Exploring Roots of Unity (and other complex numbers)

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2017-02-03 09:04

This example demonstrates the distribution of roots of unity on the complex plane and their relationship with regular polygons. Core math concepts include complex numbers, trigonometric functions, circumcircle, similarity transformation. Constructed using GeoGebra commands like Textfield, Circle, Angle, Sequence. Suitable for high school and above learners to understand the connection between complex operations and geometric shapes.

GeoGebra Commands & Parameters

Textfield(Z) → InputBox1
Textfield(n) → InputBox2
Circle((0, 0), Mag^(1 / n)) → c
Angle(A, B, Z) → α
Sequence(Angle[A, B, Rotate[A, α / n + ((2 * k) * π / n), B]], k, 1, n) → list1
Sequence(Vector[(0, 0), (cos(α / n + ((2 * k) * π / n)), sin(α / n + ((2 * k) * π / n)))], k, 1, n) → list2
Sequence(Vector[(0, 0), ((Mag^(1 / n) * cos(α / n + ((2 * k) * π / n) + ((a * k) * (α / n + ((2 * k) * π / n))))), (Mag^(1 / n) * sin(α / n + ((2 * k) * π / n) + ((a * k) * (α / n + ((2 * k) * π / n))))))], k, 1, n) → list3
Sequence(Vector[(0, 0), ((Mag^(1 / n) * cos(α / n + ((2 * k) * π / n) + ((a * (n - 1)) * (α / n + ((2 * k) * π / n))))), (Mag^(1 / n) * sin(α / n + ((2 * k) * π / n) + ((a * (n - 1)) * (α / n + ((2 * k) * π / n))))))], k, 0, n - 1) → list4
Sequence(Vector[(0, 0), ((Mag^(((a * (n - 1)) + 1) / n) * cos(α / n + ((2 * k) * π / n) + ((a * (n - 1)) * (α / n + ((2 * k) * π / n))))), (Mag^(((a * (n - 1)) + 1) / n) * sin(α / n + ((2 * k) * π / n) + ((a * (n - 1)) * (α / n + ((2 * k) * π / n))))))], k, 0, n - 1) → list5
Sequence(Vector[(0, 0), ((Mag^(((a * (n - 1)) + 1) / n) * cos(α / n + ((2 * k) * π / n) + ((a * (n - 1)) * (α / n + ((2 * k) * π / n))))), (Mag^(((a * (n - 1)) + 1) / n) * sin(α / n + ((2 * k) * π / n) + ((a * (n - 1)) * (α / n + ((2 * k) * π / n))))))], k, 0, 0) → list6