Properties of the Digital Root and its Extension to Rational Numbers -- an Algebraic Approach

TL;DR

This paper provides an algebraic analysis of the digital root's invariance under division for natural bases, covering both repeating and non-repeating fractions, with illustrative examples and known results.

Contribution

It offers a self-contained algebraic approach to digital roots extended to rational numbers, including invariance rules and fractional representations.

Findings

Digital root invariance under division established for arbitrary bases

Analysis includes both finite and infinite base representations

Provides illustrative examples and discusses known fractional representation results

Abstract

This paper contains an algebraic constructive and self-contained account of the invariance rule of the digital root under division for an arbitrary natural basis representation. Both the cases of repeating and non-repeating fractionals are treated. In the preliminary section some known results such as the uniqueness in the representation of a fraction are discussed for both the finite and infinite bases cases. Simple examples are introduced throughout the text for illustrative purposes.