Mastering Complex Numbers: Rectangular to Polar Form and Operations

1. Section 6.5 Notes

2. 1st Day • Today we will learn how to change from rectangular (standard) complex form to trigonometric (polar) complex form and vice versus.

3. A complex number • z = a + bi • can be represented as a point (a, b) in a coordinate plane called the complex plane. • The horizontal axis is called the real axis and the vertical axis is called the imaginary axis.

4. imaginary axis real axis

5. Example 1 Graph. • 2 + 3i • -1 – 2i

6. I R

7. In Algebra II you learned how to add, subtract, multiply, and divide complex numbers. • In Pre-Calculus you will learn how to work with powers and roots of complex numbers. • To do this you must write the complex numbers in trigonometric form (or polar form).

8. On the next slide you will see how we change a rectangular (standard) complex number into a trigonometric (polar) complex number.

9. I a + bi (a, b) r b θ R a

10. The trigonometric form of the complex number z= a + bi is • z = r(cosθ + isin θ) • where a = r cosθ, b = r sin θ, • In most cases 0 ≤ θ < 2π or 0° ≤ θ < 360°.

11. There is a shortcut for writing a trigonometric complex number. The shortcut is • z = rcisθ • = r(cos θ + isin θ)

12. Example 2 Write the complex number z= 6 – 6i in trigonometric (polar) form in radians. • The point is in what quadrant? 4th quadrant

13. Find r. • Find θ. Remember θ is in the 4th quad.

14. Example 3 • Represent the complex number graphically and then find the rectangular (standard) form of the number. No rounding. • z = 6(cos 135° + isin 135°)

15. 6 135°

16. Find a and b.

17. Now we will learn how to multiply and divide complex numbers in trigonometric (polar) form.

18. Product of Complex Numbers

19. Example 4 • Find the product z1z2 of the complex numbers. Write your answer in standard form.

21. Quotient of Complex Numbers

22. Example 5

23. END OF 1st DAY

24. 2nd Day • Today we will learn to: • 1. Raise complex numbers to a power. • 2. Find the roots of complex numbers.

25. Multiply: • (4 + 2i)10

26. DeMoivre’s Theorem • If z = r(cosθ + isinθ) is a complex number and n is a positive integer, then

27. Example 6 In what quadrant is this complex number? 2nd Quadrant

28. Change to polar form. • Find θ.

30. The nth Root of a Complex Number • The complex number u = a + bi is an nth root of the complex number z if • z = un= (a + bi)n

31. Finding the nth Root of a Complex Number

32. For a positive integer n, the complex number • z = r(cosθ + i sin θ) • has exactly n distinct nth roots given by or

33. Example 7 • Find all the fourth roots of 1. • This means: x4 = 1. • Change 1 into a polar complex number. • 1 = cos(0π) + isin(0π) • r = 1, n = 4 and k = 0, 1, 2, 3

34. k = 0

35. k = 1

36. k = 2

37. k = 3

38. Notice that the roots in example 2 all have a magnitude of 1 and are equally spaced around the unit circle. Also notice that the complex roots occur in conjugate pairs. • The n distinct nth roots of 1 are called the nth roots of unity.

39. Example 8 • Find the three cube roots of z= -6 + 6i to the nearest thousandth. • This means x3= -6 + 6i.

40. Change to polar complex form in degrees. • Now find the three cube roots.

41. k = 0

42. k = 1

43. k = 2