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Combined Complex Multiplication Demonstration

Combined Complex Multiplication Demonstration
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2019-11-01 13:47
VectorComplex NumberGeometric ConstructionGraphing
This example demonstrates the geometric meaning of complex multiplication in the plane, showing the effect of rotation and scaling on complex multiplication operations. Core mathematical concepts include complex numbers, vectors, rotation, scaling, and plane geometric transformations. The construction uses GeoGebra commands Polygon, Rotate, Dilate, Translate, and involves objects such as h, g, f, d, C, A, B, t1, c, a, b, E, D, t2, e, a_1, d_1, t3, A', B', C', t1', c', a', b', t4, t5, A'_1, D, E, t2, e', a'_1, d, A'_2, B'_1, C'_1, t1'_1, c'_1, a'_2. It is suitable for high school mathematics teaching, helping students understand the geometric representation of complex operations and is ideal for students interested in complex numbers and geometric transformations.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → t1, c, a, b
  2. Polygon(A, D, E) → t2, e, a_1, d_1
  3. Rotate(Polygon[Dilate[A, c], Dilate[D, c], Dilate[E, c]], Angle[B]) → t3
  4. Dilate(A, g) → A'
  5. Dilate(B, g) → B'
  6. Dilate(C, g) → C'
  7. Polygon(A', B', C') → t1', c', a', b'
  8. Translate(Rotate[Polygon[Dilate[A, h], Dilate[B, h], Dilate[C, h]], π / 2], B') → t4
  9. Translate(Polygon[Rotate[A, π / 2], Rotate[B, π / 2], Rotate[C, π / 2]], B') → t5
  10. Rotate(A, Angle[B, A, C]) → A'_1
  11. Rotate(D, Angle[B, A, C]) → D'
  12. Rotate(E, Angle[B, A, C]) → E'
  13. Polygon(A'_1, D', E') → t2', e', a'_1, d'
  14. Rotate(A, Angle[D]) → A'_2
  15. Rotate(B, Angle[D]) → B'_1
  16. Rotate(C, Angle[D]) → C'_1
  17. Polygon(A'_2, B'_1, C'_1) → t1'_1, c'_1, a'_2, b'_1
  18. Dilate(A'_2, e) → F
  19. Dilate(B'_1, e) → G
  20. Dilate(C'_1, e) → H
  21. Polygon(F, G, H) → t6, h_1, f_1, g_1
  22. Rotate(A, Angle[B, A, C]) → A'_3
  23. Rotate(D, Angle[B, A, C]) → D'_1
  24. Rotate(E, Angle[B, A, C]) → E'_1
  25. Polygon(A'_3, D'_1, E'_1) → t2'_1, e'_1, a'_3, d'_1