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

Authors

Benoit Rittaud
Laurent Vivier

Volume

21

Issue

1-2

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 18th 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.

First Page

3

Last Page

34

Recommended Citation

Rittaud, Benoit and Vivier, Laurent (2024) "The Field Q and the Equality 0.999. . . = 1 from Combinatorics of Circular Words and History of Practical Arithmetics," The Mathematics Enthusiast: Vol. 21 : No. 1 , Article 2.
DOI: https://doi.org/10.54870/1551-3440.1614
Available at: https://scholarworks.umt.edu/tme/vol21/iss1/2

Digital Object Identifier (DOI)

10.54870/1551-3440.1614

Publisher

University of Montana, Maureen and Mike Mansfield Library