1. 3.4 Complex Numbers
2 Timothy 3:16-17 All Scripture is breathed out by
God and profitable for teaching, for reproof, for
correction, and for training in righteousness, that the
man of God may be competent, equipped for every good
work.

2. A complex number has the form
a+bi

3. A complex number has the form
a+bi
real
part

4. A complex number has the form
a+bi
real
part imaginary
part

5. A complex number has the form
a+bi
i = −1
real
part imaginary
part

6. A complex number has the form
a+bi
i = −1
real
part imaginary
part
Examples:

7. A complex number has the form
a+bi
i = −1
real
part imaginary
part
Examples:
5+3i Real Part: 5 Imaginary Part: 3

8. A complex number has the form
a+bi
i = −1
real
part imaginary
part
Examples:
5+3i Real Part: 5 Imaginary Part: 3
-4i Real Part: 0 Imaginary Part: -4

9. A complex number has the form
a+bi
i = −1
real
part imaginary
part
Examples:
5+3i Real Part: 5 Imaginary Part: 3
-4i Real Part: 0 Imaginary Part: -4
12 Real Part: 12 Imaginary Part: 0

10. 7 ± 3i means we have two numbers
7+3i 7-3i

11. 7 ± 3i means we have two numbers
7+3i 7-3i
these are called conjugates of each other

12. 7 ± 3i means we have two numbers
7+3i 7-3i
these are called conjugates of each other
Examples:

13. 7 ± 3i means we have two numbers
7+3i 7-3i
these are called conjugates of each other
Examples: a+bi

14. 7 ± 3i means we have two numbers
7+3i 7-3i
these are called conjugates of each other
Examples: a+bi a-bi

15. 7 ± 3i means we have two numbers
7+3i 7-3i
these are called conjugates of each other
Examples: a+bi a-bi
7− 2

16. 7 ± 3i means we have two numbers
7+3i 7-3i
these are called conjugates of each other
Examples: a+bi a-bi
7− 2 7+ 2

17. 7 ± 3i means we have two numbers
7+3i 7-3i
these are called conjugates of each other
Examples: a+bi a-bi
7− 2 7+ 2
1− sin θ

18. 7 ± 3i means we have two numbers
7+3i 7-3i
these are called conjugates of each other
Examples: a+bi a-bi
7− 2 7+ 2
1− sin θ 1+ sin θ

19. 7 ± 3i means we have two numbers
7+3i 7-3i
these are called conjugates of each other
Examples: a+bi a-bi
7− 2 7+ 2
1− sin θ 1+ sin θ
Complex Conjugate Theorem
If a+bi is a root of an equation
Then a-bi is also a root

20. Arithmetic with Complex Numbers

21. Arithmetic with Complex Numbers
We can
add, subtract, multiply, divide
complex numbers

22. Arithmetic with Complex Numbers
We can
add, subtract, multiply, divide
complex numbers
You need to know how to do these operations
a) on your calculator
b) by hand

23. Addition ( a + bi ) + ( c + di ) = ( a + c ) + (b + d ) i

24. Addition ( a + bi ) + ( c + di ) = ( a + c ) + (b + d ) i
Example: ( 8 − 4i ) + ( 2 + i )

25. Addition ( a + bi ) + ( c + di ) = ( a + c ) + (b + d ) i
Example: ( 8 − 4i ) + ( 2 + i )
10 − 3i

26. Addition ( a + bi ) + ( c + di ) = ( a + c ) + (b + d ) i
Example: ( 8 − 4i ) + ( 2 + i )
10 − 3i
and do this on the calculator

27. Addition ( a + bi ) + ( c + di ) = ( a + c ) + (b + d ) i
Example: ( 8 − 4i ) + ( 2 + i )
10 − 3i
and do this on the calculator
Subtraction ( a + bi ) − ( c + di ) = ( a − c ) + (b − d ) i

28. Addition ( a + bi ) + ( c + di ) = ( a + c ) + (b + d ) i
Example: ( 8 − 4i ) + ( 2 + i )
10 − 3i
and do this on the calculator
Subtraction ( a + bi ) − ( c + di ) = ( a − c ) + (b − d ) i
Example: ( 8 − 4i ) − ( 2 + i )

29. Addition ( a + bi ) + ( c + di ) = ( a + c ) + (b + d ) i
Example: ( 8 − 4i ) + ( 2 + i )
10 − 3i
and do this on the calculator
Subtraction ( a + bi ) − ( c + di ) = ( a − c ) + (b − d ) i
Example: ( 8 − 4i ) − ( 2 + i )
6 − 5i

30. Addition ( a + bi ) + ( c + di ) = ( a + c ) + (b + d ) i
Example: ( 8 − 4i ) + ( 2 + i )
10 − 3i
and do this on the calculator
Subtraction ( a + bi ) − ( c + di ) = ( a − c ) + (b − d ) i
Example: ( 8 − 4i ) − ( 2 + i )
6 − 5i
verify on the calculator

31. Multiplication Use FOIL

32. Multiplication Use FOIL
Example: ( 8 − 4i )( 2 + i )

33. Multiplication Use FOIL
Example: ( 8 − 4i )( 2 + i )
2
16 + 8i − 8i − 4i

34. Multiplication Use FOIL
Example: ( 8 − 4i )( 2 + i )
2
16 + 8i − 8i − 4i
2
i = −1 ⋅ −1 = −1

35. Multiplication Use FOIL
Example: ( 8 − 4i )( 2 + i )
2
16 + 8i − 8i − 4i
2
i = −1 ⋅ −1 = −1
16 − 4(−1)

36. Multiplication Use FOIL
Example: ( 8 − 4i )( 2 + i )
2
16 + 8i − 8i − 4i
2
i = −1 ⋅ −1 = −1
16 − 4(−1)
20

37. Multiplication Use FOIL
Example: ( 8 − 4i )( 2 + i )
2
16 + 8i − 8i − 4i
2
i = −1 ⋅ −1 = −1
16 − 4(−1)
20
and do this on the calculator

38. Example: ( 4 + 6i )( 2 + 8i )

39. Example: ( 4 + 6i )( 2 + 8i )
2
8 + 32i + 12i + 48i

40. Example: ( 4 + 6i )( 2 + 8i )
2
8 + 32i + 12i + 48i
−40 + 44i

41. Example: ( 4 + 6i )( 2 + 8i )
2
8 + 32i + 12i + 48i
−40 + 44i
verify on calculator

42. Division multiply by conjugate of denominator

43. Division multiply by conjugate of denominator
8 − 4i
2+i

44. Division multiply by conjugate of denominator
8 − 4i
2+i
8 − 4i 2 − i
⋅
2+i 2−i

45. Division multiply by conjugate of denominator
8 − 4i
2+i
8 − 4i 2 − i
⋅
2+i 2−i
2
16 − 16i + 4i
2
4−i

46. Division multiply by conjugate of denominator
8 − 4i
2+i 12 − 16i
8 − 4i 2 − i 5
⋅
2+i 2−i
2
16 − 16i + 4i
2
4−i

47. Division multiply by conjugate of denominator
8 − 4i
2+i 12 − 16i
8 − 4i 2 − i 5
⋅
2+i 2−i 12 16i 12 16
2
− or − i
16 − 16i + 4i 5 5 5 5
2
4−i

48. Division multiply by conjugate of denominator
8 − 4i
2+i 12 − 16i
8 − 4i 2 − i 5
⋅
2+i 2−i 12 16i 12 16
2
− or − i
16 − 16i + 4i 5 5 5 5
2
4−i
verify on calculator

49. Solve
2
1) 9x + 4 = 0

50. Solve
2
1) 9x + 4 = 0
2
9x = −4

51. Solve
2
1) 9x + 4 = 0
2
9x = −4
2 4
x =−
9

52. Solve
2 2 4
1) 9x + 4 = 0 x = −
2
9
9x = −4
2 4
x =−
9

53. Solve
2 2 4
1) 9x + 4 = 0 x = −
2
9
9x = −4
4
2 4 x= ⋅ −1
x =− 9
9

54. Solve
2 2 4
1) 9x + 4 = 0 x = −
2
9
9x = −4
4
2 4 x= ⋅ −1
x =− 9
9
2
x=± i
3

55. Solve
2 2 4
1) 9x + 4 = 0 x = −
2
9
9x = −4
4
2 4 x= ⋅ −1
x =− 9
9
2
x=± i
3
2
Discuss the graph of y = 9x + 4

56. Solve
2 2 4
1) 9x + 4 = 0 x = −
2
9
9x = −4
4
2 4 x= ⋅ −1
x =− 9
9
2
x=± i
3
2
Discuss the graph of y = 9x + 4
a) parabola opening up; vertex at (0,4)
b) no x-intercepts so no Real solutions

57. 2
2) Solve 2x − 2x + 1 = 0

58. 2
2) Solve 2x − 2x + 1 = 0
Could try Master Product, but it doesn’t factor.
Use QF ... remember ... by hand; on calculator.

59. 2
2) Solve 2x − 2x + 1 = 0
Could try Master Product, but it doesn’t factor.
Use QF ... remember ... by hand; on calculator.
2 ± 4 − 4(2)
x=
4

60. 2
2) Solve 2x − 2x + 1 = 0
Could try Master Product, but it doesn’t factor.
Use QF ... remember ... by hand; on calculator.
2 ± 4 − 4(2)
x=
4
2 ± −4
x=
4

61. 2
2) Solve 2x − 2x + 1 = 0
Could try Master Product, but it doesn’t factor.
Use QF ... remember ... by hand; on calculator.
2 ± 4 − 4(2)
x=
4
2 ± −4
x=
4
2 ± 2i
x=
4

62. 2
2) Solve 2x − 2x + 1 = 0
Could try Master Product, but it doesn’t factor.
Use QF ... remember ... by hand; on calculator.
2 ± 4 − 4(2) 1 ± 1i
x= x=
4 2
2 ± −4
x=
4
2 ± 2i
x=
4

63. 2
2) Solve 2x − 2x + 1 = 0
Could try Master Product, but it doesn’t factor.
Use QF ... remember ... by hand; on calculator.
2 ± 4 − 4(2) 1 ± 1i
x= x=
4 2
2 ± −4 1 1
x= x= ± i
4 2 2
2 ± 2i
x=
4

64. 2
2) Solve 2x − 2x + 1 = 0
Could try Master Product, but it doesn’t factor.
Use QF ... remember ... by hand; on calculator.
2 ± 4 − 4(2) 1 ± 1i
x= x=
4 2
2 ± −4 1 1
x= x= ± i
4 2 2
2 ± 2i a) do on calculator
x=
4 b) graph

65. HW #5
Quiz Tomorrow!
“Not everything that is faced can be changed
but nothing can be changed until it is faced.”
James Baldwin