1. Complex Numbers 2

2. Complex Numbers

3. Complex Numbers What is truth?

4. Complex Numbers Who uses them in real life?

5. Complex Numbers Who uses them in real life? Here’s a hint….

6. Complex Numbers Who uses them in real life? Here’s a hint….

7. Complex Numbers Who uses them in real life?
The navigation system in the space shuttle depends on complex numbers!

8. Can you see a problem here?-2
Can you see a problem here?

9. -2
Who goes first?

10. Complex numbers do not have order-2
Complex numbers do not have order

11. What is a complex number?
It is a tool to solve an equation.

12. What is a complex number?
It is a tool to solve an equation.
It has been used to solve equations for the last 200 years or so.

13. What is a complex number?
It is a tool to solve an equation.
It has been used to solve equations for the last 200 years or so.
It is defined to be i such that ;

14. What is a complex number?
It is a tool to solve an equation.
It has been used to solve equations for the last 200 years or so.
It is defined to be i such that ;
Or in other words;

15. Complex
i is an imaginary number

16. Complex
i is an imaginary number
Or a complex number

17. Complex i is an imaginary number Or a complex number
Or an unreal number

18. Complex? i is an imaginary number Or a complex number
Or an unreal number
The terms are inter-changeable
unreal
complex
imaginary

19. Some observations
In the beginning there were counting numbers 1 2

20. Some observations In the beginning there were counting numbers
And then we needed integers 1 2

21. Some observations In the beginning there were counting numbers
And then we needed integers 1 2 -1 -3

22. Some observations In the beginning there were counting numbers
And then we needed integers
And rationals 1
0.41 2 -1 -3

23. Some observations In the beginning there were counting numbers
And then we needed integers
And rationals
And irrationals 1
0.41 2 -1 -3

24. Some observations In the beginning there were counting numbers
And then we needed integers
And rationals
And irrationals
And reals 1
0.41 2 -1 -3

25. So where do unreals fit in ?
We have always used them. 6 is not just 6 it is 6 + 0i. Complex numbers incorporate all numbers.
3 + 4i 2i 1
0.41 2 -1 -3

26. A number such as 3i is a purely imaginary number

27. A number such as 3i is a purely imaginary number
A number such as 6 is a purely real number

28. A number such as 3i is a purely imaginary number
A number such as 6 is a purely real number
6 + 3i is a complex number

29. A number such as 3i is a purely imaginary number
A number such as 6 is a purely real number
6 + 3i is a complex number
x + iy is the general form of a complex number

30. A number such as 3i is a purely imaginary number
A number such as 6 is a purely real number
6 + 3i is a complex number
x + iy is the general form of a complex number
If x + iy = 6 – 4i then x = 6 and y = -4

31. A number such as 3i is a purely imaginary number
A number such as 6 is a purely real number
6 + 3i is a complex number
x + iy is the general form of a complex number
If x + iy = 6 – 4i then x = 6 and y = – 4
The ‘real part’ of 6 – 4i is 6

32. Worked Examples
Simplify

33. Worked Examples
Simplify

34. Worked Examples
Simplify
Evaluate

35. Worked Examples
Simplify
Evaluate

36. Worked Examples
3. Simplify

37. Worked Examples
3. Simplify

38. Worked Examples
3. Simplify
4. Simplify

39. Worked Examples
3. Simplify
4. Simplify

40. Worked Examples
3. Simplify
4. Simplify
5. Simplify

41. Addition Subtraction Multiplication
3. Simplify
4. Simplify
5. Simplify

42. Division
6. Simplify

43. Division
6. Simplify
The trick is to make the denominator real:

44. Division
6. Simplify
The trick is to make the denominator real:

45. Solving Quadratic Functions

46. Powers of i

47. Powers of i

48. Powers of i

49. Powers of i

50. Powers of i

51. Developing useful rules

52. Developing useful rules

53. Developing useful rules

54. Developing useful rules

55. Argand Diagrams
Jean Robert Argand was a Swiss amateur mathematician. He was an accountant book-keeper.

56. Argand Diagrams
Jean Robert Argand was a Swiss amateur mathematician. He was an accountant book-keeper.
He is remembered for 2 things
His ‘Argand Diagram’

57. Argand Diagrams
Jean Robert Argand was a Swiss amateur mathematician. He was an accountant book-keeper.
He is remembered for 2 things
His ‘Argand Diagram’
His work on the ‘bell curve’

58. Argand Diagrams
Jean Robert Argand was a Swiss amateur mathematician. He was an accountant book-keeper.
He is remembered for 2 things
His ‘Argand Diagram’
His work on the ‘bell curve’
Very little is known about Argand. No likeness has survived.

59. Argand Diagrams x y 1 2 3
2 + 3i

60. Argand Diagrams x y 1 2 3
2 + 3i
We can represent complex numbers as a point.

61. Argand Diagrams x y 1 2 3

62. Argand Diagrams y x We can represent complex numbers as a vector. 1 23 A O
We can represent complex numbers as a vector.

63. Argand Diagrams x y 1 2 3 B A O

64. Argand Diagrams C x y 1 2 3 B A O

65. Argand Diagrams C x y 1 2 3 B A O

66. Argand Diagrams C x y 1 2 3 B A O

67. Argand Diagrams C x y 1 2 3 B A O

68. Argand Diagrams C x y 1 2 3 B A O

69. Argand Diagrams C x y 1 2 3 B A O

70. Argand Diagrams C x y 1 2 3 B A O

71. De Moivre
Abraham De Moivre was a French Protestant who moved to England in search of religious freedom.
He was most famous for his work on probability and was an acquaintance of Isaac Newton.
His theorem was possibly suggested to him by Newton.

72. This remarkable formula works for all values of n.
De Moivre’s Theorem
This remarkable formula works for all values of n.

73. Enter Leonhard Euler…..

74. Euler who was the first to use i for complex numbers had several great ideas. One of them was that
eiq = cos q + i sin q
Here is an amazing proof….

94. One last amazing result
Have you ever thought about ii ?

95. One last amazing result
What if I told you that ii is a real number?

99. ii =

100. ii =

101. So ii is an infinite number of real numbers

102. The End