A Complex Primitive $N$th Root Of Unity: A Very Elementary Approach

TL;DR

This paper introduces an elementary method to construct a primitive nth root of unity in the complex plane without using advanced mathematical tools like exponential functions, trigonometry, or group theory.

Contribution

It provides a novel, simplified approach to defining primitive roots of unity, avoiding traditional complex analysis and algebraic techniques.

Findings

Constructs primitive roots without exponential or trigonometric functions

Uses an indirect, elementary method for the construction

Simplifies understanding of roots of unity for educational purposes

Abstract

This paper presents a primitive $n$th root of unity in $\mathbb C$. The approach is very elementary and avoids the following: the complex exponential function, trigonometry, and group theory. It also avoids differentiation, integration, and series. The presentation of the primitive is indirect.