √ — ab. Your friend concludes √ — −4 ⋅ √ — –9 = √ — 36 = 6. Is your friend correct? Explain. (3 + 2i )(5 − i ) = 15 − 3i + 10i − 2i 2 = 15 + 7i − 2i 2 = −2i 2 + 7i + 15 ✗ (4 + 6i )2 = (4)2 + (6i )2 = 16 + 36i 2 = 16 + (36)(−1) = −20 ✗