Pre-Calculus -...
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Slide 1 / 106 Pre-Calculus Polar & Complex Numbers www.njctl.org 2015-03-23 Slide 2 / 106 Table of Contents Complex Numbers Geometry of Complex Numbers Complex Numbers: Powers Complex Numbers: Roots Polar Number Properties Polar Equations and Graphs Polar: Rose Curves and Spirals click on the topic to go to that section Slide 3 / 106
Transcript of Pre-Calculus -...
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Slide 1 / 106
Pre-Calculus
Polar & Complex Numbers
www.njctl.org
2015-03-23
Slide 2 / 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
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Complex Numbers
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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
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2 Simplify
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B
C
D
Complex Numbers
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5 Simplify
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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
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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
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6 Simplify
A i
B -1
C -i
D 1
Complex Numbers
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7 Simplify
A i
B -1
C -i
D 1
Complex Numbers
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8 Simplify
A i
B -1
C -i
D 1
Complex Numbers
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9 Simplify
A i
B -1
C -i
D 1
Complex Numbers
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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
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Examples:
Complex Numbers
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Examples (in the complex form the real term comes first)
Complex Numbers
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Examples
Complex Numbers
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10 Simplify:
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B
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D
Complex Numbers
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11 Simplify:
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D
Complex Numbers
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12 Simplify:
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Complex Numbers
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13 Simplify:
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Complex Numbers
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14 Simplify:
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Complex Numbers
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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
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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
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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
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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
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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 Numbers
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Simplify
AnswersComplex Numbers
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15 Simplify
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B
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Complex Numbers
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17 Simplify
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Complex Numbers
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Simplify:
Complex Numbers
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19 Simplify:
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B
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D
Complex Numbers
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20 Simplify:
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Complex Numbers
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21 Simplify:
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Complex Numbers
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A Complex Number is written in the form:
a is the real part b is the imaginary part
Complex Numbers
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22 Which point is -5 + 3i ?
i
AB
CD
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23 Which point is 3 - 5i ?
i
AB
CD
Complex Numbers
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24 Points B and C are
i
AB
C
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A Additive InverseB Multiplicitive InverseC ConjugatesD Opposites
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Polar Number Properties
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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
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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
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25 Which is another way to name [5, ]
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Polar Properties
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26 Which is another way to name [4, ]
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Polar Properties
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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
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29 Which of the following is equivalent to
A
B
CD They are all equivalent.
Polar Properties
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Geometry of Complex Numbers
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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
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30 Let w = 4 + 2i and z= -3 +5i, how far to the right of the origin is w + z?
Geometry of Complex Numbers
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31 Let w = 4 + 2i and z= -3 +5i, how far above the origin is w + z?
Geometry of Complex Numbers
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32 Let w = 4 + 2i and z= -3 +5i, how far from the origin isz+w?
Geometry of Complex Numbers
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33 Let w = 4 + 2i and z= -3 +5i, what is the angle of rotation, in degrees, of w+z?
Geometry of Complex Numbers
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34 Let w = 4 + 2i and z= -3 +5i, how far from the origin is wz?
Geometry of Complex Numbers
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35 Let w = 4 + 2i and z= -3 +5i, what is the angle of rotation, in degrees, is zw?
Geometry of Complex Numbers
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Polar Equations and Graphs
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Polar coordinates are graphed on polar grid.
Polar Equations and Graphs
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Rectangular Polar
r
r=f(#) r=f(#)
Polar Equations and Graphs
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2 4 6 8 10 12
Graph [7,3# /4]
Polar Equations and Graphs
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2 4 6 8 10 12
Graph r = 9
Polar Equations and Graphs
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2 4 6 8 10 12
Graph θ = π/4
Polar Equations and Graphs
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2 4 6 8 10 12
Graph r = 2sinθ
Polar Equations and Graphs
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2 4 6 8 10 12
Graph r = 1 + 2sin θ
This graph is called a limacon?
Polar Equations and Graphs
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Polar:Rose Curves and Spirals
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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
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36 What is the length of the 'petal' of r = 6 cos
Rose Curves and Spirals
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38 What is the length of the 'petal' of r = 2 cos
Rose Curves and Spirals
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Limacon,
Rose Curves and Spirals
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Complex Numbers: Powers
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Examples: Compute the power of complex number. Write your answer in the same form as the original.
Powers
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40 How far is from the origin?
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41 What is position relative to the x-axis?
Powers
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Examples: Compute the power of complex number. Write your answer in the same form as the original.
Powers
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44 How far is (5,6)4 from the origin?
Powers
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45 What is (5,6)4 position relative to the x-axis?
Powers
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Examples: Compute the power of complex number. Write your answer in the same form as the original.
Powers
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46 How far is (-2 + 7i)6 from the origin?
Powers
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47 What is (-2 + 7i)6 position relative to the x-axis?
Powers
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Complex Numbers: Roots
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Finding Roots of Complex Numbers
Use rules for exponents and DeMoivre's Theorem.Example: Find the cube root of -8i
Roots
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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
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48 When calculating the fourth root of 3i, how many roots are there?
Roots
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49 When calculating the fourth root of 3i, how far,in radians, will the space be between roots?
Roots
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50 When calculating the fourth root of 3i, what is the root's position when k=0?
Roots
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51 When calculating the fourth root of 3i, what is the radius?
Roots
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