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The Geometry of i - A Complex Relationship

The Geometry of i - A Complex Relationship
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2015-07-31 10:50
AngleVectorTrigonometryComplex NumberGeometric Construction
In collections Complex Math
This example demonstrates the geometric representation of the imaginary unit i and its relationship with the Pythagorean theorem. It involves core mathematical concepts like complex numbers, geometric interpretation, and the Pythagorean theorem. The construction uses GeoGebra commands/objects such as Line, Mirror, Segment, Angle, Point, and Vector. Suitable for high school mathematics students to understand the geometric meaning and construction methods of complex numbers.

GeoGebra Commands & Parameters

  1. Line(C, E) → a
  2. Mirror(D, a) → D'
  3. Mirror(A, a) → A'
  4. Line(D, F) → b
  5. Mirror(A', b) → A''
  6. Mirror(A, b) → A'_1
  7. Segment(A'_1, A'') → c
  8. Segment(A'', A') → d
  9. Segment(A', A) → e
  10. Segment(A, A'_1) → f
  11. Segment(A, A'') → g
  12. Segment(D, D') → h
  13. Segment(A', A'_1) → m
  14. Segment(B, G) → j
  15. Segment(B, A'') → k
  16. Angle(h, k) → α
  17. Segment(B, A') → l
  18. Segment(B, D') → n
  19. Segment(D', A'') → q
  20. Segment(T, U) → a_1
  21. Segment(T_1, U_1) → a_2
  22. Segment(T_2, U_2) → a_3
  23. Segment(T_3, U_3) → a_4
  24. Segment(T_4, U_4) → a_5
  25. Point(a_5) → V
  26. Point(a_4) → W
  27. Segment(V, W) → b_1
  28. Segment(Z, A_1) → c_1
  29. Segment(Z_1, A_2) → c_2
  30. Vector((w, w)) → r_1
  31. Segment(L_1, M_1) → p_1
  32. Segment(L_3, M_3) → p_3
  33. Segment(L_4, M_4) → p_4
  34. Segment(P_1, Q_1) → p
  35. Vector(R_1, S_1) → u_1
  36. Vector(V_1, W_1) → v_1
  37. Segment(B_2, C_2) → f_1
  38. Segment(D_2, E_2) → i_1
  39. Segment(F_2, G_2) → j_1
  40. Segment(L_5, M_5) → p_2
  41. Segment(L_6, M_6) → p_5
  42. Segment(L_7, M_7) → p_6
  43. Segment(L_8, M_8) → p_7
  44. Segment(L_9, M_9) → p_8
  45. Vector(H_2, I_2) → w_1
  46. Vector(J_2, I_2) → k_1