1. 6.5 Complex Numbers in Polar Form: DeMoivre’s Theorem
Objectives
Fine absolute value of a complex #
Write complex # in polar form
Convert a complex # from polar to rectangular form
Plot complex numbers in the complex plane
Find products & quotients of complex numbers in polar form
Find powers of complex # in polar form
Find roots of complex # in polar form

2. Complex number = z = a + bi
a is a real number
bi is an imaginary number
Together, the sum, a+bi is a COMPLEX #
Complex plane has a real axis (horizontal) and an imaginary axis (vertical)
2 – 5i is found in the 4th quadrant of the complex plane (horiz = 2, vert = -5)
Absolute value of 2 – 5i refers to the distance this pt. is from the origin (continued)

3. Find the absolute value
Since the horizontal component = 2 and vertical = -5, we can consider the distance to that point as the same as the length of the hypotenuse of a right triangle with those respective legs

4. z = 2 + 3i
Plot the above point and find the absolute value.

5. z = i
Plot the above point and find the absolute value.

6. Expressing complex numbers in polar form
z = a + bi

7. Express z = -5 + 3i in complex form

8. Express z = -1 + i in complex form

9. Express z = -6 + 3i in complex form

10. Product & Quotient of complex numbers

11. Product & Quotient of complex numbers

12. Product & Quotient of complex numbers

13. Product & Quotient of complex numbers

14. Product & Quotient of complex numbers

15. Multiplying complex numbers together leads to raising a complex number to a given power
If r is multiplied by itself n times, it creates
If the angle, theta, is added to itself n times, it creates the new angle, (n times theta)
THUS,

16. Multiplying complex numbers together leads to raising a complex number to a given power
Find

17. Multiplying complex numbers together leads to raising a complex number to a given power
Find

18. Multiplying complex numbers together leads to raising a complex number to a given power
Find

19. Multiplying complex numbers together leads to raising a complex number to a given power
Find

20. Taking a root (DeMoivre’s Theorem)
Radian
Degree
Where k = 0,1,2,…n-1

21. If you’re working with degrees add 360/n to the angle measure to complete the circle.
Example: Find the 6th roots of z= i
Express in polar form, find the 1st root, then add 60 degrees successively to find the other 5 roots.

22. Example: Find the 4th roots of z= -1 + 1i

23. Example: Find the 4th roots of

24. Example: Find the cube roots of 8, in degrees.

25. Example: Find the cube roots of 27, in radians.