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Complex Roots by Mr Hardman

Complex Roots by Mr Hardman
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2015-08-09 07:19
FunctionCoordinate GeometryComplex Number
This example illustrates the concept of complex roots. It involves the geometric representation and algebraic properties of complex numbers. Using GeoGebra commands such as Sequence, Point, Distance, etc., it dynamically demonstrates the properties of complex roots. Suitable for high school mathematics education, particularly in complex numbers and coordinate geometry.

GeoGebra Commands & Parameters

  1. Sequence((a^(1 / n) * (cos(((2 * π) * k / n)) + (ί * sin(((2 * π) * k / n))))), k, 1, n) → list2
  2. Point(yAxis) → A
  3. Point(list2) → B
  4. Distance(B, (0, 0)) → f
  5. Textfield(a) → inputBox1
  6. Intersect(xAxis, yAxis) → C
  7. Point(list2) → D
  8. Point(list2) → E
  9. Angle(D, C, E) → α
  10. Intersect(e, xAxis, 2) → F
  11. Point(list2) → G
  12. Point(list2) → H
  13. Point(list2) → I
  14. Point(list2) → J
  15. Angle(Element[list2, n], (0, 0), Element[list2, 1]) → δ
  16. Segment(J, A) → d
  17. Point(list2) → K
  18. Point(list2) → L
  19. Segment(Element[list2, n], (0, 0)) → j
  20. Polygon(list2) → poly1
  21. FractionText(arg(a) / π) → text10
  22. FractionText(θ) → text11
  23. FractionText(2 / n) → text12
  24. FractionText(Element[list4, 1]) → text14
  25. FractionText(Element[list4, 2]) → text15
  26. FractionText(Element[list4, 3]) → text16
  27. FractionText(Element[list4, 4]) → text20
  28. FractionText(Element[list4, 5]) → text21
  29. FractionText(Element[list4, 6]) → text22
  30. FractionText(Element[list4, 7]) → text23
  31. FractionText(Element[list4, 8]) → text24
  32. FractionText(Element[list4, 9]) → text25
  33. FractionText(Element[list4, 10]) → text26
  34. Sequence((a_1^(1 / n) * (cos(((2 * π) * k / n)) + (ί * sin(((2 * π) * k / n))))), k, 1, n) → list2_1
  35. Point(list2_1) → B_1
  36. Textfield(a_1) → inputBox1_1
  37. Point(list2_1) → D_1
  38. Point(list2_1) → E_1
  39. Point(list2_1) → G_1
  40. Point(list2_1) → H_1
  41. Point(list2_1) → I_1
  42. Point(list2_1) → J_1
  43. Point(list2_1) → K_1
  44. Point(list2_1) → L_1
  45. Polygon(list2_1) → poly1_1
  46. FractionText(Element[list4_1, 3]) → text16_1
  47. Sequence((a_2^(1 / n) * (cos(((2 * π) * k / n)) + (ί * sin(((2 * π) * k / n))))), k, 1, n) → list2_2
  48. Point(list2_2) → B_2
  49. Textfield(a_2) → inputBox1_2
  50. Point(list2_2) → D_2
  51. Point(list2_2) → E_2
  52. Point(list2_2) → G_2
  53. Point(list2_2) → H_2
  54. Point(list2_2) → I_2
  55. Point(list2_2) → J_2
  56. Point(list2_2) → K_2
  57. Point(list2_2) → L_2
  58. Polygon(list2_2) → poly1_2
  59. FractionText(Element[list4_2, 4]) → text20_1
  60. Sequence((a_3^(1 / n) * (cos(((2 * π) * k / n)) + (ί * sin(((2 * π) * k / n))))), k, 1, n) → list2_3
  61. Point(list2_3) → B_3
  62. Textfield(a_3) → inputBox1_3
  63. Point(list2_3) → D_3
  64. Point(list2_3) → E_3
  65. Point(list2_3) → G_3
  66. Point(list2_3) → H_3
  67. Point(list2_3) → I_3
  68. Point(list2_3) → J_3
  69. Point(list2_3) → K_3
  70. Point(list2_3) → L_3
  71. Polygon(list2_3) → poly1_3
  72. FractionText(Element[list4_3, 6]) → text22_1
  73. Sequence((a_4^(1 / n) * (cos(((2 * π) * k / n)) + (ί * sin(((2 * π) * k / n))))), k, 1, n) → list2_4
  74. Point(list2_4) → B_4
  75. Textfield(a_4) → inputBox1_4
  76. Point(list2_4) → D_4
  77. Point(list2_4) → E_4
  78. Point(list2_4) → G_4
  79. Point(list2_4) → H_4
  80. Point(list2_4) → I_4
  81. Point(list2_4) → J_4
  82. Point(list2_4) → K_4
  83. Point(list2_4) → L_4
  84. Polygon(list2_4) → poly1_4
  85. FractionText(Element[list4_4, 6]) → text22_2
  86. Sequence((a_5^(1 / n) * (cos(((2 * π) * k / n)) + (ί * sin(((2 * π) * k / n))))), k, 1, n) → list2_5
  87. Point(list2_5) → B_5
  88. Textfield(a_5) → inputBox1_5
  89. Point(list2_5) → D_5
  90. Point(list2_5) → E_5
  91. Point(list2_5) → G_5
  92. Point(list2_5) → H_5
  93. Point(list2_5) → I_5
  94. Point(list2_5) → J_5
  95. Point(list2_5) → K_5
  96. Point(list2_5) → L_5
  97. Polygon(list2_5) → poly1_5
  98. FractionText(Element[list4_5, 7]) → text23_1
  99. Sequence((a_6^(1 / n) * (cos(((2 * π) * k / n)) + (ί * sin(((2 * π) * k / n))))), k, 1, n) → list2_6
  100. Point(list2_6) → B_6
  101. Textfield(a_6) → inputBox1_6
  102. Point(list2_6) → D_6
  103. Point(list2_6) → E_6
  104. Point(list2_6) → G_6
  105. Point(list2_6) → H_6
  106. Point(list2_6) → I_6
  107. Point(list2_6) → J_6
  108. Point(list2_6) → K_6
  109. Point(list2_6) → L_6
  110. Polygon(list2_6) → poly1_6
  111. FractionText(Element[list4_6, 8]) → text24_1
  112. Sequence((a_7^(1 / n) * (cos(((2 * π) * k / n)) + (ί * sin(((2 * π) * k / n))))), k, 1, n) → list2_7
  113. Point(list2_7) → B_7
  114. Textfield(a_7) → inputBox1_7
  115. Point(list2_7) → D_7
  116. Point(list2_7) → E_7
  117. Point(list2_7) → G_7
  118. Point(list2_7) → H_7
  119. Point(list2_7) → I_7
  120. Point(list2_7) → J_7
  121. Point(list2_7) → K_7
  122. Point(list2_7) → L_7
  123. Polygon(list2_7) → poly1_7
  124. FractionText(Element[list4_7, 10]) → text25_1
  125. Vector((0, 0), a) → u

AI Drawing Prompt Reference

Draw the geometric representation of the n-th roots of a complex number. Start by taking a point A on the x-axis, then generate n complex roots based on the given complex number a and the definition of n-th roots, and plot them on the complex plane to form a regular polygon. Calculate the distance f from each root to the origin, and mark the angle α. Draw line segments from the origin to each root, and calculate the angle δ between each root and its adjacent root. Finally, draw a polygon containing all the roots and label the argument of each root.
The prompt above was compiled from the original drawing commands and is for reference only.