Educational Studies in Mathematics

, Volume 74, Issue 3, pp 259–273 | Cite as

Zooming in on infinitesimal 1–.9.. in a post-triumvirate era

  • Karin Usadi Katz
  • Mikhail G. KatzEmail author


The view of infinity as a metaphor, a basic premise of modern cognitive theory of embodied knowledge, suggests in particular that there may be alternative ways in which one could formalize mathematical ideas about infinity. We discuss the key ideas about infinitesimals via a proceptual analysis of the meaning of the ellipsis “...” in the real formula \(\hbox{.999\ldots = 1}\). Infinitesimal-enriched number systems accommodate quantities in the half-open interval [0,1) whose extended decimal expansion starts with an unlimited number of repeated digits 9. Do such quantities pose a challenge to the unital evaluation of the symbol “.999...”? We present some non-standard thoughts on the ambiguity of the ellipsis in the context of the cognitive concept of generic limit of B. Cornu and D. Tall. We analyze the vigorous debates among mathematicians concerning the idea of infinitesimals.


Decimal representation Generic limit Hypernatural number Infinitesimal Limit Unital evaluation 



We are grateful to David Ebin and David Tall for a careful reading of an earlier version of the manuscript and for making numerous helpful suggestions. We thank the editor of the article and the anonymous referees for an exceptionally thorough analysis of the shortcomings of the version originally submitted, resulting, through numerous intermediate versions, in a more focused text.


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© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Bar Ilan UniversityRamat GanIsrael

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