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: 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 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 Teacher Teacher
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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
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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.
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 NumbersTe
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rTe
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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:
A
B
C
D
Complex Numbers
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11 Simplify:
A
B
C
D
Complex Numbers
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12 Simplify:
A
B
C
D
Complex Numbers
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13 Simplify:
A
B
C
D
Complex Numbers
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14 Simplify:
A
B
C
D
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)
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 NumbersTe
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Simplify
AnswersComplex Numbers
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15 Simplify
A
B
C
D
Complex Numbers
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17 Simplify
A
B
C
D
Complex Numbers
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Simplify:
Complex Numbers
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19 Simplify:
A
B
C
D
Complex Numbers
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20 Simplify:
A
B
C
D
Complex Numbers
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21 Simplify:
A
B
C
D
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
Complex Numbers
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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
D
A Additive InverseB Multiplicitive InverseC ConjugatesD Opposites
Complex Numbers
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Polar Number Properties
Return to Table of Contents
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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: