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Complex Numbers

2. Complex Numbers. ALGEBRA 2 LESSON 5-6. 1. Simplify – –169. 2. Find |–1 + i |. –13 i. 5-6. Pages 274–276 Exercises. 1. 2 i 2. i 7 3. i 15 4. 9 i 5. 5 i 2 6. 4 i 7. 4 i 2 8. 9 i . 9. –10 i 10. 6 i 2 11. 2 + i 3

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Complex Numbers

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  1. 2 Complex Numbers ALGEBRA 2 LESSON 5-6 1. Simplify – –169. 2. Find |–1 + i| –13i 5-6

  2. Pages 274–276 Exercises 1. 2i 2. i 7 3. i 15 4. 9i 5. 5i 2 6. 4i 7. 4i 2 8. 9i 9. –10i 10. 6i 2 11. 2 + i 3 12. 8 + 2i 2 13. 6 – 2i 7 14. 3 + 2i 15. 7 – 5i 16. 2 + i 17. –2 – 5i 2 18. 4 + 6i 2 19. 2 20. 13 21. 2 2 22. 17 23. 3 5 24. –4i Complex Numbers ALGEBRA 2 LESSON 5-6 5-6

  3. 25. –5 + 3i 26. –9 – i 27. 3 + 2i 28. 4 – 7i 29. 6 + 3i 30. 1 – 7i 31. 7 + 4i 32. –2 – 3i 33. 10 + 6i 34. –7 – 10i 35. 10 36. 26 – 7i 37. 9 + 58i 38. 9 – 23i 39. –36 40. 65 + 72i 41. ± 5i 42. ± 43. ± 44. ± i 7 45. ±6i 46. ± 47. –i, –1 – i, i 48. –2i, –4 – 2i, 12 + 14i 49. 1 – i, 1 – 3i, –7 – 7i 50. ± i 65 51. ±7i 8i 3 3 i 15 3 i 2 2 Complex Numbers ALGEBRA 2 LESSON 5-6 5-6

  4. 52. ± i 53. No; the test scores were real numbers. He added the scores and divided by the number of scores. The set of real numbers is closed with respect to addition and division so he should have gotten a real number. 54. a.A: –5, B: 3 + 2i, C: 2 – i, D: 3i, E: –6 – 4i, F: –1 + 5i b. 5, –3 – 2i, –2 + i, –3i, 6 + 4i, 1 – 5i 55. a. Check students’ work. b. a circle with radius 10 and center at the origin 56. –5, 5 57. 288i 58. –1 + 5i 59. 10 – 4i 60. 8 – 2i 61. 11 – 5i Complex Numbers ALGEBRA 2 LESSON 5-6 5-6

  5. Complex Numbers • You can apply all the operations of real numbers to complex numbers • Addition/ Additive inverse • Multiplication

  6. Complex Numbers ALGEBRA 2 LESSON 5-6 Find the additive inverse of –7 – 9i. –7 – 9i –(–7 – 9i) Find the opposite. 7 + 9iSimplify. 5-6

  7. Complex Numbers ALGEBRA 2 LESSON 5-6 Simplify (3 + 6i) – (4 – 8i). (3 + 6i) – (4 – 8i) = 3 + (–4) + 6i + 8i Use commutative and associative properties. = –1 + 14i Simplify. 5-6

  8. Complex Numbers ALGEBRA 2 LESSON 5-6 Find each product. a. (3i)(8i) (3i)(8i) = 24i2Multiply the real numbers. = 24(–1)Substitute –1 for i2. = –24 Multiply. b. (3 – 7i)(2 – 4i) (3 – 7i)(2 – 4i) = 6 – 14i – 12i + 28i2 Multiply the binomials. = 6 – 26i + 28(–1) Substitute –1 for i2. = –22 – 26iSimplify. 5-6

  9. x= ±i 6 Find the square root of each side. Check: 9x2 + 54 = 0 9x2 + 54 = 0 9(i 6)2 + 54 0 9(i(– 6))2 + 54 0 9(6)i2 + 54 0 9(6i2) + 54 0 54(–1) –54 54(–1) –54 54 = 54 –54 = –54 Complex Numbers ALGEBRA 2 LESSON 5-6 Solve 9x2 + 54 = 0. 9x2 + 54 = 0 9x2 = –54 Isolate x2. x2 = –6 5-6

  10. Complex Numbers • Functions of the form f(z) = z2 + c where c is a complex number generate fractal graphs on the complex plane like the one at the left. To test if z belongs to the graph, use z as the first input value and repeatedly use each output as the next input. If the output values do not become infinitely large, then z is on the graph. • f(z) = z2 + a+bi

  11. Use z = 0 as the first input value. f(0) = 02 – 4i = –4i f(–4i)= (–4i)2 – 4i First output becomes second input. Evaluate for z = –4i. = –16 – 4i = [(–16)2 + (–16)(–4i) + (–16)(–4i) + (–4i)2] – 4i = (256 + 128i – 16) – 4i = 240 + 124i Complex Numbers ALGEBRA 2 LESSON 5-6 Find the first three output values for f(z) = z2 – 4i. Use z = 0 as the first input value. f(–16 – 4i)= (–16 – 4i)2 – 4i Second output becomes third input. Evaluate for z = –16 – 4i. The first three output values are –4i, –16 – 4i, 240 + 124i. 5-6

  12. Assignment 43 • Page 274 34-48 even, 56-64 even

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