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arXiv:0811.0164math.HO

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

Karin Usadi Katz, Mikhail G. Katz

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.

arXiv comments
21 pages, 3 figures
Journal reference
DOI
Subjects
math.HOHistory and Overview

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  1. 0.9999 ≊ 1 1 comment

    • FWIW, there's an old Arxiv paper with this same argument: https://arxiv.org/abs/0811.0164 It feels intuitively correct is what I'll say in its favor.

      A_D_E_P_TView on HNParent context

  2. Far From 'Infinitesimal': A Mathematical Paradox's Role In History 1 comment

    • Of course that argument only applies when talking about the standard real number system. http://arxiv.org/pdf/1007.3018.pdf?origin=publication_detail http://arxiv.org/pdf/0811.0164.pdf ...of course you can make a pretty good case that uncountable, uncomputable, unnameable…

      GregBuchholzView on HNParent context

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