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

Karin Usadi Katz, Mikhail G. Katz

Research output: Contribution to journalArticlepeer-review

18 Scopus citations


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 .999... = 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.

Original languageEnglish
Pages (from-to)259-273
Number of pages15
JournalEducational Studies in Mathematics
Issue number3
StatePublished - 2010


  • Decimal representation
  • Generic limit
  • Hypernatural number
  • Infinitesimal
  • Limit
  • Unital evaluation


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