and

1

• 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 v-25
• Take the square root on both sides.
• The solution set is 5i.

4
Another Example

• Solve x2 54 0
• The solution set is

5
Example Products and Quotients

• Divide

• Multiply

6
Addition and Subtraction of Complex Numbers

• For complex numbers a bi and c di,
• Examples

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
(7 3i)2

• (2 - 4i)(3 5i)

9
Powers of i

• i1 i i5 i i9 i
• i2 -1 i6 -1 i10 -1
• i3 -i i7 -i i11 -i
• i4 1 i8 1 i12 1
• and so on.

10
Simplifying Examples
i-4

• i17
• i4 1
• i17 (i4)4 i
• 1 i
• i

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 ?
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 ? i sin ? is sometimes abbreviated cis ?.
• 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 ? by using the equation
  
• 
  choosing the quadrant indicated in Step 1.

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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 Moivres 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 ? is in degrees, then the
  nonzero complex number r(cos ? i sin ?) 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 ? 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, ?), 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 ? 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 ?

38
Example

• Graph r 2 cos 3?

39
Example

• Convert r -3 cos ? - sin ? into a rectangular
  equation.

40
Circles and Lemniscates
41
Limacons
42
Rose Curves