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Powers of Complex Numbers and Their Conjugates

Powers of Complex Numbers and Their Conjugates
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2019-08-10 02:29
CircleVectorComplex NumberGeometric Construction
This example demonstrates the visualization of powers of complex numbers and their conjugates. It involves core mathematical concepts such as complex numbers, complex conjugates, polar form of complex numbers, rotation, and reflection transformations. The construction utilizes GeoGebra commands and objects like Intersect, Circle, Rotate, Angle, ToComplex, ToPolar, Vector, and Mirror. It is suitable for high school and above students to understand the geometric meaning and properties of complex number operations.

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