Pre-Calculus Polar & Complex Numbers

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2 Slide 2 / 106 Pre-Calculus Polar & Complex Numbers

3 Slide 3 / 106 Table of Contents click on the topic to go to that section Complex Numbers Polar Number Properties Geometry of Complex Numbers Polar Equations and Graphs Polar: Rose Curves and Spirals Complex Numbers: Powers Complex Numbers: Roots

4 Slide 4 / 106 Complex Numbers Return to Table of Contents

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6 Slide 6 / 106 Complex Numbers 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?

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9 Slide 9 / 106 Complex Numbers 2 Simplify A B C D

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12 Slide 12 / 106 Complex Numbers 5 Simplify A B C D

13 Slide 13 / 106 Complex Numbers 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 i 2 i 3 i 4 i 5 =i 4 i i 6 = i 4 i 2 i 7 = i 4 i 3 i 8 = i 4 i 4 i 9 = i 4 i 4 i i 10 = i 4 i 4 i 2 i 11 = i 4 i 4 i 3 i 12 = i 4 i 4 i 4 i 13 = i 4 i 4 i 4 i i 14 = i 4 i 4 i 4 i 2 i 15 = i 4 i 4 i 4 i 3 i 16 = i 4 i 4 i 4 i i raised to a power can be rewritten as a product of i 4 's and an i to the 1 st to the 4 th. Since each i 4 = 1, we need only be concerned with the non-power of 4.

14 Slide 14 / 106 Complex Numbers To simplify an i without writing out the table say i 87, divide by 4. The number of times 4 goes in evenly gives you that many i 4 's. The remainder is the reduced power. Simplify. Example: Simplify

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16 Slide 16 / 106 Complex Numbers 6 Simplify A i B -1 C -i D 1

17 Slide 17 / 106 Complex Numbers 7 Simplify A i B -1 C -i D 1

18 Slide 18 / 106 Complex Numbers 8 Simplify A i B -1 C -i D 1

19 Slide 19 / 106 Complex Numbers 9 Simplify A i B -1 C -i D 1

20 Slide 20 / 106 Complex Numbers Recall: 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:

21 Slide 21 / 106 Complex Numbers Examples:

22 Slide 22 / 106 Complex Numbers Examples (in the complex form the real term comes first)

23 Slide 23 / 106 Complex Numbers Examples

24 Slide 24 / 106 Complex Numbers 10 Simplify: A B C D

25 Slide 25 / 106 Complex Numbers 11 Simplify: A B C D

26 Slide 26 / 106 Complex Numbers 12 Simplify: A B C D

27 Slide 27 / 106 Complex Numbers 13 Simplify: A B C D

28 Slide 28 / 106 Complex Numbers 14 Simplify: A B C D

29 Slide 29 / 106 Complex Numbers 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.

30 Slide 30 / 106 Complex Numbers Conductors Some 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?

31 Slide 31 / 106 Complex Numbers Current vs Resistance & Voltage 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

32 Slide 32 / 106 Complex Numbers Ohm's Law V is for voltage, measured in volts, and is potential of a circuit. Z is for impedance, measured in ohms ( ), which is the opposition to the flow of current. The total impedance of a circuit is a complex number. I is for current, measured in amps, the rate of flow of a circuit.

33 Slide 33 / 106 Complex Numbers Application: Suppose two AC currents are connected in a series. One with i 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?

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35 Complex Numbers Simplify Slide 35 / 106 Answers

36 Slide 36 / 106 Complex Numbers 15 Simplify A B C D

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38 Slide 38 / 106 Complex Numbers 17 Simplify A B C D

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40 Slide 40 / 106 Complex Numbers Simplify:

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42 Slide 42 / 106 Complex Numbers 19 Simplify: A B C D

43 Slide 43 / 106 Complex Numbers 20 Simplify: A B C D

44 Slide 44 / 106 Complex Numbers 21 Simplify: A B C D

45 Slide 45 / 106 Complex Numbers A Complex Number is written in the form: a is the real part b is the imaginary part

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47 Slide 47 / 106 Complex Numbers 22 Which point is i? i B A C D

48 Slide 48 / 106 Complex Numbers 23 Which point is 3-5i? i B A C D

49 Slide 49 / 106 Complex Numbers 24 Points B and C are A Additive Inverse B Multiplicitive Inverse C Conjugates D Opposites i B A C D

50 Slide 50 / 106 Polar Number Properties Return to Table of Contents

51 Slide 51 / 106 Polar Properties 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

52 Slide 52 / 106 Polar Properties Point A can be described with polar coordinates 4 ways: > A # r Example: [4, π / 3 ] [4, -5π / 3 ] [-4, 4# / 3 ] [-4, -2# / 3 ]

53 Slide 53 / 106 Polar Properties 25 Which is another way to name [5, ] A B C D

54 Slide 54 / 106 Polar Properties 26 Which is another way to name [4, ] A B C D

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57 Slide 57 / 106 Polar Properties Polar Properties Example: Complete the table Complex Rectangular Polar Trigonometric (3,4) [5, 2# / 3 ] 3(cos # / 4 +isin # / 4 ) 4+i

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61 Slide 61 / 106 Polar Properties 29 Which of the following is equivalent to A B C D They are all equivalent.

62 Slide 62 / 106 Geometry of Complex Numbers Return to Table of Contents

63 Slide 63 / 106 Geometry of Complex Numbers Geometric Addition Let u= a + bi and v= c + di be complex numbers. then u+v=(a+c) + (b+d)i Geometric Multiplication Let u and v be complex numbers. Written in polar form u = [r,#] and v = [s,# ], then uv=[rs, #+# ]

64 Slide 64 / 106 Geometry of Complex Numbers 30 Let w = 4 + 2i and z= -3 +5i, how far to the right of the origin is w + z?

65 Slide 65 / 106 Geometry of Complex Numbers 31 Let w = 4 + 2i and z= -3 +5i, how far above the origin is w + z?

66 Slide 66 / 106 Geometry of Complex Numbers 32 Let w = 4 + 2i and z= -3 +5i, how far from the origin is z+w?

67 Slide 67 / 106 Geometry of Complex Numbers 33 Let w = 4 + 2i and z= -3 +5i, what is the angle of rotation, in degrees, of w+z?

68 Slide 68 / 106 Geometry of Complex Numbers 34 Let w = 4 + 2i and z= -3 +5i, how far from the origin is wz?

69 Slide 69 / 106 Geometry of Complex Numbers 35 Let w = 4 + 2i and z= -3 +5i, what is the angle of rotation, in degrees, is zw?

70 Slide 70 / 106 Polar Equations and Graphs Return to Table of Contents

71 Slide 71 / 106 Polar Equations and Graphs Polar coordinates are graphed on polar grid.

72 Slide 72 / 106 Polar Equations and Graphs Rectangular Polar r r=f(#) r=f(#)

73 Slide 73 / 106 Polar Equations and Graphs Graph [7, 3# / 4 ]

74 Slide 74 / 106 Polar Equations and Graphs Graph r =

75 Slide 75 / 106 Polar Equations and Graphs Graph θ = π /

76 Slide 76 / 106 Polar Equations and Graphs Graph r = 2sinθ

77 Slide 77 / 106 Polar Equations and Graphs Graph r = 1 + 2sin θ This graph is called a limacon?

78 Slide 78 / 106 Polar: Rose Curves and Spirals Return to Table of Contents

79 Slide 79 / 106 Rose Curves and Spirals Rose Curves r = 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'

80 Slide 80 / 106 Rose Curves and Spirals 36 What is the length of the 'petal' of r = 6 cos

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82 Slide 82 / 106 Rose Curves and Spirals 38 What is the length of the 'petal' of r = 2 cos

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85 Slide 85 / 106 Rose Curves and Spirals Limacon,

86 Slide 86 / 106 Complex Numbers: Powers Return to Table of Contents

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88 Slide 88 / 106 Powers Examples: Compute the power of complex number. Write your answer in the same form as the original.

89 Slide 89 / 106 Powers 40 How far is from the origin?

90 Slide 90 / 106 Powers 41 What is position relative to the x-axis?

91 Slide 91 / 106 Powers Examples: Compute the power of complex number. Write your answer in the same form as the original.

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95 Slide 95 / 106 Powers 44 How far is (5,6) 4 from the origin?

96 Slide 96 / 106 Powers 45 What is (5,6) 4 position relative to the x-axis?

97 Slide 97 / 106 Powers Examples: Compute the power of complex number. Write your answer in the same form as the original.

98 Slide 98 / 106 Powers 46 How far is (-2 + 7i) 6 from the origin?

99 Slide 99 / 106 Powers 47 What is (-2 + 7i) 6 position relative to the x-axis?

100 Slide 100 / 106 Complex Numbers: Roots Return to Table of Contents

101 Slide 101 / 106 Roots Finding Roots of Complex Numbers Use rules for exponents and DeMoivre's Theorem. Example: Find the cube root of -8i

102 Slide 102 / 106 Roots Notice there were 3 roots because of the cube root, so k=0, 1, 2. In general the n th root will have n roots and k=0, 1, 2,..., n-1

103 Slide 103 / 106 Roots 48 When calculating the fourth root of 3i, how many roots are there?

104 Slide 104 / 106 Roots 49 When calculating the fourth root of 3i, how far,in radians, will the space be between roots?

105 Slide 105 / 106 Roots 50 When calculating the fourth root of 3i, what is the root's position when k=0?

106 Slide 106 / 106 Roots 51 When calculating the fourth root of 3i, what is the radius?

INVESTIGATION ONE: WHAT DOES A VOLTMETER DO? How Are Values of Circuit Variables Measured?

INVESTIGATION ONE: WHAT DOES A VOLTMETER DO? How Are Values of Circuit Variables Measured? How Are Values of Circuit Variables Measured? INTRODUCTION People who use electric circuits for practical purposes often need to measure quantitative values of electric pressure difference and flow rate