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Fourth Roots of a Complex Number

Fourth Roots of a Complex Number
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2014-04-17 00:34
AngleComplex NumberGeometric ConstructionSlider AnimationGraphing
In collections Roots of Complex Numbers
This GeoGebra worksheet visualizes the four fourth roots of a complex number N on the complex plane. It demonstrates the polar form of complex numbers, De Moivre's theorem, and the symmetric distribution of roots on a circle centered at the origin. The construction uses complex number objects (N, A), a slider k to select root indices, and generates points R_0 through R_3 via the root formula. Segments from the origin to each root and angle labels are included to highlight the equal angular spacing. Suitable for high school or undergraduate students learning complex analysis, providing an interactive way to explore the geometric meaning of complex roots.

GeoGebra Commands & Parameters

  1. Point N(0.0, 1.0)
  2. arg(N)
  3. Numeric p(4.0)
  4. (A + 0) / p
  5. (A + (2 * π)) / p
  6. (A + ((2 * π) * 2)) / p
  7. abs(N)^(1 / p)
  8. (r * ℯ^((ί * A_1)))
  9. (r * ℯ^((ί * A_0)))
  10. (r * ℯ^((ί * A_2)))
  11. Boolean a(true)
  12. A / °
  13. Boolean b(true)
  14. A_0 / °
  15. A_1 / °
  16. A_2 / °
  17. abs(N)
  18. Boolean c(true)
  19. Boolean d(true)
  20. Boolean e(true)
  21. (A + ((2 * π) * 3)) / p
  22. A_3 / °
  23. (r * ℯ^((ί * A_3)))
  24. Boolean f(true)

AI Drawing Prompt Reference

Draw a complex plane with a point representing a complex number. Show its four fourth roots as points on a circle centered at the origin, with radius equal to the modulus of the complex number. The roots are equally spaced by 90 degrees. Draw line segments from the origin to each root and label the angle of each segment with the positive real axis. Include a slider to select a root index and display the coordinates of the selected root. Also display the principal argument, modulus, and the four angle values to illustrate polar form and De Moivre's theorem.
The prompt above was compiled from the original drawing commands and is for reference only.