A strict non-standard inequality .999... < 1

TL;DR

This paper explores the ambiguity of the decimal expansion .999... and argues, using non-standard analysis, that it can be infinitesimally less than 1, challenging the conventional equality in standard real numbers.

Contribution

It introduces a non-standard hyperinteger framework to model .999... as an infinitesimal less than 1, providing a rigorous justification for students' intuition.

Findings

.999... can be modeled as infinitesimally less than 1 in non-standard analysis.

The standard part function's application is questioned as an unacknowledged assumption.

Students' resistance to .999... = 1 can be justified through non-standard models.

Abstract

Is .999... equal to 1? Lightstone's decimal expansions yield an infinity of numbers in [0,1] whose expansion starts with an unbounded number of digits "9". We present some non-standard thoughts on the ambiguity of an ellipsis, modeling the cognitive concept of generic limit of B. Cornu and D. Tall. A choice of a non-standard hyperinteger H specifies an H-infinite extended decimal string of 9s, corresponding to an infinitesimally diminished hyperreal value. In our model, the student resistance to the unital evaluation of .999... is directed against an unspoken and unacknowledged application of the standard part function, namely the stripping away of a ghost of an infinitesimal, to echo George Berkeley. So long as the number system has not been specified, the students' hunch that .999... can fall infinitesimally short of 1, can be justified in a mathematically rigorous fashion.