Complex Numbers

1. SECTION 3-3
Complex Numbers

2. ESSENTIAL QUESTIONS
➤ How do you perform operations with pure imaginary
numbers?
➤ How do you perform operations with imaginary
numbers?

3. VOCABULARY
11. Imaginary Number/Unit:
2. Pure Imaginary Number:
3. Square Root Property:

4. VOCABULARY
11. Imaginary Number/Unit: Allows for us to take
the square root of a negative number
2. Pure Imaginary Number:
3. Square Root Property:

5. VOCABULARY
11. Imaginary Number/Unit: Allows for us to take
the square root of a negative number i = −1
2. Pure Imaginary Number:
3. Square Root Property:

6. VOCABULARY
11. Imaginary Number/Unit: Allows for us to take
the square root of a negative number i = −1
2. Pure Imaginary Number: Numbers that are
square roots of negative real numbers
3. Square Root Property:

7. VOCABULARY
11. Imaginary Number/Unit: Allows for us to take
the square root of a negative number i = −1
2. Pure Imaginary Number: Numbers that are
square roots of negative real numbers
5i, − 7i, i 13
3. Square Root Property:

8. VOCABULARY
11. Imaginary Number/Unit: Allows for us to take
the square root of a negative number i = −1
2. Pure Imaginary Number: Numbers that are
square roots of negative real numbers
5i, − 7i, i 13
3. Square Root Property: To negate something
that is squared in an equation, you can square
root both sides of an equation; this will
provide two answers (positive and negative)

9. VOCABULARY
4. Complex Number:
5. Complex Conjugate:

10. VOCABULARY
4. Complex Number: A number that is a
combination of a real and an imaginary part
5. Complex Conjugate:

11. VOCABULARY
4. Complex Number: A number that is a
combination of a real and an imaginary part
4 + 3i
5. Complex Conjugate:

12. VOCABULARY
4. Complex Number: A number that is a
combination of a real and an imaginary part
4 + 3i
5. Complex Conjugate:
Real part

13. VOCABULARY
4. Complex Number: A number that is a
combination of a real and an imaginary part
4 + 3i
5. Complex Conjugate:
Real part Imaginary part

14. VOCABULARY
4. Complex Number: A number that is a
combination of a real and an imaginary part
4 + 3i
5. Complex Conjugate: Two complex numbers that
when multiplied will result in a real number
Real part Imaginary part

15. VOCABULARY
4. Complex Number: A number that is a
combination of a real and an imaginary part
4 + 3i
5. Complex Conjugate: Two complex numbers that
when multiplied will result in a real number
a + bi, a − bi
Real part Imaginary part

16. EXAMPLE 1
a. −28
Simplify.
b. −32
c. −81
d. −17

17. EXAMPLE 1
a. −28
Simplify.
−1i 4 i 7
b. −32
c. −81
d. −17

18. EXAMPLE 1
a. −28
Simplify.
−1i 4 i 7
i i 2 i 7
b. −32
c. −81
d. −17

19. EXAMPLE 1
a. −28
Simplify.
−1i 4 i 7
i i 2 i 7
2i 7
b. −32
c. −81
d. −17

20. EXAMPLE 1
a. −28
Simplify.
−1i 4 i 7
i i 2 i 7
2i 7
b. −32
−1i 16 i 2
c. −81
d. −17

21. EXAMPLE 1
a. −28
Simplify.
−1i 4 i 7
i i 2 i 7
2i 7
b. −32
−1i 16 i 2
4i 2
c. −81
d. −17

22. EXAMPLE 1
a. −28
Simplify.
−1i 4 i 7
i i 2 i 7
2i 7
b. −32
−1i 16 i 2
4i 2
c. −81
−1i 81
d. −17

23. EXAMPLE 1
a. −28
Simplify.
−1i 4 i 7
i i 2 i 7
2i 7
b. −32
−1i 16 i 2
4i 2
c. −81
−1i 81
9i
d. −17

24. EXAMPLE 1
a. −28
Simplify.
−1i 4 i 7
i i 2 i 7
2i 7
b. −32
−1i 16 i 2
4i 2
c. −81
−1i 81
9i
d. −17
−1i 17

25. EXAMPLE 1
a. −28
Simplify.
−1i 4 i 7
i i 2 i 7
2i 7
b. −32
−1i 16 i 2
4i 2
c. −81
−1i 81
9i
d. −17
−1i 17
i 17

26. EXAMPLE 2
a. − 3i i 2i
Simplify.
b. −12 i −2 c. i32

27. EXAMPLE 2
a. − 3i i 2i
Simplify.
−6i2
b. −12 i −2 c. i32

28. EXAMPLE 2
a. − 3i i 2i
Simplify.
−6i2
−6(−1)
b. −12 i −2 c. i32

29. EXAMPLE 2
a. − 3i i 2i
Simplify.
−6i2
−6(−1)
6
b. −12 i −2 c. i32

30. EXAMPLE 2
a. − 3i i 2i
Simplify.
−6i2
−6(−1)
6
b. −12 i −2
−1i 12 i −1i 2
c. i32

31. EXAMPLE 2
a. − 3i i 2i
Simplify.
−6i2
−6(−1)
6
b. −12 i −2
−1i 12 i −1i 2
−1i 24
c. i32

32. EXAMPLE 2
a. − 3i i 2i
Simplify.
−6i2
−6(−1)
6
b. −12 i −2
−1i 12 i −1i 2
−1i 24
c. i32
−1i 4 i 6

33. EXAMPLE 2
a. − 3i i 2i
Simplify.
−6i2
−6(−1)
6
b. −12 i −2
−1i 12 i −1i 2
−1i 24
c. i32
−1i 4 i 6
−2 6

34. EXAMPLE 2
a. − 3i i 2i
Simplify.
−6i2
−6(−1)
6
b. −12 i −2
−1i 12 i −1i 2
−1i 24
c. i32
(i2
)16
−1i 4 i 6
−2 6

35. EXAMPLE 2
a. − 3i i 2i
Simplify.
−6i2
−6(−1)
6
b. −12 i −2
−1i 12 i −1i 2
−1i 24
c. i32
(i2
)16
(−1)16
−1i 4 i 6
−2 6

36. EXAMPLE 2
a. − 3i i 2i
Simplify.
−6i2
−6(−1)
6
b. −12 i −2
−1i 12 i −1i 2
−1i 24
c. i32
(i2
)16
(−1)16
−1i 4 i 6
−2 6
1

37. EXAMPLE 3
5x2
+ 20 = 0
Solve.

38. EXAMPLE 3
5x2
+ 20 = 0
Solve.
−20 −20

39. EXAMPLE 3
5x2
+ 20 = 0
Solve.
5x2
= −20
−20 −20

40. EXAMPLE 3
5x2
+ 20 = 0
Solve.
5x2
= −20
−20 −20
5 5

41. EXAMPLE 3
5x2
+ 20 = 0
Solve.
5x2
= −20
x2
= −4
−20 −20
5 5

42. EXAMPLE 3
5x2
+ 20 = 0
Solve.
5x2
= −20
x2
= −4
−20 −20
5 5
x2
= ± −4

43. EXAMPLE 3
5x2
+ 20 = 0
Solve.
5x2
= −20
x2
= −4
−20 −20
5 5
x2
= ± −4
x = ±2i

44. EXAMPLE 4
2x + yi = −14 − 3i
Find the values of x and y that make the equation
true.

45. EXAMPLE 4
2x + yi = −14 − 3i
Find the values of x and y that make the equation
true.
2x = −14

46. EXAMPLE 4
2x + yi = −14 − 3i
Find the values of x and y that make the equation
true.
2x = −14 yi = −3i

47. EXAMPLE 4
2x + yi = −14 − 3i
Find the values of x and y that make the equation
true.
2x = −14 yi = −3i
2 2

48. EXAMPLE 4
2x + yi = −14 − 3i
Find the values of x and y that make the equation
true.
2x = −14 yi = −3i
2 2
x = −7

49. EXAMPLE 4
2x + yi = −14 − 3i
Find the values of x and y that make the equation
true.
2x = −14 yi = −3i
2 2
x = −7
i i

50. EXAMPLE 4
2x + yi = −14 − 3i
Find the values of x and y that make the equation
true.
2x = −14 yi = −3i
2 2
x = −7
i i
y = −3

51. EXAMPLE 5
Simplify.
a. (3 + 5i)+(2 − 4i) b. (6 − 6i)−(3 − 7i)

52. EXAMPLE 5
Simplify.
a. (3 + 5i)+(2 − 4i)
3 + 5i + 2 − 4i
b. (6 − 6i)−(3 − 7i)

53. EXAMPLE 5
Simplify.
a. (3 + 5i)+(2 − 4i)
3 + 5i + 2 − 4i
3 + 2 + 5i − 4i
b. (6 − 6i)−(3 − 7i)

54. EXAMPLE 5
Simplify.
a. (3 + 5i)+(2 − 4i)
3 + 5i + 2 − 4i
3 + 2 + 5i − 4i
5 + i
b. (6 − 6i)−(3 − 7i)

55. EXAMPLE 5
Simplify.
a. (3 + 5i)+(2 − 4i)
3 + 5i + 2 − 4i
3 + 2 + 5i − 4i
5 + i
b. (6 − 6i)−(3 − 7i)
6 − 6i − 3 + 7i

56. EXAMPLE 5
Simplify.
a. (3 + 5i)+(2 − 4i)
3 + 5i + 2 − 4i
3 + 2 + 5i − 4i
5 + i
b. (6 − 6i)−(3 − 7i)
6 − 6i − 3 + 7i
3 + i

57. EXAMPLE 5
Simplify.
c. (4 + 3i)(5 − 2i) d. (2 + 7i)(2 − 7i)

58. EXAMPLE 5
Simplify.
c. (4 + 3i)(5 − 2i)
20 − 8i + 15i − 6i2
d. (2 + 7i)(2 − 7i)

59. EXAMPLE 5
Simplify.
c. (4 + 3i)(5 − 2i)
20 − 8i + 15i − 6i2
20 + 7i − 6(−1)
d. (2 + 7i)(2 − 7i)

60. EXAMPLE 5
Simplify.
c. (4 + 3i)(5 − 2i)
20 − 8i + 15i − 6i2
20 + 7i − 6(−1)
26 + 7i
d. (2 + 7i)(2 − 7i)

61. EXAMPLE 5
Simplify.
c. (4 + 3i)(5 − 2i)
20 − 8i + 15i − 6i2
20 + 7i − 6(−1)
26 + 7i
d. (2 + 7i)(2 − 7i)
4 − 14i + 14i − 49i2

62. EXAMPLE 5
Simplify.
c. (4 + 3i)(5 − 2i)
20 − 8i + 15i − 6i2
20 + 7i − 6(−1)
26 + 7i
d. (2 + 7i)(2 − 7i)
4 − 14i + 14i − 49i2
4 − 49(−1)

63. EXAMPLE 5
Simplify.
c. (4 + 3i)(5 − 2i)
20 − 8i + 15i − 6i2
20 + 7i − 6(−1)
26 + 7i
d. (2 + 7i)(2 − 7i)
4 − 14i + 14i − 49i2
4 − 49(−1)
53

64. EXAMPLE 6
In an AC circuit, the voltage E, the current I,
and the impedance Z are related by the formula
E = IZ. Find the voltage in a circuit with current
1 + 4i amps and impedance 3 - 6i ohms.
E = IZ

65. EXAMPLE 6
In an AC circuit, the voltage E, the current I,
and the impedance Z are related by the formula
E = IZ. Find the voltage in a circuit with current
1 + 4i amps and impedance 3 - 6i ohms.
E = IZ
E = (1+ 4i)(3 − 6i)

66. EXAMPLE 6
In an AC circuit, the voltage E, the current I,
and the impedance Z are related by the formula
E = IZ. Find the voltage in a circuit with current
1 + 4i amps and impedance 3 - 6i ohms.
E = IZ
E = (1+ 4i)(3 − 6i)
3 − 6i + 12i − 24i2

67. EXAMPLE 6
In an AC circuit, the voltage E, the current I,
and the impedance Z are related by the formula
E = IZ. Find the voltage in a circuit with current
1 + 4i amps and impedance 3 - 6i ohms.
E = IZ
E = (1+ 4i)(3 − 6i)
3 − 6i + 12i − 24i2
3 + 6i + 24

68. EXAMPLE 6
In an AC circuit, the voltage E, the current I,
and the impedance Z are related by the formula
E = IZ. Find the voltage in a circuit with current
1 + 4i amps and impedance 3 - 6i ohms.
E = IZ
E = (1+ 4i)(3 − 6i)
3 − 6i + 12i − 24i2
3 + 6i + 24
27 + 6i

69. EXAMPLE 6
In an AC circuit, the voltage E, the current I,
and the impedance Z are related by the formula
E = IZ. Find the voltage in a circuit with current
1 + 4i amps and impedance 3 - 6i ohms.
E = IZ
E = (1+ 4i)(3 − 6i)
3 − 6i + 12i − 24i2
3 + 6i + 24
27 + 6i volts

70. EXAMPLE 7
Simplify.
a.
5i
3 + 2i
b.
5 + i
2i

71. EXAMPLE 7
Simplify.
a.
5i
3 + 2i
15i − 10i2
9 − 6i + 6i − 4i2
b.
5 + i
2i

72. EXAMPLE 7
Simplify.
a.
5i
3 + 2i
15i − 10i2
9 − 6i + 6i − 4i2
10 + 15i
9 + 4
b.
5 + i
2i

73. EXAMPLE 7
Simplify.
a.
5i
3 + 2i
15i − 10i2
9 − 6i + 6i − 4i2
10 + 15i
9 + 4
10 + 15i
13
b.
5 + i
2i

74. EXAMPLE 7
Simplify.
a.
5i
3 + 2i
15i − 10i2
9 − 6i + 6i − 4i2
10 + 15i
9 + 4
10 + 15i
13
10
13
+
15
13
i
b.
5 + i
2i

75. EXAMPLE 7
Simplify.
a.
5i
3 + 2i
15i − 10i2
9 − 6i + 6i − 4i2
10 + 15i
9 + 4
10 + 15i
13
10
13
+
15
13
i
b.
5 + i
2i
(5 + i)
(2i)
i
i
i

76. EXAMPLE 7
Simplify.
a.
5i
3 + 2i
15i − 10i2
9 − 6i + 6i − 4i2
10 + 15i
9 + 4
10 + 15i
13
10
13
+
15
13
i
b.
5 + i
2i
(5 + i)
(2i)
i
i
i
5i + i2
2i2

77. EXAMPLE 7
Simplify.
a.
5i
3 + 2i
15i − 10i2
9 − 6i + 6i − 4i2
10 + 15i
9 + 4
10 + 15i
13
10
13
+
15
13
i
b.
5 + i
2i
(5 + i)
(2i)
i
i
i
5i + i2
2i2
−1+ 5i
−2

78. EXAMPLE 7
Simplify.
a.
5i
3 + 2i
15i − 10i2
9 − 6i + 6i − 4i2
10 + 15i
9 + 4
10 + 15i
13
10
13
+
15
13
i
b.
5 + i
2i
(5 + i)
(2i)
i
i
i
5i + i2
2i2
−1+ 5i
−2
1
2
−
5
2
i