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Demonstration of Roots of Complex Numbers

Demonstration of Roots of Complex Numbers
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2016-12-12 02:55
VectorComplex NumberTransformationGeometric Construction
This example demonstrates how to find the n-th roots of a complex number on the complex plane through rotation and vector operations. It involves core mathematical concepts such as complex numbers, argument, rotation, and geometric construction. Key GeoGebra commands and objects used include Rotate, Angle, Vector, and Segment to construct and visualize the positions and properties of complex roots. Suitable for high school students learning the geometric interpretation of complex number operations.

GeoGebra Commands & Parameters

  1. Rotate(X, θ, O) → C
  2. Angle(X, O, C) → t
  3. Vector(C) → c
  4. Segment(O, X) → u
  5. Segment(O, X) → a
  6. Rotate(O, t / n, O) → O'
  7. Rotate(X, t / n, O) → X'
  8. Segment(O', X') → z_1
  9. Rotate(O, t / n + (2 * 3.141592653589793 / n), O) → O'_1
  10. Rotate(X, t / n + (2 * 3.141592653589793 / n), O) → X'_1
  11. Segment(O'_1, X'_1) → z_2
  12. Rotate(O, t / n + (4 * 3.141592653589793 / n), O) → O'_2
  13. Rotate(X, t / n + (4 * 3.141592653589793 / n), O) → X'_2
  14. Segment(O'_2, X'_2) → z_3
  15. Rotate(O, t / n + (6 * 3.141592653589793 / n), O) → O'_3
  16. Rotate(X, t / n + (6 * 3.141592653589793 / n), O) → X'_3
  17. Segment(O'_3, X'_3) → z_4
  18. Rotate(O, t / n + (8 * 3.141592653589793 / n), O) → O'_4
  19. Rotate(X, t / n + (8 * 3.141592653589793 / n), O) → X'_4
  20. Segment(O'_4, X'_4) → z_5
  21. Rotate(O, t / n + (10 * 3.141592653589793 / n), O) → O'_5
  22. Rotate(X, t / n + (10 * 3.141592653589793 / n), O) → X'_5
  23. Segment(O'_5, X'_5) → z_6
  24. Rotate(O, t / n + (12 * 3.141592653589793 / n), O) → O'_6
  25. Rotate(X, t / n + (12 * 3.141592653589793 / n), O) → X'_6
  26. Segment(O'_6, X'_6) → z_7
  27. Rotate(O, t / n + (14 * 3.141592653589793 / n), O) → O'_7
  28. Rotate(X, t / n + (14 * 3.141592653589793 / n), O) → X'_7
  29. Segment(O'_7, X'_7) → z_8