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Nth Roots of a Complex Number

Nth Roots of a Complex Number
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2020-09-26 22:55
Complex NumberTransformationGeometric Construction
This example demonstrates how to plot all nth roots of a complex number and visually represents their distribution. Key math concepts include complex number modulus, argument, polar representation, nth root formulas, and similarity transformations. Construction involves commands like Length, Sequence, Segment, Angle, Element, with objects such as z, mod_z, arg_z, n, list1, etc. It is suitable for high school and above students to understand the connection between complex operations and geometric representations, and for teaching complex number theory.

GeoGebra Commands & Parameters

  1. Length(z) → mod_z
  2. Sequence((mod_z^(1 / n) * (cos(arg_z + ((2 * k) * π / n)) + (ί * sin(arg_z + ((2 * k) * π / n))))), k, 0, n - 1) → list1
  3. Sequence(Segment[Element[list1, k], Element[list1, k + 1]], k, 1, n - 1) → list2
  4. Segment(Element[list1, n], Element[list1, 1]) → a
  5. Sequence(Segment[(0, 0), Element[list1, j]], j, 1, n) → list3
  6. Angle(Element[list3, 1], Element[list3, 2]) → α
  7. Element(list1, A1) → B1
  8. Element(list1, A2) → B2
  9. Element(list1, A3) → B3
  10. Element(list1, A4) → B4
  11. Element(list1, A5) → B5
  12. Element(list1, A6) → B6
  13. Element(list1, A7) → B7
  14. Element(list1, A8) → B8
  15. Element(list1, A9) → B9
  16. Element(list1, A10) → B10
  17. Element(list1, A11) → B11
  18. Element(list1, A12) → B12
  19. Element(list1, A13) → B13
  20. Element(list1, A14) → B14
  21. Element(list1, A15) → B15
  22. Element(list1, A16) → B16
  23. Element(list1, A17) → B17
  24. Element(list1, A18) → B18
  25. Element(list1, A19) → B19
  26. Element(list1, A20) → B20

AI Drawing Prompt Reference

Plot all nth roots of a complex number and show their distribution. First, draw the complex number z, then calculate its modulus mod_z and argument arg_z. Next, compute the nth roots and connect them to form a polygon. Finally, draw vectors from the origin to each nth root and label one angle α.
The prompt above was compiled from the original drawing commands and is for reference only.