1. Complex Numbers

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

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

4. 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.

5. 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 ;

6. 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;

7. Complex
i is an imaginary number

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

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

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

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

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

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

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

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

16. 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

17. 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

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

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

20. 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

21. 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

22. 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

23. 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

24. Worked Examples
Simplify

25. Worked Examples
Simplify

26. Worked Examples
Simplify
Evaluate

27. Worked Examples
Simplify
Evaluate

28. Worked Examples
3. Simplify

29. Worked Examples
3. Simplify

30. Worked Examples
3. Simplify
4. Simplify

31. Worked Examples
3. Simplify
4. Simplify

32. Worked Examples
3. Simplify
4. Simplify
5. Simplify

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

34. Division
6. Simplify

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

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

37. Solving Quadratic Functions

38. Powers of i

39. Powers of i

40. Powers of i

41. Powers of i

42. Powers of i

43. Useful rules

44. Useful rules

45. Useful rules

46. Useful rules

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

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

49. Argand Diagrams x y 1 2 3

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

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

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

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

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

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

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

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

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

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

60. Now we prove that ,
e iq = cos q + i sin q

80. The End