The document discusses operations on complex numbers, including multiplication, raising to powers, and De Moivre's theorem. It provides examples of multiplying complex numbers algebraically and in polar form. It also gives examples of raising complex numbers to powers using the rules zn = rnθn and De Moivre's theorem, which states that (r cis θ)n = rn cis nθ.

1. Operations on Complex Numbers
Mathematics 4
November 29, 2011
Mathematics 4 () Operations on Complex Numbers November 29, 2011 1 / 18

2. Review of Multiplication of Complex Numbers
Find the product of 4 + 4i and −2 − 3i
1. 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. Review of Multiplication of Complex Numbers
Find the product of 4 + 4i and −2 − 3i
2. 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. Review of Multiplication of Complex Numbers
Rule for Multiplication of Complex Numbers in Polar Form
Given:
z1 = r1 cis α z2 = r2 cis β
Mathematics 4 () Operations on Complex Numbers November 29, 2011 4 / 18

5. Review of Multiplication of Complex Numbers
Rule for Multiplication of Complex Numbers in Polar Form
Given:
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. Raising Complex Numbers to a Power
Given: z = r cis θ
z0 =
Mathematics 4 () Operations on Complex Numbers November 29, 2011 5 / 18

7. Raising Complex Numbers to a Power
Given: z = r cis θ
z0 = 1
Mathematics 4 () Operations on Complex Numbers November 29, 2011 5 / 18

8. Raising Complex Numbers to a Power
Given: z = r cis θ
z0 = 1
z1 =
Mathematics 4 () Operations on Complex Numbers November 29, 2011 5 / 18

9. Raising Complex Numbers to a Power
Given: z = r cis θ
z0 = 1
z 1 = r cis θ
Mathematics 4 () Operations on Complex Numbers November 29, 2011 5 / 18

10. Raising Complex Numbers to a Power
Given: z = r cis θ
z0 = 1
z 1 = r cis θ
z2 =
Mathematics 4 () Operations on Complex Numbers November 29, 2011 5 / 18

11. Raising Complex Numbers to a Power
Given: 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. Raising Complex Numbers to a Power
Given: 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. Raising Complex Numbers to a Power
Given: 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. Raising Complex Numbers to a Power
Given: 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. Raising Complex Numbers to a Power
Given: 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. Raising Complex Numbers to a Power
Given: 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. Raising Complex Numbers to a Power
De Moivre’s Theorem
(r cis θ)n = rn cis(n · θ)
Mathematics 4 () Operations on Complex Numbers November 29, 2011 6 / 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. 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. De Moivre’s√Theorem
Example 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. De Moivre’s√Theorem
Example 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. De Moivre’s√Theorem
Example 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. De Moivre’s√Theorem
Example 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. De Moivre’s√Theorem
Example 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. De Moivre’s√Theorem
Example 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. De Moivre’s√Theorem
Example 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. De Moivre’s√Theorem
Example 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. De Moivre’s√Theorem
Example 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. De Moivre’s√Theorem
Example 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. De Moivre’s√Theorem
Example 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. De Moivre’s Theorem
Example 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. De Moivre’s Theorem
Example 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. De Moivre’s Theorem
Example 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. De Moivre’s Theorem
Example 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. De Moivre’s Theorem
Example 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. De Moivre’s Theorem
Example 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. De Moivre’s Theorem
Example 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. De Moivre’s Theorem
Example 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. De Moivre’s Theorem
Example 3: Find (1 − i)8
√
(1 − i)8 = ( 2 cis 315o )8
Mathematics 4 () Operations on Complex Numbers November 29, 2011 11 / 18

40. De Moivre’s Theorem
Example 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. De Moivre’s Theorem
Example 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. De Moivre’s Theorem
Example 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. De Moivre’s Theorem
Example 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. De Moivre’s Theorem
Example 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. De Moivre’s Theorem
Example 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. De Moivre’s Theorem
Example 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. De Moivre’s Theorem
Example 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. De Moivre’s Theorem
Example 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. De Moivre’s Theorem
Example 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. De Moivre’s Theorem
Example 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. De Moivre’s Theorem
De 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. De Moivre’s Theorem
De 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
Using De Moivre’s Theorem and expressing the complex number in polar
form:
r3 cis 3θ = 4 cis 240o
Therefore:
r3 = 4 and 3θ = 240o + k · 360o , k ∈ Z
Mathematics 4 () Operations on Complex Numbers November 29, 2011 13 / 18

53. De Moivre’s Theorem
De 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
Using De Moivre’s Theorem and expressing the complex number in polar
form:
r3 cis 3θ = 4 cis 240o
Therefore:
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. Finding the nth roots of complex numbers
For 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. Finding the nth roots of complex numbers
Example 1: Find the fourth roots of 16 cis 120o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 15 / 18

56. Finding the nth roots of complex numbers
Example 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. Finding the nth roots of complex numbers
Example 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. Finding the nth roots of complex numbers
Example 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. Finding the nth roots of complex numbers
Example 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. Finding the nth roots of complex numbers
Example 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. Finding the nth roots of complex numbers
Example 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. Finding the nth roots of complex numbers
Example 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. Finding the nth roots of complex numbers
Example 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. Finding the nth roots of complex numbers
Example 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. Finding the nth roots of complex numbers
Example 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. Finding the nth roots of complex numbers
Example 2: Find the cube roots of −8
Mathematics 4 () Operations on Complex Numbers November 29, 2011 17 / 18

67. Finding the nth roots of complex numbers
Example 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. Finding the nth roots of complex numbers
Example 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. Finding the nth roots of complex numbers
Example 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. Finding the nth roots of complex numbers
Example 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. Finding the nth roots of complex numbers
Example 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. Finding the nth roots of complex numbers
Example 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. Finding the nth roots of complex numbers
Example 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. Finding the nth roots of complex numbers
Example 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