1. 9.3 Complex Numbers; The Complex Plane; Polar Form of Complex Numbers

2. 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.

3. The Complex Plane
Imaginary Axis
Real Axis O

4. 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).

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

8. is sometimes abbreviated as

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

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

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

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

13. Find r:

14. Imaginary Axis 4 -3
Real Axis

15. Find the reference angle () by solving

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

18. Write in standard (rectangular) form.

19. Lesson Overview 9-7A

20. Product Theorem

21. Lesson Overview 9-7B

22. Quotient Theorem

25. 5-Minute Check Lesson 9-8A

26. 5-Minute Check Lesson 9-8B

27. Powers and Roots of Complex Numbers

28. DeMoivre’s Theorem

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

31. Lesson Overview 9-8A

32. Lesson Overview 9-8B

36. Theorem Finding Complex Roots

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

41. 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

42. Complex Plane

43. Polar plane

44. Polar plane

46. 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

47. 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

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

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