Visualizing Modular Addition and Inverse on Circular Representation

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2018-09-19 04:42

This GeoGebra applet dynamically demonstrates modular addition and modular inverse. It uses Circle command to map modular arithmetic results onto points on a circle's circumference, creating an intuitive cyclic representation. Sequence and Take commands generate number sequences and extract subsequences, while Mirror command performs point reflection to visually illustrate modular inverse. SelectedElement enables interactive element selection. The core mathematical concepts include modular arithmetic, cyclic group structure, and the existence condition for modular inverse (gcd(a,n)=1). Suitable for high school advanced math or introductory number theory courses, also serves as an intuitive entry to understanding cyclic groups in abstract algebra.

GeoGebra Commands & Parameters

SelectedElement(pnt) → a
Sequence((cos(w), sin(w)), w, 0, (2 * π) - (2 * π / a), (2 * π / a)) → L_1
Sequence(Segment[L_1(w), (cos((((w + n - 1) * 2) * π / a)), sin((((w + n - 1) * 2) * π / a)))], w, 1, a) → L_2
Take(L_2, 1, m) → L_3
Circle((0, 0), 1) → c
Mirror(L_3, c) → L_4

AI Drawing Prompt Reference

Create a visualization showing the cyclic structure of modular arithmetic. First draw a unit circle in the coordinate system, then place several evenly distributed points around the circle's circumference (the number determined by parameter a), representing remainders modulo a. Connect these points with line segments following modular addition rules, starting from each point to the position after adding n-1, forming a closed loop pattern. Finally reflect this connection pattern to the opposite side of the circle to demonstrate the concept of modular inverse through symmetry. The entire graphic should clearly present the mathematical structure of cyclic groups.

The prompt above was compiled from the original drawing commands and is for reference only.