The Field Q and the Equality 0.999 . . . = 1 from Combinatorics of Circular Words and History of Practical Arithmetics

TL;DR

This paper explores the equality 0.999...=1 using circular words, revealing historical perspectives and algebraic structures that justify the equality within the rational numbers, and reconstructs the field Q from these concepts.

Contribution

It introduces a novel approach using circular words to analyze the equality 0.999...=1 and reconstructs the field Q from these structures, linking history and algebra.

Findings

Circular words provide an algebraic framework for the equality 0.999...=1.

Historical analysis shows early assertions of the equality by 18th-century educators.

Reconstruction of Q demonstrates algebraic integers are either integers or irrational.

Abstract

We reconsider the classical equality 0.999. .. = 1 with the tool of circular words, that is: finite words whose last letter is assumed to be followed by the first one. Such circular words are naturally embedded with algebraic structures that enlight this problematic equality, allowing it to be considered in Q rather than in R. We comment early history of such structures, that involves English teachers and accountants of the first part of the xviii th century, who appear to be the firsts to assert the equality 0.999. .. = 1. Their level of understanding show links with Dubinsky et al.'s apos theory in mathematics education. Eventually, we rebuilt the field Q from circular words, and provide an original proof of the fact that an algebraic integer is either an integer or an irrational number.