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# Powers and Roots of Complex numbers

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### Powers and Roots of Complex numbers

1. 1. Operations on Complex Numbers Mathematics 4 November 29, 2011Mathematics 4 () Operations on Complex Numbers November 29, 2011 1 / 18
2. 2. Review of Multiplication of Complex NumbersFind the product of 4 + 4i and −2 − 3i1. Multiply Algebraically (4 + 4i)(−2 − 3i) = −8 − 12i − 8i − 12i2 = −8 − 20i + 12 = 4 − 20i Mathematics 4 () Operations on Complex Numbers November 29, 2011 2 / 18
3. 3. Review of Multiplication of Complex NumbersFind the product of 4 + 4i and −2 − 3i2. Multiply in their polar forms √ √ (4 + 4i)(−2 − 3i) = (4 2 cis 45o ) · ( 13 cis 236.31o ) √ √ = (4 2 · 13) cis(45 + 236.31)o √ = 4 26 cis 281.31o = 4 − 20i Mathematics 4 () Operations on Complex Numbers November 29, 2011 3 / 18
4. 4. Review of Multiplication of Complex NumbersRule for Multiplication of Complex Numbers in Polar FormGiven: z1 = r1 cis α z2 = r2 cis β Mathematics 4 () Operations on Complex Numbers November 29, 2011 4 / 18
5. 5. Review of Multiplication of Complex NumbersRule for Multiplication of Complex Numbers in Polar FormGiven: z1 = r1 cis α z2 = r2 cis β z1 · z2 = (r1 · r2 ) cis(α + β) Mathematics 4 () Operations on Complex Numbers November 29, 2011 4 / 18
6. 6. Raising Complex Numbers to a PowerGiven: z = r cis θ z0 = Mathematics 4 () Operations on Complex Numbers November 29, 2011 5 / 18
7. 7. Raising Complex Numbers to a PowerGiven: z = r cis θ z0 = 1 Mathematics 4 () Operations on Complex Numbers November 29, 2011 5 / 18
8. 8. Raising Complex Numbers to a PowerGiven: z = r cis θ z0 = 1 z1 = Mathematics 4 () Operations on Complex Numbers November 29, 2011 5 / 18
9. 9. Raising Complex Numbers to a PowerGiven: z = r cis θ z0 = 1 z 1 = r cis θ Mathematics 4 () Operations on Complex Numbers November 29, 2011 5 / 18
10. 10. Raising Complex Numbers to a PowerGiven: z = r cis θ z0 = 1 z 1 = r cis θ z2 = Mathematics 4 () Operations on Complex Numbers November 29, 2011 5 / 18
11. 11. Raising Complex Numbers to a PowerGiven: z = r cis θ z0 = 1 z 1 = r cis θ z 2 = (r cis θ) · (r cis θ) Mathematics 4 () Operations on Complex Numbers November 29, 2011 5 / 18
12. 12. Raising Complex Numbers to a PowerGiven: z = r cis θ z0 = 1 z 1 = r cis θ z 2 = (r cis θ) · (r cis θ) = r2 cis 2θ Mathematics 4 () Operations on Complex Numbers November 29, 2011 5 / 18
13. 13. Raising Complex Numbers to a PowerGiven: z = r cis θ z0 =1 z1 = r cis θ z2 = (r cis θ) · (r cis θ) = r2 cis 2θ z3 = Mathematics 4 () Operations on Complex Numbers November 29, 2011 5 / 18
14. 14. Raising Complex Numbers to a PowerGiven: z = r cis θ z0 = 1 z1 = r cis θ z2 = (r cis θ) · (r cis θ) = r2 cis 2θ z3 = (r2 cis 2θ) · (r cis θ) Mathematics 4 () Operations on Complex Numbers November 29, 2011 5 / 18
15. 15. Raising Complex Numbers to a PowerGiven: z = r cis θ z0 = 1 z1 = r cis θ z2 = (r cis θ) · (r cis θ) = r2 cis 2θ z3 = (r2 cis 2θ) · (r cis θ) = r3 cis 3θ Mathematics 4 () Operations on Complex Numbers November 29, 2011 5 / 18
16. 16. Raising Complex Numbers to a PowerGiven: z = r cis θ z0 = 1 z1 = r cis θ z2 = (r cis θ) · (r cis θ) = r2 cis 2θ z3 = (r2 cis 2θ) · (r cis θ) = r3 cis 3θ Mathematics 4 () Operations on Complex Numbers November 29, 2011 5 / 18
17. 17. Raising Complex Numbers to a PowerDe Moivre’s Theorem (r cis θ)n = rn cis(n · θ) Mathematics 4 () Operations on Complex Numbers November 29, 2011 6 / 18
18. 18. De Moivre’s Theorem √Example 1: Find ( 2 cis 20o )10 √ √ ( 2 cis 20o )10 = ( 2)10 cis(10 · 20)o Mathematics 4 () Operations on Complex Numbers November 29, 2011 7 / 18
19. 19. De Moivre’s Theorem √Example 1: Find ( 2 cis 20o )10 √ √ ( 2 cis 20o )10 = ( 2)10 cis(10 · 20)o √ ( 2 cis 20o )10 = 32 cis 200o Mathematics 4 () Operations on Complex Numbers November 29, 2011 7 / 18
20. 20. De Moivre’s√TheoremExample 1: Find ( 2 cis 20o )10 √ ( 2 cis 20o )0 = 1 cis 0 Mathematics 4 () Operations on Complex Numbers November 29, 2011 8 / 18
21. 21. De Moivre’s√TheoremExample 1: Find ( 2 cis 20o )10 √ √ ( 2 cis 20o )1 = 2 cis 20o Mathematics 4 () Operations on Complex Numbers November 29, 2011 8 / 18
22. 22. De Moivre’s√TheoremExample 1: Find ( 2 cis 20o )10 √ ( 2 cis 20o )2 = 2 cis 400 Mathematics 4 () Operations on Complex Numbers November 29, 2011 8 / 18
23. 23. De Moivre’s√TheoremExample 1: Find ( 2 cis 20o )10 √ √ ( 2 cis 20o )3 = 2 2 cis 60o Mathematics 4 () Operations on Complex Numbers November 29, 2011 8 / 18
24. 24. De Moivre’s√TheoremExample 1: Find ( 2 cis 20o )10 √ ( 2 cis 20o )4 = 4 cis 80o Mathematics 4 () Operations on Complex Numbers November 29, 2011 8 / 18
25. 25. De Moivre’s√TheoremExample 1: Find ( 2 cis 20o )10 √ √ ( 2 cis 20o )5 = 4 2 cis 100o Mathematics 4 () Operations on Complex Numbers November 29, 2011 8 / 18
26. 26. De Moivre’s√TheoremExample 1: Find ( 2 cis 20o )10 √ ( 2 cis 20o )6 = 8 cis 120o Mathematics 4 () Operations on Complex Numbers November 29, 2011 8 / 18
27. 27. De Moivre’s√TheoremExample 1: Find ( 2 cis 20o )10 √ √ ( 2 cis 20o )7 = 8 2 cis 140o Mathematics 4 () Operations on Complex Numbers November 29, 2011 8 / 18
28. 28. De Moivre’s√TheoremExample 1: Find ( 2 cis 20o )10 √ ( 2 cis 20o )8 = 16 cis 160o Mathematics 4 () Operations on Complex Numbers November 29, 2011 8 / 18
29. 29. De Moivre’s√TheoremExample 1: Find ( 2 cis 20o )10 √ √ ( 2 cis 20o )9 = 16 2 cis 180o Mathematics 4 () Operations on Complex Numbers November 29, 2011 8 / 18
30. 30. De Moivre’s√TheoremExample 1: Find ( 2 cis 20o )10 √ ( 2 cis 20o )10 = 32 cis 200o Mathematics 4 () Operations on Complex Numbers November 29, 2011 8 / 18
31. 31. De Moivre’s TheoremExample 2: Find (3 cis 120o )5 (3 cis 120o )5 = 35 cis(5 · 120)o Mathematics 4 () Operations on Complex Numbers November 29, 2011 9 / 18
32. 32. De Moivre’s TheoremExample 2: Find (3 cis 120o )5 (3 cis 120o )5 = 35 cis(5 · 120)o (3 cis 120o )5 = 243 cis 600o = 243 cis 240o Mathematics 4 () Operations on Complex Numbers November 29, 2011 9 / 18
33. 33. De Moivre’s TheoremExample 2: Find (3 cis 120o )5 (3 cis 120o )0 = 1 cis 0 Mathematics 4 () Operations on Complex Numbers November 29, 2011 10 / 18
34. 34. De Moivre’s TheoremExample 2: Find (3 cis 120o )5 (3 cis 120o )1 = 3 cis 120o Mathematics 4 () Operations on Complex Numbers November 29, 2011 10 / 18
35. 35. De Moivre’s TheoremExample 2: Find (3 cis 120o )5 (3 cis 120o )2 = 9 cis 240o Mathematics 4 () Operations on Complex Numbers November 29, 2011 10 / 18
36. 36. De Moivre’s TheoremExample 2: Find (3 cis 120o )5 (3 cis 120o )3 = 27 cis 360o Mathematics 4 () Operations on Complex Numbers November 29, 2011 10 / 18
37. 37. De Moivre’s TheoremExample 2: Find (3 cis 120o )5 (3 cis 120o )4 = 81 cis 480o Mathematics 4 () Operations on Complex Numbers November 29, 2011 10 / 18
38. 38. De Moivre’s TheoremExample 2: Find (3 cis 120o )5 (3 cis 120o )5 = 243 cis 600o = 243 cis 240o Mathematics 4 () Operations on Complex Numbers November 29, 2011 10 / 18
39. 39. De Moivre’s TheoremExample 3: Find (1 − i)8 √ (1 − i)8 = ( 2 cis 315o )8 Mathematics 4 () Operations on Complex Numbers November 29, 2011 11 / 18
40. 40. De Moivre’s TheoremExample 3: Find (1 − i)8 √ √ (1 − i)8 = ( 2 cis 315o )8 ( 2 cis 315o )8 = 16 cis 2520o = 16 cis 0 = 16 Mathematics 4 () Operations on Complex Numbers November 29, 2011 11 / 18
41. 41. De Moivre’s TheoremExample 3: Find (1 − i)8 √ (1 − i)0 = ( 2 cis(−45)o )0 = 1 cis 0 Mathematics 4 () Operations on Complex Numbers November 29, 2011 12 / 18
42. 42. De Moivre’s TheoremExample 3: Find (1 − i)8 √ √ (1 − i)1 = ( 2 cis(−45)o )1 = 2 cis(−45)o Mathematics 4 () Operations on Complex Numbers November 29, 2011 12 / 18
43. 43. De Moivre’s TheoremExample 3: Find (1 − i)8 √ (1 − i)2 = ( 2 cis(−45)o )2 = 2 cis(−90)o Mathematics 4 () Operations on Complex Numbers November 29, 2011 12 / 18
44. 44. De Moivre’s TheoremExample 3: Find (1 − i)8 √ √ (1 − i)3 = ( 2 cis(−45)o )3 = 2 2 cis(−135)o Mathematics 4 () Operations on Complex Numbers November 29, 2011 12 / 18
45. 45. De Moivre’s TheoremExample 3: Find (1 − i)8 √ (1 − i)4 = ( 2 cis(−45)o )4 = 4 cis(−180)o Mathematics 4 () Operations on Complex Numbers November 29, 2011 12 / 18
46. 46. De Moivre’s TheoremExample 3: Find (1 − i)8 √ √ (1 − i)5 = ( 2 cis(−45)o )5 = 4 2 cis(−225)o Mathematics 4 () Operations on Complex Numbers November 29, 2011 12 / 18
47. 47. De Moivre’s TheoremExample 3: Find (1 − i)8 √ (1 − i)6 = ( 2 cis(−45)o )6 = 8 cis(−270)o Mathematics 4 () Operations on Complex Numbers November 29, 2011 12 / 18
48. 48. De Moivre’s TheoremExample 3: Find (1 − i)8 √ √ (1 − i)7 = ( 2 cis(−45)o )7 = 8 2 cis(−315)o Mathematics 4 () Operations on Complex Numbers November 29, 2011 12 / 18
49. 49. De Moivre’s TheoremExample 3: Find (1 − i)8 √ (1 − i)8 = ( 2 cis(−45)o )8 = 16 cis(−360)o = 16 cis 0 Mathematics 4 () Operations on Complex Numbers November 29, 2011 12 / 18
50. 50. De Moivre’s TheoremExample 3: Find (1 − i)8 √ (1 − i)8 = ( 2 cis(−45)o )8 = 16 cis(−360)o = 16 cis 0 Mathematics 4 () Operations on Complex Numbers November 29, 2011 12 / 18
51. 51. De Moivre’s TheoremDe Moivre’s Theorem can be used to ﬁnd the nth of a complex number: √Find the three cube roots of −2 − i2 3.We wish to ﬁnd values of r and θ such that: √ (r cis θ)3 = −2 − i2 3 Mathematics 4 () Operations on Complex Numbers November 29, 2011 13 / 18
52. 52. De Moivre’s TheoremDe Moivre’s Theorem can be used to ﬁnd the nth of a complex number: √Find the three cube roots of −2 − i2 3.We wish to ﬁnd values of r and θ such that: √ (r cis θ)3 = −2 − i2 3Using De Moivre’s Theorem and expressing the complex number in polarform: r3 cis 3θ = 4 cis 240oTherefore: r3 = 4 and 3θ = 240o + k · 360o , k ∈ Z Mathematics 4 () Operations on Complex Numbers November 29, 2011 13 / 18
53. 53. De Moivre’s TheoremDe Moivre’s Theorem can be used to ﬁnd the nth of a complex number: √Find the three cube roots of −2 − i2 3.We wish to ﬁnd values of r and θ such that: √ (r cis θ)3 = −2 − i2 3Using De Moivre’s Theorem and expressing the complex number in polarform: r3 cis 3θ = 4 cis 240oTherefore: r3 = √ 4 and 3θ = 240o + k · 360o , k ∈ Z r= 34 and θ = 80o + k · 120o , k ∈ Z Mathematics 4 () Operations on Complex Numbers November 29, 2011 13 / 18
54. 54. Finding the nth roots of complex numbersFor any complex number r cis θ and n ∈ Z+ :The nth roots of r cis θ is given by: √ n r cis θk θ + k360o θk = , k = 0, 1, 2, ...(n − 1) n Mathematics 4 () Operations on Complex Numbers November 29, 2011 14 / 18
55. 55. Finding the nth roots of complex numbersExample 1: Find the fourth roots of 16 cis 120o Mathematics 4 () Operations on Complex Numbers November 29, 2011 15 / 18
56. 56. Finding the nth roots of complex numbersExample 1: Find the fourth roots of 16 cis 120o r4 cis 4θ = 16 cis 120o Mathematics 4 () Operations on Complex Numbers November 29, 2011 15 / 18
57. 57. Finding the nth roots of complex numbersExample 1: Find the fourth roots of 16 cis 120o r4 cis 4θ = 16 cis 120o r4 = 16 and 4θ = 120o + k360o Mathematics 4 () Operations on Complex Numbers November 29, 2011 15 / 18
58. 58. Finding the nth roots of complex numbersExample 1: Find the fourth roots of 16 cis 120o r4 cis 4θ = 16 cis 120o r4 = 16 and 4θ = 120o + k360o r=2 and θ = 30o + k90o Mathematics 4 () Operations on Complex Numbers November 29, 2011 15 / 18
59. 59. Finding the nth roots of complex numbersExample 1: Find the fourth roots of 16 cis 120o r4 cis 4θ = 16 cis 120o r4 = 16 and 4θ = 120o + k360o r=2 and θ = 30o + k90o 2 cis 30o 2 cis 120o 2 cis 210o 2 cis 300o Mathematics 4 () Operations on Complex Numbers November 29, 2011 15 / 18
60. 60. Finding the nth roots of complex numbersExample 1: Find the fourth roots of 16 cis 120o (2 cis 30)0 (2 cis 210)0 (2 cis 120)0 (2 cis 300)0 Mathematics 4 () Operations on Complex Numbers November 29, 2011 16 / 18
61. 61. Finding the nth roots of complex numbersExample 1: Find the fourth roots of 16 cis 120o (2 cis 30)1 (2 cis 210)1 (2 cis 120)1 (2 cis 300)1 Mathematics 4 () Operations on Complex Numbers November 29, 2011 16 / 18
62. 62. Finding the nth roots of complex numbersExample 1: Find the fourth roots of 16 cis 120o (2 cis 30)2 (2 cis 210)2 (2 cis 120)2 (2 cis 300)2 Mathematics 4 () Operations on Complex Numbers November 29, 2011 16 / 18
63. 63. Finding the nth roots of complex numbersExample 1: Find the fourth roots of 16 cis 120o (2 cis 30)3 (2 cis 210)3 (2 cis 120)3 (2 cis 300)3 Mathematics 4 () Operations on Complex Numbers November 29, 2011 16 / 18
64. 64. Finding the nth roots of complex numbersExample 1: Find the fourth roots of 16 cis 120o (2 cis 30)4 (2 cis 210)4 (2 cis 120)4 (2 cis 300)4 Mathematics 4 () Operations on Complex Numbers November 29, 2011 16 / 18
65. 65. Finding the nth roots of complex numbersExample 1: Find the fourth roots of 16 cis 120o (2 cis 30)4 (2 cis 210)4 (2 cis 120)4 (2 cis 300)4 Mathematics 4 () Operations on Complex Numbers November 29, 2011 16 / 18
66. 66. Finding the nth roots of complex numbersExample 2: Find the cube roots of −8 Mathematics 4 () Operations on Complex Numbers November 29, 2011 17 / 18
67. 67. Finding the nth roots of complex numbersExample 2: Find the cube roots of −8 r3 cis 3θ = −8 = 8 cis 180o Mathematics 4 () Operations on Complex Numbers November 29, 2011 17 / 18
68. 68. Finding the nth roots of complex numbersExample 2: Find the cube roots of −8 r3 cis 3θ = −8 = 8 cis 180o r3 = 8 and 3θ = 180o + k360o Mathematics 4 () Operations on Complex Numbers November 29, 2011 17 / 18
69. 69. Finding the nth roots of complex numbersExample 2: Find the cube roots of −8 r3 cis 3θ = −8 = 8 cis 180o r3 = 8 and 3θ = 180o + k360o r=2 and θ = 60o + k120o Mathematics 4 () Operations on Complex Numbers November 29, 2011 17 / 18
70. 70. Finding the nth roots of complex numbersExample 2: Find the cube roots of −8 r3 cis 3θ = −8 = 8 cis 180o r3 = 8 and 3θ = 180o + k360o r=2 and θ = 60o + k120o 2 cis 60o 2 cis 180o = −2 2 cis 300o Mathematics 4 () Operations on Complex Numbers November 29, 2011 17 / 18
71. 71. Finding the nth roots of complex numbersExample 2: Find the cube roots of −8 (2 cis 60)0 (2 cis 180)0 (2 cis 300)0 Mathematics 4 () Operations on Complex Numbers November 29, 2011 18 / 18
72. 72. Finding the nth roots of complex numbersExample 2: Find the cube roots of −8 (2 cis 60)1 (2 cis 180)1 (2 cis 300)1 Mathematics 4 () Operations on Complex Numbers November 29, 2011 18 / 18
73. 73. Finding the nth roots of complex numbersExample 2: Find the cube roots of −8 (2 cis 60)2 (2 cis 180)2 (2 cis 300)2 Mathematics 4 () Operations on Complex Numbers November 29, 2011 18 / 18
74. 74. Finding the nth roots of complex numbersExample 2: Find the cube roots of −8 (2 cis 60)3 (2 cis 180)3 (2 cis 300)3 Mathematics 4 () Operations on Complex Numbers November 29, 2011 18 / 18