Educational Studies in Mathematics

, Volume 74, Issue 3, pp 259–273

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


DOI: 10.1007/s10649-010-9239-4

Cite this article as:
Katz, K.U. & Katz, M.G. Educ Stud Math (2010) 74: 259. doi:10.1007/s10649-010-9239-4


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 

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Bar Ilan UniversityRamat GanIsrael

Personalised recommendations