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Power of a Complex Number

Power of a Complex Number
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2022-01-28 18:29
VectorComplex NumberGeometric ConstructionLocus
This example demonstrates the geometric representation of complex number power operations in the plane. Key math concepts include polar form of complex numbers, Euler's formula, rotation, and scaling. Construction highlights involve commands like Intersect, Point, Segment, Vector, Element, If, Sequence, Textfield, and TableText to create complex number representations, rotational animations, and loci. Suitable for high school and above students to understand the geometric meaning and dynamic changes of complex operations.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(yAxis) → B
  3. Point(yAxis) → C
  4. Segment(B, C) → h
  5. Point(xAxis) → D
  6. Point(xAxis) → E
  7. Segment(D, E) → i
  8. Vector((a, b)) → p
  9. Element(Sequence1, 1) → B2
  10. Element(Sequence1, 2) → B3
  11. Element(Sequence1, 3) → B4
  12. Element(Sequence1, 4) → B5
  13. Element(Sequence1, 5) → B6
  14. Element(Sequence1, 6) → B7
  15. Element(Sequence1, 7) → B8
  16. Element(Sequence1, 8) → B9
  17. Element(Sequence1, 9) → B10
  18. Element(Sequence1, 10) → B11
  19. If(k ≥ 0, B2 + "") → C14
  20. If(k ≥ 1, B3 + "") → C15
  21. If(k ≥ 2, B4 + "") → C16
  22. If(k ≥ 3, B5 + "") → C17
  23. If(k ≥ 4, B6 + "") → C18
  24. If(k ≥ 5, B7 + "") → C19
  25. If(k ≥ 6, B8 + "") → C20
  26. If(k ≥ 7, B9 + "") → C21
  27. If(k ≥ 8, B10 + "") → C22
  28. If(k ≥ 9, B11 + "") → C23
  29. Sequence(Element[Sequence3, α + 1], α, 0, k) → SequenceA, {"1", "2", "3", "4", "5"}
  30. Sequence(Element[rcisθ, α + 1], α, 0, k) → Sequencercisθ, {" 2.24\;cis\left(-0.89\right)", " 5\;cis\left(-1.77\right)", " 11.18\;cis\left(-2.66\right)", " 25\;cis\left(-3.54\right)", " 55.9\;cis\left(-4.43\right)"}
  31. Sequence(Element[Powers, l + 1], l, 1, k) → l1, {-0.9999999999999991 - 4.898979485566357ί, -9.899494936611665 - 5.196152422706635ί, -23.000000000000007 + 9.797958971132704ί, -15.556349186104075 + 53.693575034635195ί}
  32. Sequence(Vector[p^(Ω + 1)], Ω, 1, k) → u, {(-0.9999999999999991, -4.898979485566357), (-9.899494936611665, -5.196152422706635), (-23.000000000000007, 9.797958971132704), (-15.556349186104075, 53.693575034635195)}
  33. Sequence(Element[Powers, α + 1], α, 0, k) → PowersB, {1.4142135623730951 - 1.7320508075688772ί, -0.9999999999999991 - 4.898979485566357ί, -9.899494936611665 - 5.196152422706635ί, -23.000000000000007 + 9.797958971132704ί, -15.556349186104075 + 53.693575034635195ί}
  34. Textfield(a) → InputBox1
  35. Textfield(b) → InputBox2
  36. TableText({Append["\blue{n}", SequenceA], Append["\blue{rcis}", Sequencercisθ], Append["\blue{a+bi}", PowersB]}, "vc|_") → text1

AI Drawing Prompt Reference

Construct a complex-plane Cartesian diagram for complex-number power operations. Start by marking several points on the x- and y-axes, connect two pairs with line segments, and use one resulting direction as the vector reference. Then, based on a chosen initial complex number, generate sequences of points for consecutive powers and the corresponding vector positions as the parameter runs from 1 to the chosen limit. In parallel, generate another sequence using the polar form (rcisθ) for the same exponents. Finally, add a table-style text display showing n, the rcis expression values, and the corresponding a+bi (real-imaginary) values for each step.
The prompt above was compiled from the original drawing commands and is for reference only.