Editing Infinitesimal (section)

== Infinitesimals in teaching ==
Calculus textbooks based on infinitesimals include the classic ''[[Calculus Made Easy]]'' by [[Silvanus P. Thompson]] (bearing the motto "What one fool can do another can"<ref>{{Cite book|url=https://archive.org/details/CalculusMadeEasy/page/n4|title=Calculus Made Easy|last=Thompson|first=Silvanus P.|publisher=The Macmillan Company|year=1914|edition=Second|location=New York|author-link=Silvanus P. Thompson}}</ref>) and the German text ''Mathematik fur Mittlere Technische Fachschulen der Maschinenindustrie'' by R. Neuendorff.<ref>R Neuendorff (1912) ''Lehrbuch der Mathematik fur Mittlere Technische Fachschulen der Maschinenindustrie'', Verlag Julius Springer, Berlin.</ref> 

Pioneering works based on [[Abraham Robinson]]'s infinitesimals include texts by [[Keith Stroyan|Stroyan]] (dating from 1972) and [[Howard Jerome Keisler]] ([[Elementary Calculus: An Infinitesimal Approach]]). Students easily relate to the intuitive notion of an infinitesimal difference 1-"[[0.999...]]", where "0.999..." differs from its standard meaning as the real number 1, and is reinterpreted as an infinite terminating extended decimal that is strictly less than 1.<ref>{{Cite journal|last=Ely|first=Robert|year=2010|title=Nonstandard student conceptions about infinitesimals|url=http://u.cs.biu.ac.il/~katzmik/ely10.pdf|url-status=live|journal=[[Journal for Research in Mathematics Education]]|volume=41|issue=2|pages=117–146|doi=10.5951/jresematheduc.41.2.0117|jstor=20720128|archive-url=https://web.archive.org/web/20190506124205/http://u.cs.biu.ac.il/~katzmik/ely10.pdf|archive-date=2019-05-06}}</ref><ref>{{cite journal|last1=Katz|first1=Karin Usadi|last2=Katz|first2=Mikhail G.|author-link2=Mikhail Katz|date=2010|title=When is .999... less than1?|url=http://www.math.umt.edu/tmme/vol7no1/TMME_vol7no1_2010_article1_pp.3_30.pdf|url-status=dead|journal=[[The Mathematics Enthusiast|The Montana Mathematics Enthusiast]]|volume=7|issue=1|pages=3–30|doi=10.54870/1551-3440.1381|arxiv=1007.3018|s2cid=11544878|issn=1551-3440|archive-url=https://web.archive.org/web/20121207075126/http://www.math.umt.edu/tmme/vol7no1/TMME_vol7no1_2010_article1_pp.3_30.pdf|archive-date=2012-12-07|access-date=2012-12-07}}</ref>

Another elementary calculus text that uses the theory of infinitesimals as developed by Robinson is ''Infinitesimal Calculus'' by Henle and Kleinberg, originally published in 1979.<ref>{{cite book|title=Infinitesimal Calculus|url=https://archive.org/details/infinitesimalcal0000henl|url-access=registration|last1=Henle|first1=James M.|last2=Kleinberg|first2=Eugene|publisher=The MIT Press, rereleased by Dover|year=1979|isbn=978-0-262-08097-2}}</ref> The authors introduce the language of first-order logic, and demonstrate the construction of a first order model of the hyperreal numbers. The text provides an introduction to the basics of integral and differential calculus in one dimension, including sequences and series of functions. In an Appendix, they also treat the extension of their model to the ''hyperhyper''reals, and demonstrate some applications for the extended model.

An elementary calculus text based on smooth infinitesimal analysis is  Bell, John L. (2008). A Primer of Infinitesimal Analysis, 2nd Edition. Cambridge University Press. ISBN 9780521887182.

A more recent calculus text utilizing infinitesimals is Dawson, C. Bryan (2022), Calculus Set Free: Infinitesimals to the Rescue, Oxford University Press.  ISBN 9780192895608.