1. Complex Numbers, the Complex Plane & Demoivre’s Theorem

2. Complex Numbers are numbers in the form of
where a and b are real numbers and i, the imaginary unit, is defined as follows:
And the powers of i are as follows:

3. The value of in, where n is any number can be found by dividing n by 4 and then dealing only with the remainder. Why?
Examples: 1)
Then from the chart on the previous slide 2)
Then from the chart on the previous slide

4. In a complex number
a is the real part and bi is the imaginary part.
When b=0, the complex number is a real number.
When a0, and b0, as in 5+8i, the complex number is an imaginary number.
When a=0, and b0, as in 5i, the complex number is a pure imaginary number.

5. Lesson Overview 9-5A

6. Lesson Overview 9-5B

7. 5-Minute Check Lesson 9-6A

8. The Complex Plane
Imaginary Axis
Real Axis O

9. Let
be a complex number.
The magnitude or modulus of z, denoted by
is defined
As the distance from the origin to the point (x, y).

10. Imaginary Axis y
|z|
Real Axis O x

13. is sometimes abbreviated as

14. Imaginary Axis 4
z =-3 + 4i
Real Axis -3

15. z = i is in Quadrant II
x = -3 and y = 4

16. z =-3 + 4i 4
Find the reference angle () by solving -3

17. z =-3 + 4i 4 -3

18. Find r:

19. Imaginary Axis 4 -3
Real Axis

20. Find the reference angle () by solving

22. Find the cosine of 330 and substitute the value.
Find the sine of 330 and substitute the value.
Distribute the r

23. Write in standard (rectangular) form.

24. Lesson Overview 9-7A

25. Product Theorem

26. Lesson Overview 9-7B

27. Quotient Theorem

30. 5-Minute Check Lesson 9-8A

31. 5-Minute Check Lesson 9-8B

32. Powers and Roots of Complex Numbers

33. DeMoivre’s Theorem

35. What if you wanted to perform the operation below?

36. Lesson Overview 9-8A

37. Lesson Overview 9-8B

41. Theorem Finding Complex Roots

42. Find the complex fifth roots of
The five complex roots are:
for k = 0, 1, 2, 3,4 .

46. The roots of a complex number a cyclical in nature.
That is, when the points are plotted on a polar plane or a complex plane, the points are evenly spaced around the origin

47. Complex Plane

48. Polar plane

49. Polar plane

51. To find the principle root, use DeMoivre’s theorem using rational exponents.
That is, to find the principle pth root of
Raise it to the power

52. Example
You may assume it is the principle root you are seeking unless specifically stated otherwise.
Find
First express as a complex number in standard form.
Then change to polar form

53. Since we are looking for the cube root, use DeMoivre’s Theorem and raise it to the power

54. Example:
Find the 4th root of
Change to polar form
Apply DeMoivre’s Theorem