Embed Size (px)
Citation preview
Slide 1 / 106 Slide 2 / 106
Pre-Calculus
Polar & Complex Numbers
www.njctl.org
2015-03-23
Slide 3 / 106
Table of Contents
Complex Numbers
Geometry of Complex Numbers
Complex Numbers: PowersComplex Numbers: Roots
Polar Number Properties
Polar Equations and GraphsPolar: Rose Curves and Spirals
click on the topic to go to that section
Slide 4 / 106
Complex Numbers
Return to Table of Contents
Slide 5 / 106 Slide 6 / 106
Operations, such as addition and division, can be done with i.Treat i like any other variable, except at the end make sure i is at most to the first power.
Use the following substitutions:
Why do they work?
Complex Numbers
Teac
her
Teac
her
Slide 7 / 106 Slide 8 / 106
Slide 9 / 106
2 Simplify
A
B
C
D
Complex NumbersTe
ache
rTe
ache
r
Slide 10 / 106
Slide 11 / 106 Slide 12 / 106
5 Simplify
A
B
C
D
Complex Numbers
Teac
her
Teac
her
Slide 13 / 106
Higher order i's can be simplified down to a power of 1 to 4, which can be simplified into i, -1, -i, or 1.
i i2 i3 i4
i5 =i4 i i6 = i4 i2 i7 = i4 i3 i8 = i4 i4
i9 = i4 i4 i i10 = i4 i4 i2 i11 = i4 i4 i3 i12 = i4 i4 i4
i13 = i4 i4 i4 i i14 = i4 i4 i4 i2 i15 = i4 i4 i4 i3 i16 = i4 i4 i4 i4
... ... ... ...
i raised to a power can be rewritten as a product of i4 's and an i to the 1st to the 4th.
Since each i4 = 1, we need only be concerned with the non-power of 4.
Complex Numbers
Slide 14 / 106
To simplify an i without writing out the table say i87, divide by 4.
The number of times 4 goes in evenly gives you that many i4 's.The remainder is the reduced power. Simplify.
Example: Simplify
Complex Numbers
Teac
her
Teac
her
Slide 15 / 106 Slide 16 / 106
6 Simplify
A i
B -1
C -i
D 1
Complex Numbers
Teac
her
Teac
her
Slide 17 / 106
7 Simplify
A i
B -1
C -i
D 1
Complex Numbers
Teac
her
Teac
her
Slide 18 / 106
8 Simplify
A i
B -1
C -i
D 1
Complex Numbers
Teac
her
Teac
her
Slide 19 / 106
9 Simplify
A i
B -1
C -i
D 1
Complex Numbers
Teac
her
Teac
her
Slide 20 / 106
Operations, such as addition and division, can be done with i.Treat i like any other variable, except at the end make sure i is at most to the first power.
Use the following substitutions:
Recall:
Complex Numbers
Slide 21 / 106
Examples:
Complex NumbersTe
ache
rTe
ache
r
Slide 22 / 106
Examples (in the complex form the real term comes first)
Complex Numbers
Teac
her
Teac
her
Slide 23 / 106
Examples
Complex Numbers
Teac
her
Teac
her
Slide 24 / 106
10 Simplify:
A
B
C
D
Complex Numbers
Teac
her
Teac
her
Slide 25 / 106
11 Simplify:
A
B
C
D
Complex Numbers
Teac
her
Teac
her
Slide 26 / 106
12 Simplify:
A
B
C
D
Complex Numbers
Teac
her
Teac
her
Slide 27 / 106
13 Simplify:
A
B
C
D
Complex Numbers
Teac
her
Teac
her
Slide 28 / 106
14 Simplify:
A
B
C
D
Complex Numbers
Teac
her
Teac
her
Slide 29 / 106
What pushes current through the circuit?
Batteries (just one source)
A battery acts like a pump, pushing charge through the circuit. It is the circuit's energy source.
Charges do not experience an electrical force unless there is a difference in electrical potential (voltage).
Therefore, batteries have a potential difference between their terminals. The positive terminal is at a higher voltage than the negative terminal.
Complex Numbers
Slide 30 / 106
ConductorsSome conductors "conduct" better or worse than others.
Reminder: conducting means a material allows for the free flow of electrons.
The flow of electrons is just another name for current.
Another way to look at it is that some conductors resist current to a greater or lesser extent.
We call this resistance, R.
Resistance is measured in ohms which is noted by the Greek symbol omega (Ω)
How will resistance affect current?
Complex Numbers
Slide 31 / 106
Raising resistance reduces current.
Raising voltage increases current.
We can combine these relationships in what we call "Ohm's Law".
I = V/R R=Volts / current (I)
Units: You can see that one # = Volts/Amps
Current vs Resistance & Voltage
Complex Numbers
Slide 32 / 106
Ohm's Law
V is for voltage , measured in volts , and is potentia l of a circuit.
Z is for impedance , measured in ohms ( ), which is the oppos ition to the flow of current.
The tota l impedance of a circuit is a complex number.
I is for current, measured in amps , the ra te of flow of a circuit.
Complex Numbers
Slide 33 / 106
Application: Suppose two AC currents are connected in a series. One with -4 + 3i ohms and the other with 7 - 2i ohms. What is the total impedance of the circuit?
If the voltage across the two circuits is 12 volts, what is the current?
Complex NumbersTe
ache
rTe
ache
r
Slide 34 / 106
Slide 35 / 106
Simplify
AnswersComplex Numbers
Teac
her
Teac
her
Slide 36 / 106
15 Simplify
A
B
C
D
Complex Numbers
Teac
her
Teac
her
Slide 37 / 106 Slide 38 / 106
17 Simplify
A
B
C
D
Complex Numbers
Teac
her
Teac
her
Slide 39 / 106 Slide 40 / 106
Simplify:
Complex Numbers
Teac
her
Teac
her
Slide 41 / 106 Slide 42 / 106
19 Simplify:
A
B
C
D
Complex Numbers
Teac
her
Teac
her
Slide 43 / 106
20 Simplify:
A
B
C
D
Complex Numbers
Teac
her
Teac
her
Slide 44 / 106
21 Simplify:
A
B
C
D
Complex Numbers
Teac
her
Teac
her
Slide 45 / 106
A Complex Number is written in the form:
a is the real part b is the imaginary part
Complex Numbers
Slide 46 / 106
Slide 47 / 106
22 Which point is -5 + 3i ?
i
AB
CD
Complex Numbers
Teac
her
Teac
her
Slide 48 / 106
23 Which point is 3 - 5i ?
i
AB
CD
Complex Numbers
Teac
her
Teac
her
Slide 49 / 106
24 Points B and C are
i
AB
C
D
A Additive InverseB Multiplicitive InverseC ConjugatesD Opposites
Complex Numbers
Teac
her
Teac
her
Slide 50 / 106
Polar Number Properties
Return to Table of Contents
Slide 51 / 106
Rectangular Coordinates, (x,y), describe a points horizontal displacement by vertical displacement in a plane.
Polar Coordinates, [r, #], describe a points distance from a pole, the origin, by the angular rotation to the point.
r
>
#
Polar Properties
Slide 52 / 106
r
>
#
Point A can be described with polar coordinates 4 ways:
A
Example:[4,π/3][4,-5π/3][-4,4# /3][-4,-2# /3]
Polar Properties
Slide 53 / 106
25 Which is another way to name [5, ]
A
B
C
D
Polar Properties
Teac
her
Teac
her
Slide 54 / 106
26 Which is another way to name [4, ]
A
B
C
D
Polar Properties
Teac
her
Teac
her
Slide 55 / 106 Slide 56 / 106
Slide 57 / 106
Example: Complete the table
Complex Rectangula r Pola r Trigonometric
(3,4)
[5 , 2# /3]
3(cos # /4 +is in # /4)
4+i
Polar Properties
Polar Properties
Slide 58 / 106
Slide 59 / 106 Slide 60 / 106
Slide 61 / 106
29 Which of the following is equivalent to
A
B
CD They are all equivalent.
Polar Properties
Teac
her
Teac
her
Slide 62 / 106
Geometry of Complex Numbers
Return to Table of Contents
Slide 63 / 106
Geometric Multiplication Let u and v be complex numbers. Written in polar form u = [r,#] and v = [s,# ], then
uv=[rs, #+# ]
Geometric Addition Let u= a + bi and v= c + di be complex numbers. then
u+v=(a+c) + (b+d)i
Geometry of Complex Numbers
Slide 64 / 106
30 Let w = 4 + 2i and z= -3 +5i, how far to the right of the origin is w + z?
Geometry of Complex Numbers
Teac
her
Teac
her
Slide 65 / 106
31 Let w = 4 + 2i and z= -3 +5i, how far above the origin is w + z?
Geometry of Complex Numbers
Teac
her
Teac
her
Slide 66 / 106
32 Let w = 4 + 2i and z= -3 +5i, how far from the origin isz+w?
Geometry of Complex Numbers
Teac
her
Teac
her
Slide 67 / 106
33 Let w = 4 + 2i and z= -3 +5i, what is the angle of rotation, in degrees, of w+z?
Geometry of Complex Numbers
Teac
her
Teac
her
Slide 68 / 106
34 Let w = 4 + 2i and z= -3 +5i, how far from the origin is wz?
Geometry of Complex Numbers
Teac
her
Teac
her
Slide 69 / 106
35 Let w = 4 + 2i and z= -3 +5i, what is the angle of rotation, in degrees, is zw?
Geometry of Complex NumbersTe
ache
rTe
ache
r
Slide 70 / 106
Polar Equations and Graphs
Return to Table of Contents
Slide 71 / 106
Polar coordinates are graphed on polar grid.
Polar Equations and Graphs
Slide 72 / 106
Rectangular Polar
r
r=f(#) r=f(#)
Polar Equations and Graphs
Slide 73 / 106
2 4 6 8 10 12
Graph [7,3# /4]
Polar Equations and Graphs
Teac
her
Teac
her
Slide 74 / 106
2 4 6 8 10 12
Graph r = 9
Polar Equations and Graphs
Teac
her
Teac
her
Slide 75 / 106
2 4 6 8 10 12
Graph θ = π/4
Polar Equations and GraphsTe
ache
rTe
ache
r
Slide 76 / 106
2 4 6 8 10 12
Graph r = 2sinθ
Polar Equations and Graphs
Teac
her
Teac
her
Slide 77 / 106
2 4 6 8 10 12
Graph r = 1 + 2sin θ
This graph is called a limacon?
Polar Equations and Graphs
Teac
her
Teac
her
Slide 78 / 106
Polar:Rose Curves and Spirals
Return to Table of Contents
Slide 79 / 106
Rose Curvesr = a sin(nθ)r = a cos(nθ)
a is the length of the 'petals'if n is even there are 2n 'petals'if n is odd there are n 'petals'
Rose Curves and Spirals
Slide 80 / 106
36 What is the length of the 'petal' of r = 6 cos
Rose Curves and Spirals
Teac
her
Teac
her
Slide 81 / 106 Slide 82 / 106
38 What is the length of the 'petal' of r = 2 cos
Rose Curves and Spirals
Teac
her
Teac
her
Slide 83 / 106 Slide 84 / 106
Slide 85 / 106
Limacon,
Rose Curves and Spirals
Slide 86 / 106
Complex Numbers: Powers
Return to Table of Contents
Slide 87 / 106 Slide 88 / 106
Examples: Compute the power of complex number. Write your answer in the same form as the original.
Powers
Teac
her
Teac
her
Slide 89 / 106
40 How far is from the origin?
Powers
Teac
her
Teac
her
Slide 90 / 106
41 What is position relative to the x-axis?
Powers
Teac
her
Teac
her
Slide 91 / 106
Examples: Compute the power of complex number. Write your answer in the same form as the original.
Powers
Teac
her
Teac
her
Slide 92 / 106
Slide 93 / 106 Slide 94 / 106
Slide 95 / 106
44 How far is (5,6)4 from the origin?
Powers
Teac
her
Teac
her
Slide 96 / 106
45 What is (5,6)4 position relative to the x-axis?
Powers
Teac
her
Teac
her
Slide 97 / 106
Examples: Compute the power of complex number. Write your answer in the same form as the original.
Powers
Teac
her
Teac
her
Slide 98 / 106
46 How far is (-2 + 7i)6 from the origin?
Powers
Teac
her
Teac
her
Slide 99 / 106
47 What is (-2 + 7i)6 position relative to the x-axis?
PowersTe
ache
rTe
ache
r
Slide 100 / 106
Complex Numbers: Roots
Return to Table of Contents
Slide 101 / 106
Finding Roots of Complex Numbers
Use rules for exponents and DeMoivre's Theorem.Example: Find the cube root of -8i
Roots
Slide 102 / 106
Notice there were 3 roots because of the cube root, so k=0, 1, 2.
In general the nth root will have n roots and k=0, 1, 2, ..., n-1
Roots
Slide 103 / 106
48 When calculating the fourth root of 3i, how many roots are there?
Roots
Teac
her
Teac
her
Slide 104 / 106
49 When calculating the fourth root of 3i, how far,in radians, will the space be between roots?
Roots
Teac
her
Teac
her
Slide 105 / 106
50 When calculating the fourth root of 3i, what is the root's position when k=0?
RootsTe
ache
rTe
ache
r
Slide 106 / 106
51 When calculating the fourth root of 3i, what is the radius?
Roots
Teac
her
Teac
her
