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Complex Number Roots Visualization

Complex Number Roots Visualization
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2022-01-28 18:26
VectorTrigonometrySequenceComplex NumberGeometric Construction
This example demonstrates how to visualize the n-th roots of complex numbers and their geometric distribution. Key math concepts include complex numbers, root locations, polar coordinates, and trigonometric functions. Construction highlights involve commands like Sequence, Element, Intersect, Vector, Point, Segment, If, TableText, and Polygon from GeoGebra. Suitable for high school and above students to understand complex operations and geometric intuition.

GeoGebra Commands & Parameters

  1. Sequence((j^(1 / n) * (cos((atan(b / a) + ((2 * q) * π)) / n) + (ί * sin((atan(b / a) + ((2 * q) * π)) / n)))), q, 0, k) → Sequence2, {1.0311581284881526 - 0.0997039294804784ί, 0.7208682482097111 + 0.7440275050185309ί, -0.13225012600310243 + 1.0274910528433818ί, -0.8857814579246445 + 0.5372328808793527ί, -0.9723012853800516 - 0.3575726079404533ί, -0.32665841361255604 - 0.9831186298291604ί, 0.5649649062224906 - 0.8683562714911719ί}
  2. Element(Sequence2, 1) → A2
  3. Element(Sequence2, 2) → A3
  4. Element(Sequence2, 3) → A4
  5. Element(Sequence2, 4) → A5
  6. Element(Sequence2, 5) → A6
  7. Element(Sequence2, 6) → A7
  8. Element(Sequence2, 7) → A8
  9. Element(Sequence2, 8) → A9
  10. Element(Sequence2, 9) → A10
  11. Element(Sequence2, 10) → A11
  12. Intersect(xAxis, yAxis) → A
  13. Vector(A, A2) → u
  14. Vector(A, A3) → v
  15. Vector(A, A4) → w
  16. Vector(A, A5) → a_{1}
  17. Vector(A, A6) → b_{1}
  18. Vector(A, A7) → c
  19. Vector(A, A8) → d
  20. Vector(A, A9) → e
  21. Vector(A, A10) → f
  22. Vector(A, A11) → g
  23. Point(yAxis) → B
  24. Point(yAxis) → C
  25. Segment(B, C) → h
  26. Point(xAxis) → D
  27. Point(xAxis) → E
  28. Segment(D, E) → i
  29. Vector((a, b)) → p
  30. Intersect(eq1, i) → F, G
  31. Intersect(eq1, h) → H, I
  32. Element(Sequence1, 1) → B2
  33. Element(Sequence1, 2) → B3
  34. Element(Sequence1, 3) → B4
  35. Element(Sequence1, 4) → B5
  36. Element(Sequence1, 5) → B6
  37. Element(Sequence1, 6) → B7
  38. Element(Sequence1, 7) → B8
  39. Element(Sequence1, 8) → B9
  40. Element(Sequence1, 9) → B10
  41. Element(Sequence1, 10) → B11
  42. If(k ≥ 0, B2 + "") → C14
  43. If(k ≥ 1, B3 + "") → C15
  44. If(k ≥ 2, B4 + "") → C16
  45. If(k ≥ 3, B5 + "") → C17
  46. If(k ≥ 4, B6 + "") → C18
  47. If(k ≥ 5, B7 + "") → C19
  48. If(k ≥ 6, B8 + "") → C20
  49. If(k ≥ 7, B9 + "") → C21
  50. If(k ≥ 8, B10 + "") → C22
  51. If(k ≥ 9, B11 + "") → C23
  52. Sequence(Element[Sequence3, α + 1], α, 0, k) → SequenceA, {"z_1", "z_2", "z_3", "z_4", "z_5", "z_6", "z_7"}
  53. Sequence(Element[Sequence2, α + 1], α, 0, k) → SequenceB, {1.0311581284881526 - 0.0997039294804784ί, 0.7208682482097111 + 0.7440275050185309ί, -0.13225012600310243 + 1.0274910528433818ί, -0.8857814579246445 + 0.5372328808793527ί, -0.9723012853800516 - 0.3575726079404533ί, -0.32665841361255604 - 0.9831186298291604ί, 0.5649649062224906 - 0.8683562714911719ί}
  54. Sequence(Element[Sequence4, α + 1], α, 0, k) → SequenceC, {" 1.04×\left [cos\left(-0.1\right)+i\;sin\left( -0.1 \right)\right ]\cr", " 1.04×\left [cos\left(0.8\right)+i\;sin\left( 0.8 \right)\right ]\cr", " 1.04×\left [cos\left(2.15\right)+i\;sin\left( 1.7 \right)\right ]\cr", " 1.04×\left [cos\left(2.6\right)+i\;sin\left( 2.6 \right)\right ]\cr", " 1.04×\left [cos\left(3.49\right)+i\;sin\left( 3.49 \right)\right ]\cr", " 1.04×\left [cos\left(4.39\right)+i\;sin\left( 4.39 \right)\right ]\cr", " 1.04×\left [cos\left(5.29\right)+i\;sin\left( 5.29 \right)\right ]\cr"}
  55. TableText({Append["\large\red{z}", SequenceA], Append["\large\red{a+bi}", SequenceB], Append["\large\red{rcis(θ)}", SequenceC]}, "vc|_") → text24
  56. Polygon(A2, A3, n) → poly1, t, r_1, K, L, M, N, O, P, Q, S, f_1, g_1, h_1, i_1, l_1, m_1, p_1, q_1

AI Drawing Prompt Reference

Draw the visualization of n-th roots of complex numbers, including point A at the intersection of the x-axis and y-axis, and 10 vectors originating from A, representing the n-th roots of complex numbers. Use trigonometric functions and polar coordinates to define these vectors, and show their relationships through intersections and segments, forming the geometric distribution of complex roots.
The prompt above was compiled from the original drawing commands and is for reference only.