1. Complex Numbers and i is the imaginary unit
Numbers in the form a + bi are called complex numbers
a is the real part
b is the imaginary part

2. Examples
a) b) c)
d) e)

3. Example: Solving Quadratic Equations
Solve x = √-25
Take the square root on both sides.
The solution set is {±5i}.

4. Another Example
Solve: x = 0
The solution set is

5. Example: Products and Quotients
Multiply:
Divide:

6. Addition and Subtraction of Complex Numbers
For complex numbers a + bi and c + di,
Examples
(10 − 4i) − (5 − 2i)
= (10 − 5) + [−4 − (−2)]i
= 5 − 2i
(4 − 6i) + (−3 + 7i)
= [4 + (−3)] + [−6 + 7]i
= 1 + i

7. Multiplication of Complex Numbers
For complex numbers a + bi and c + di,
The product of two complex numbers is found by multiplying as if the numbers were binomials and using the fact that i2 = −1.

8. Examples: Multiplying
(2 − 4i)(3 + 5i)
(7 + 3i)2

9. Powers of i i1 = i i5 = i i9 = i i2 = −1 i6 = −1 i10 = −1
and so on.

10. Simplifying Examples
i17
i4 = 1
i17 = (i4)4 • i
= 1 • i
= i
i−4

11. Property of Complex Conjugates
For real numbers a and b,
(a + bi)(a − bi) = a2 + b2.
The product of a complex number and its conjugate is always a real number.
Example

12. Relationships Among x, y, r, and θ

13. Trigonometric (Polar) Form of a Complex Number
The expression
is called the trigonometric form or (polar form) of the complex number x + yi. The expression
cos θ + i sin θ is sometimes abbreviated cis θ.
Using this notation

14. Example Express 2(cos 120° + i sin 120°) in rectangular form.
Notice that the real part is negative and the imaginary part is positive, this is consistent with 120 degrees being a quadrant II angle.

15. Converting from Rectangular Form to Trigonometric Form
Step 1 Sketch a graph of the number x + yi in the complex plane.
Step 2 Find r by using the equation
Step 3 Find θ by using the equation
choosing the quadrant indicated in Step 1.

16. Example Example: Find trigonometric notation for −1 − i.
First, find r.
Thus,

17. Product Theorem If are any two complex numbers, then
In compact form, this is written

18. Example: Product
Find the product of

19. Quotient Theorem If
are any two complex numbers, where then

20. Example: Quotient
Find the quotient.

21. De Moivre’s Theorem
If is a complex number, and if n is any real number, then
In compact form, this is written

22. Example: Find (−1 − i)5 and express the result in rectangular form.
First, find trigonometric notation for −1 − i
Theorem

23. nth Roots
For a positive integer n, the complex number a + bi is an nth root of the complex number x + yi if

24. nth Root Theorem
If n is any positive integer, r is a positive real number, and θ is in degrees, then the nonzero complex number r(cos θ + i sin θ) has exactly n distinct nth roots, given by
where

25. Example: Square Roots

26. Example: Fourth Root
Find all fourth roots of Write the roots in rectangular form.
Write in trigonometric form.
Here r = 16 and θ = 120°. The fourth roots of this number have absolute value

27. Example: Fourth Root continued
There are four fourth roots, let k = 0, 1, 2 and 3.
Using these angles, the fourth roots are

28. Example: Fourth Root continued
Written in rectangular form
The graphs of the roots are all on a circle that has center at the origin and radius 2.

29. Polar Coordinate System
The polar coordinate system is based on a point, called the pole, and a ray, called the polar axis.

30. Rectangular and Polar Coordinates
If a point has rectangular coordinates (x, y) and polar coordinates (r, θ), then these coordinates are related as follows.

31. Example
Plot the point on a polar coordinate system. Then determine the rectangular coordinates of the point.
P(2, 30°)
r = 2 and θ = 30°, so point P is located 2 units from the origin in the positive direction making a 30° angle with the polar axis.

32. Example continued Using the conversion formulas:
The rectangular coordinates are

33. Example
Convert (4, 2) to polar coordinates.

34. The following slides are extension work for Complex Numbers …..

35. Rectangular and Polar Equations
To convert a rectangular equation into a polar equation, use
and
and solve for r.
For the linear equation
you will get the polar equation

36. Example
Convert x + 2y = 10 into a polar equation.
x + 2y = 10

37. Example Graph r = −2 sin θ 1 1.414 2 -1 -1.414 r 330 315 270 180 150-1
-1.414 r
330
315
270
180
150
135 θ
-1.732
120 -2 90 60 45 30

38. Example
Graph r = 2 cos 3θ
−1.41 −2
1.41 2 r 90 75 60 45 30 15 θ

39. Example
Convert r = −3 cos θ − sin θ into a rectangular equation.

40. Circles and Lemniscates

41. Limacons

42. Rose Curves