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Powers of Complex Numbers by Judah L Schwartz (2)

Powers of Complex Numbers by Judah L Schwartz (2)
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2018-12-11 00:05
FunctionComplex NumberTransformationGeometric Construction
This example demonstrates the geometric interpretation of powers of complex numbers, visually showing how multiplication of complex numbers corresponds to rotation and dilation in the complex plane. Key math concepts include polar form of complex numbers, Euler's formula, and the geometric properties of complex multiplication. GeoGebra commands like Intersect, Circle, Rotate, Angle, ToComplex, ToPolar, Vector, and Mirror are used to construct visualizations of complex numbers, rotations, and scalings. Suitable for high school and above mathematics learners to understand the intuitive representation of complex operations and transformations in the complex plane.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Circle(A, 1) → c
  3. Rotate(C, (360 / N)°, A) → Nroot
  4. Angle(C, A, Nroot) → α_1
  5. ToComplex(Vector[Nroot]) → u
  6. ToPolar(Vector[Nroot]) → v
  7. Vector(NrootPower) → w
  8. ToPolar(w) → z
  9. Angle(C, A, NrootPower) → β
  10. ToComplex(w) → a
  11. ToComplex(Vector[Z]) → CZ
  12. Mirror(Z, xAxis) → ZC
  13. Mirror(ZC, c) → ZCI
  14. ToComplex(Vector[ZC]) → CZC
  15. ToComplex(Vector[ZCI]) → CZCI
  16. Vector(Zpower) → h
  17. Vector(z_1) → j
  18. Vector(z_2) → k
  19. Vector(z_3show) → l
  20. Vector(z_2show) → m
  21. Mirror(Z, c) → ZRI