MathPlotAI Logo MathPlotAI

Nth Roots of Unity by moebiuscurve

Nth Roots of Unity by moebiuscurve
Loading the math board and drawing, please wait…
2017-04-06 10:23
CircleVectorTrigonometrySequenceComplex Number
This example demonstrates the geometric distribution of nth roots of unity in the complex plane. It involves core math concepts like complex numbers, trigonometry, unit circle, and sequences. Constructed using GeoGebra commands such as Intersect, Circle, Vector, Sequence, Point, and Text to plot the unit circle, vectors, root points, and labels. Suitable for high school and above students to understand the connection between complex operations and geometric representations.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → O
  2. Circle(O, 1) → c
  3. Vector(O, Z) → ZVector
  4. Circle(O, Z) → ZCircle
  5. Sequence((Z^(1 / n) * ℯ^((((ί * 2) * π) * k / n))), k, 0, n - 1) → Roots
  6. Circle(O, abs(ZVector)^(1 / n)) → WCircle
  7. Sequence(Text["W_{" + ((k * "}")), Element[Roots, k], n > 0, n > 0], k, 1, n) → RootsLabels
  8. Sequence(Vector[O, Element[Roots, k]], k, 1, n) → RootsVectors
  9. Point(WCircle) → W
  10. Sequence(Circle[O, abs(W^(k))], k, 2, n - 1) → WCircles
  11. Sequence(Vector[O, W^(k)], k, 1, n) → WVectors
  12. Vector(O, (1, 0)) → u
  13. Sequence(Text["W^{" + (k) + "}", W^(k), n > 0, n > 0], k, 2, n) → WLabels
  14. Text("W", W) → textW
  15. Sequence(Text["W^{" + (k) + "}, \angle" + (Angle[u, Vector[W^(k)]]), W^(k), n > 0, n > 0], k, 2, n) → WLabelsAngles
  16. Text("W, ∠" + (Angle[u, Vector[W]]), W) → textWAngles
  17. Text("Z, ∠" + (Angle[u, Vector[Z]]), Z) → textZAngles
  18. Text("Z", Z) → textZ
  19. Sequence(Text["W_{" + ((k * "}, \angle")) + (Angle[u, Vector[Element[Roots, k]]]), Element[Roots, k], n > 0, n > 0], k, 1, n) → RootsLabelsAngles