The document defines and provides examples of complex numbers and operations involving imaginary and complex numbers. It begins by defining imaginary and pure imaginary numbers. It then defines complex numbers as combinations of real and imaginary parts and introduces the concept of complex conjugates. The remainder of the document provides examples of simplifying expressions and solving equations involving 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