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Imaginary Number Line (i) - Time Space

Imaginary Number Line (i) - Time Space
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2015-08-17 00:58
VectorComplex NumberTransformationParametric EquationGeometric Construction
This GeoGebra example presents a dynamic geometric model incorporating an imaginary number line, aiming to visualize the relationship between the imaginary unit i, the real axis, and spatial geometric elements, while attempting to express the abstract concept of 'time-space' graphically. Core mathematical concepts involve the complex plane, imaginary unit, vector operations, rotation transformations, and congruence/similarity in geometric constructions. Key construction steps include using commands such as Vector, Point, Polygon, Rotate, Segment, Angle, Circle, Distance, Line, and Mirror to create multiple points, segments, polygons, circles, and mirrored objects, with sliders or parameters enabling dynamic changes to show connections among the imaginary axis, real axis, angles, and areas. This example is suitable for high school and above students or math enthusiasts to understand the geometric meaning of complex numbers, algebraic expressions of rotations, and to explore interdisciplinary links between mathematics and physical spacetime concepts.

GeoGebra Commands & Parameters

  1. Vector((a, 0)) → u
  2. Vector((a, 0), (a, b)) → v
  3. Vector((0, 0), (a, b)) → w
  4. Vector((0, 0), (a, c)) → d
  5. Vector((a, 0), (a, c)) → e
  6. Vector((o, 0)) → s
  7. Vector((0, o)) → u_1
  8. Vector((o, 0), (0, o)) → v_1
  9. Point(u_1) → A
  10. Point(s) → B
  11. Polygon(C, B, A) → poly1, a_2, c_1, b_2
  12. Rotate(h_1, -80*°, C) → e_1
  13. Point(e_1) → D
  14. Point(e_1) → E
  15. Polygon(C, E, D) → poly2, d_2, c_2, e_2
  16. Segment(C, D) → g_1
  17. Segment(C, E) → i_1
  18. Angle(E, C, D) → α_i
  19. Point(d) → F
  20. Point(u) → G
  21. Circle(C, F) → p_1
  22. Circle(C, A) → q_1
  23. Point(u) → H
  24. Circle(C, H) → r_1
  25. Circle(C, z_1) → s_1
  26. Circle(C, z_2) → t_1
  27. Circle(C, z_3) → f_2
  28. Circle(C, z_4) → g_2
  29. Circle(C, z_5) → h_2
  30. Distance(C, z_1) → distanceCz1
  31. Distance(C, z_2) → distanceCz2
  32. Distance(C, z_3) → distanceCz3
  33. Distance(C, z_4) → distanceCz4
  34. Distance(C, z_5) → distanceCz5
  35. Line(I, v_1) → a_3
  36. Mirror(C, a_3) → C'
  37. Mirror(B, a_3) → B'
  38. Mirror(A, a_3) → A'
  39. Mirror(C, a_3) → C'_1
  40. Mirror(B, a_3) → B'_1
  41. Mirror(A, a_3) → A'_1
  42. Polygon(C'_1, B'_1, A'_1) → poly1', a'_1, c'_1, b'_1
  43. Point(c_1) → J
  44. Point(c'_1) → K
  45. Point(a_2) → L
  46. Point(a'_1) → M
  47. Vector(K, J) → w_2
  48. Vector(M, L) → u_3
  49. Mirror(d, a_2) → d'
  50. Mirror(w, a_2) → w'
  51. Mirror(v, a_2) → v'
  52. Mirror(e, a_2) → e'
  53. Distance(K, J) → distanceKJ
  54. Distance(K, M) → distanceKM
  55. Mirror(v_1, xAxis) → v_3
  56. Mirror(u_1, xAxis) → w_3
  57. Point(s) → N
  58. Point(u_1) → O
  59. Circle(N, O) → d_3
  60. Circle(A, x_7) → k_3
  61. Circle(A, x_6) → p_3