Using Complex Numbers - Comprehensive Course on De Moivre's Theorem, Argand Diagrams, and Complex Analysis

Using Complex Numbers - Comprehensive Course on De Moivre's Theorem, Argand Diagrams, and Complex Analysis

Eddie Woo via YouTube Direct link

Sketching (z-1)รท(z-i) (2 of 2: When it's imaginary)

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Sketching (z-1)รท(z-i) (2 of 2: When it's imaginary)

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Classroom Contents

Using Complex Numbers - Comprehensive Course on De Moivre's Theorem, Argand Diagrams, and Complex Analysis

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  1. 1 Using De Moivre's Theorem to Prove Trigonometric Identity
  2. 2 Understanding & Applying the Conjugate Root Theorem
  3. 3 Argand Diagram / Locus Question
  4. 4 Interesting Complex Polynomial Question (1 of 2: Factorisation)
  5. 5 Interesting Complex Polynomial Question (2 of 2: Trigonometric Result)
  6. 6 Complex Numbers as Points (1 of 4: Geometric Meaning of Addition)
  7. 7 Complex Numbers as Points (2 of 4: Geometric Meaning of Subtraction)
  8. 8 Complex Numbers as Points (3 of 4: Geometric Meaning of Multiplication)
  9. 9 Complex Numbers as Points (4 of 4: Second Multiplication Example)
  10. 10 De Moivre's Theorem
  11. 11 How to graph the locus of |z-1|=1
  12. 12 Complex Numbers as Vectors (1 of 3: Introduction & Addition)
  13. 13 Complex Numbers as Vectors (2 of 3: Subtraction)
  14. 14 Complex Numbers as Vectors (3 of 3: Using Geometric Properties)
  15. 15 Complex Roots (2 of 5: Expanding in Rectangular Form)
  16. 16 The Triangle Inequalities (1 of 3: Sum of Complex Numbers)
  17. 17 Graphs in the Complex Plane (1 of 4: Introductory Examples)
  18. 18 The Triangle Inequalities (2 of 3: Discussing Specific Cases)
  19. 19 The Triangle Inequalities (3 of 3: Difference of Complex Numbers)
  20. 20 Graphs in the Complex Plane (2 of 4: Graphing Complex Inequalities)
  21. 21 Graphs in the Complex Plane (3 of 4 : Shifting the Point of Reference)
  22. 22 Graphs in the Complex Plane (4 of 4: Where is the argument measured from?)
  23. 23 Further Graphs on the Complex Plane (1 of 3: Geometrical Representation of Moduli)
  24. 24 Further Graphs on the Complex Plane (2 of 3: Algebraically verifying Graphs concerning the Moduli)
  25. 25 Further Graphs on the Complex Plane (3 of 3: Geometrical Representation of Arguments)
  26. 26 Graphs on the Complex Plane (3 of 4: Geometry of arg(z)-arg(z-1))
  27. 27 Graphs on the Complex Plane (4 of 4: Exploring how the argument traced the graph)
  28. 28 Graphs on the Complex Plane [Continued] (1 of 4: What's behind the graph?)
  29. 29 Graphs on the Complex Plane [Continued] (2 of 4: Finding Regions of Inequality by Testing Points)
  30. 30 Using Inverse tan to find arguments? (1 of 2: Why it doesn't work... Sometimes)
  31. 31 Complex Roots (1 of 5: Introduction)
  32. 32 Complex Roots (3 of 5: Through Polar Form Using De Moivre's Theorem)
  33. 33 Using Inverse tan to find arguments? (2 of 2: Why it works... Sometimes)
  34. 34 Complex Roots (4 of 5: Through Polar Form Generating Solutions)
  35. 35 Complex Roots (5 of 5: Flowing Example - Solving z^6=64)
  36. 36 Complex Conjugate Root Theorem (1 of 4: Using DMT and Polar Form to solve for Complex Roots)
  37. 37 Complex Conjugate Root Theorem (2 of 4: Introduction to the Conjugate Root Theorem)
  38. 38 Complex Conjugate Root Theorem (3 of 4: Geometrical Shape represented by Conjugate Root Theorem)
  39. 39 Complex Conjugate Root Theorem (4 of 4: Using Factorisation to find patterns with Roots of Unity)
  40. 40 DMT and Trig Identities (1 of 4: Deriving multi-angle identities with compound angles)
  41. 41 DMT and Trig Identities (2 of 4: Using De Moivre's Theorem and Binomial Expansions)
  42. 42 DMT and Trig Identities (3 of 4: Deriving tan expression from cos and sin)
  43. 43 DMT and Trig Identities (4 of 4: Using Multi-angle formula to solve polynomials)
  44. 44 Complex Numbers (1 of 6: Solving Harder Complex Numbers Questions) [Student requested problem]
  45. 45 Complex Numbers (2 of 6: Solving Harder Complex Numbers Questions) [Student Requested Problem]
  46. 46 Complex Numbers (3 of 6: Harder Complex Numbers Question) [Student Requested Problem]
  47. 47 Complex Numbers (4 of 6: Harder Complex Numbers Questions) [Student Requested Problem]
  48. 48 Complex Numbers (5 of 6: Complex Numbers Proofs [Using the Conjugate])
  49. 49 Complex Numbers (6 of 6: Finishing off the Proof)
  50. 50 Roots and Coefficients (1 of 3: Using DMT & Binomial Theorem to find identities)
  51. 51 Roots and Coefficients (2 of 3: Using Trigonometry to solve polynomial problems)
  52. 52 Roots and Coefficients (3 of 3: Using the results to find a relation in cosine)
  53. 53 Extension II Assessment Review (5 of 5: De Moivre's Theorem and Polynomials)
  54. 54 Why Complex Numbers? (1 of 5: Atoms & Strings)
  55. 55 Why Complex Numbers? (2 of 5: Impossible Roots)
  56. 56 Why Complex Numbers? (3 of 5: The Imaginary Unit)
  57. 57 Why Complex Numbers? (4 of 5: Turning the key)
  58. 58 Why Complex Numbers? (5 of 5: Where to now?)
  59. 59 Complex Arithmetic (1 of 2: Addition, Subtraction & Multiplication)
  60. 60 Complex Arithmetic (2 of 2: Division)
  61. 61 Factorisation with Complex Numbers
  62. 62 Vectors (1 of 4: Outline of vectors and their ability to represent complex number)
  63. 63 Vectors (2 of 4: Representing addition & subtraction of complex numbers with vectors)
  64. 64 Vectors (3 of 4: Geometrically representing multiplication of complex numbers with vectors)
  65. 65 Vectors (4 of 4: Outlining the usefulness of vectors in representing geometry)
  66. 66 Curves and Regions on the Complex Plane (1 of 4: Introductory example plotting |z|=5 geometrically)
  67. 67 Curves and Regions on the Complex Plane (2 of 4: Deciphering terminology to plot complex numbers)
  68. 68 Curves and Regions on the Complex Plane (3 of 4: Simplifying expressions to plot on a complex plane)
  69. 69 Curves and Regions on the Complex Plane (4 of 4: Plotting simultaneous shifted complex numbers)
  70. 70 Further Curves and Regions (1 of 5: Why does Sine & the Sine Rule produce an ambiguous case?)
  71. 71 Further Curves and Regions (2 of 5: Finding the Range of |z| )
  72. 72 Further Curves and Regions (3 of 5: Finding the range of arg z)
  73. 73 Further Curves and Regions (4 of 5: Geometrical expression of expressions of arg)
  74. 74 Further Curves and Regions (5 of 5: Using Circle properties to graph an expression of args)
  75. 75 DMT & Complex Roots (1 of 4: Reviewing geometrical expression of arg equations)
  76. 76 DMT & Complex Roots (2 of 4: Using DMT to find roots of a complex polynomial)
  77. 77 DMT & Complex Roots (3 of 4: Using the fundamental theorem of algebra to justify number of roots)
  78. 78 DMT & Complex Roots (4 of 4: Solving for roots of a complex number taking advantage of DMT)
  79. 79 Complex Roots (1 of 5: Observing Complex Conjugate Root Theorem through seventh roots of unity)
  80. 80 Complex Roots (2 of 5: Using Trigonometrical Identities & Conjugates to solve an equation)
  81. 81 Complex Roots (3 of 5: Using DMT to solve an equation of roots of unity)
  82. 82 Complex Roots (4 of 5: Using Polynomial Identities to prove unity identities)
  83. 83 Complex Roots (5 of 5: Using Geometric Progression to find factors of ฯ‰^n - 1)
  84. 84 DMT and Trig Identities (1 of 4: Noticing a pattern in natural numbers)
  85. 85 DMT and Trig Identities (2 of 4: Using Trig expansion to find the sine triple angle formula)
  86. 86 DMT and Trig Identities (3 of 4: Using DMT and Polynomials to verify triple angle formula for sine)
  87. 87 DMT and Trig Identities (4 of 4: Using Trig Identities to solve polynomial equations)
  88. 88 HSC Question on Complex Numbers, Vectors & Triangle Area (1 of 2: Thinking geometrically)
  89. 89 HSC Question on Complex Numbers, Vectors & Triangle Area (2 of 2: Manipulating trigonometric terms)
  90. 90 HSC Question on Complex Numbers, Vectors & Polynomials (1 of 2: How to "explain")
  91. 91 HSC Question on Complex Numbers, Vectors & Polynomials (2 of 2: Combining results)
  92. 92 2016 HSC - Complex Numbers on Unit Circle (1 of 2: Considering Re & Im Parts)
  93. 93 2016 HSC - Complex Numbers on Unit Circle (2 of 2: Evaluating the arguments)
  94. 94 2016 HSC - Complex Identity Proof (1 of 3: Convert to polar form)
  95. 95 2016 HSC - Complex Identity Proof (2 of 3: Using binomial theorem)
  96. 96 2016 HSC - Complex Identity Proof (3 of 3: Combining results)
  97. 97 Semi Circles on Argand Diagrams (3 of 3: Oblique example)
  98. 98 Semi Circles on Argand Diagrams (2 of 3: Graphing the locus)
  99. 99 Semi Circles on Argand Diagrams (1 of 3: Relating the angles)
  100. 100 Algebraic Proof for Opposing Rays (3 of 3: Testing cases)
  101. 101 Algebraic Proof for Opposing Rays (2 of 3: Generating the equation)
  102. 102 Algebraic Proof for Opposing Rays (1 of 3: Foundational knowledge)
  103. 103 cosโดฮธ Identity
  104. 104 Graphs in the Complex Plane (3 of 3: Opposing rays)
  105. 105 Graphs in the Complex Plane (2 of 3: Algebraic method)
  106. 106 Graphs in the Complex Plane (1 of 3: Perpendicular bisector - visual method)
  107. 107 Complex Polynomial Identity Question (4 of 4: Roots & coefficients)
  108. 108 Complex Polynomial Identity Question (3 of 4: de Moivre's Theorem)
  109. 109 Complex Polynomial Identity Question (2 of 4: Difference of cubes)
  110. 110 Complex Polynomial Identity Question (1 of 4: Quadratic factors)
  111. 111 Sum & Product of Cosines (3 of 3: Drawing โ„ conclusions)
  112. 112 Sum & Product of Cosines (2 of 3: Simplifying with conjugates)
  113. 113 Sum & Product of Cosines (1 of 3: 9th roots of unity)
  114. 114 Nth Roots of a โ„‚ Number (2 of 2: Example problem)
  115. 115 Nth Roots of a โ„‚ Number (1 of 2: General form)
  116. 116 Roots of Unity (2 of 2: Insights from polar & exponential forms)
  117. 117 Roots of Unity (1 of 2: Evaluating the cube roots)
  118. 118 Complex Conjugate Root Theorem (2 of 2: Other conjugate properties)
  119. 119 Complex Conjugate Root Theorem (1 of 2: Conjugate of a sum)
  120. 120 Solving Higher Degree Trigonometric Equations (3 of 3: Finding solutions)
  121. 121 Solving Higher Degree Trigonometric Equations (2 of 3: Combining results into proof)
  122. 122 Solving Higher Degree Trigonometric Equations (1 of 3: Initial use of de Moivre's Theorem)
  123. 123 Equations with Complex Solutions (2 of 2: Solving & factorising)
  124. 124 Equations with Complex Solutions (1 of 2: Relation to square roots)
  125. 125 Polynomials with Trigonometric Solutions (3 of 3: Simplifying with identities)
  126. 126 Polynomials with Trigonometric Solutions (2 of 3: Substitute & solve)
  127. 127 Polynomials with Trigonometric Solutions (1 of 3: de Moivre's Theorem)
  128. 128 Trigonometric Expansions from Complex Numbers (3 of 3: General compound angles)
  129. 129 Trigonometric Expansions from Complex Numbers (2 of 3: Double angle results)
  130. 130 Trigonometric Expansions from Complex Numbers (1 of 3: Concept map)
  131. 131 Using de Moivre's Theorem - example question (2 of 2: Purely imaginary)
  132. 132 Using de Moivre's Theorem - example question (1 of 2: Purely real)
  133. 133 Proving de Moivre's Theorem (2 of 2: Derivation & example problem)
  134. 134 Proving de Moivre's Theorem (1 of 2: Prologue)
  135. 135 Arcs on the Complex Plane (1 of 4: Review questions)
  136. 136 Arcs on the Complex Plane (3 of 4: Identifying length and direction)
  137. 137 Arcs on the Complex Plane (2 of 4: Exploring circle properties)
  138. 138 Arcs on the Complex Plane (4 of 4: Cartesian equation)
  139. 139 Angles in the Same Segment
  140. 140 Algebraic Approach for Major Arc (1 of 2: Foundational steps)
  141. 141 Algebraic Approach for Major Arc (2 of 2: Identifying intercept & equation)
  142. 142 Max/Min Value of |z| (1 of 2: Geometric solution)
  143. 143 Max/Min Value of |z| (2 of 2: Triangle inequality)
  144. 144 Investigating de Moivre's Theorem (1 of 3: Why must we be cautious?)
  145. 145 Investigating de Moivre's Theorem (2 of 3: Infinite values for i-th powers?!)
  146. 146 Investigating de Moivre's Theorem (3 of 3: Proof by mathematical induction)
  147. 147 Varying |z| & Argz on a Locus
  148. 148 Maximising Sum of Moduli (1 of 3: Geometric approach)
  149. 149 Maximising Sum of Moduli (2 of 3: Differentiation)
  150. 150 Maximising Sum of Moduli (3 of 3: Interpreting stationary points)
  151. 151 Sketching (z-1)รท(z-i) (1 of 2: When it's real)
  152. 152 Sketching (z-1)รท(z-i) (2 of 2: When it's imaginary)
  153. 153 Complex Numbers Exam Review (1 of 4: Visualising & Manipulating Arithmetic)
  154. 154 Complex Numbers Exam Review (2 of 4: Proving i^i is real, identity proof)
  155. 155 Complex Numbers Exam Review (3 of 4: Cube roots of unity)
  156. 156 Complex Numbers Exam Review (4 of 4: Locus; polynomial identity)
  157. 157 Prove arg(zโ‚zโ‚‚) = arg(zโ‚+zโ‚‚)ยฒ (1 of 2: Preliminary thoughts)
  158. 158 Prove arg(zโ‚zโ‚‚) = arg(zโ‚+zโ‚‚)ยฒ (2 of 2: Geometric approach)
  159. 159 Complex Geometry - Square Problem (1 of 2: Complex numbers โ†’ vectors)
  160. 160 Complex Geometry - Square Problem (2 of 2: Vectors โ†’ complex numbers)
  161. 161 Complex Geometry - Equilateral Triangle (3 of 3: Vector proof)
  162. 162 Complex Geometry - Equilateral Triangle (2 of 3: Algebraic method)
  163. 163 Complex Geometry - Equilateral Triangle (1 of 3: Arithmetic proof)
  164. 164 The Basel Problem (9 of 9: Squeeze law)
  165. 165 The Basel Problem (8 of 9: Returning to trigonometric terms)
  166. 166 The Basel Problem (7 of 9: Manipulating the polynomial integral)
  167. 167 The Basel Problem (5 of 9: Telescoping sum)
  168. 168 The Basel Problem (6 of 9: Equations โ†’ inequalities)
  169. 169 The Basel Problem (4 of 9: Introducing xยฒ to the integrand)
  170. 170 The Basel Problem (3 of 9: Integration by *different* parts)
  171. 171 The Basel Problem (1 of 9: Prologue)
  172. 172 The Basel Problem (2 of 9: Recurrence relation)
  173. 173 Centre of a Major Arc (5 of 5: Algebraic proof)
  174. 174 Centre of a Major Arc (4 of 5: Inscribed equilateral triangle)
  175. 175 Centre of a Major Arc (3 of 5: Using trigonometric & vectors)
  176. 176 Centre of a Major Arc (2 of 5: Finding centre and radius)
  177. 177 Centre of a Major Arc (1 of 5: Evaluating internal angle)
  178. 178 How to graph a region on the complex plane
  179. 179 Three ways to find a parallelogram's area

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