Exploring nth Roots of Unity via Inversion

This applet visually demonstrates the geometric properties of nth roots of unity through the lens of inversion in the complex plane. By applying an inversion transformation, it offers an intuitive and dynamic exploration of how these fundamental complex numbers behave and relate to one another both algebraically and geometrically. At its core, the applet computes and visualizes the sum of the expression 1/(1-ω) + 1/(1-ω^2) + ... + 1/(1-ω^(n-1)), where ω represents the complex nth roots of unity, i.e., the solutions to the equation x^n - 1 = 0. This sum is not merely an abstract algebraic exercise; rather, it reveals a beautiful geometric pattern that emerges when each term 1/(1-ω^k) is interpreted as the image of a point under inversion mapping. Through dynamic polygons, vectors, and rays, the applet constructs visual representations that connect the algebraic properties of complex roots with their geometric counterparts. Students can observe how the nth roots of unity are evenly distributed around the unit circle and how inversion transforms these points in predictable and elegant ways. The dynamic nature of the construction allows users to adjust the parameter n and immediately see how the configuration changes, reinforcing the understanding of the underlying mathematical relationships. The applet serves as a powerful pedagogical tool for students studying complex numbers, particularly those learning about roots of unity, Möbius transformations, or geometric interpretations of complex arithmetic. By making abstract algebraic identities visible and interactive, it helps learners build deeper intuition about why certain results hold and how different areas of mathematics interconnect. The combination of symbolic computation and geometric visualization ensures that users can verify algebraic claims through direct observation, fostering a more robust and memorable understanding of the material.