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Analysis with ultrasmall numbers

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Systematic use of infinitesimals in mathematics originated with Leibniz in the 17th century. Infinitesimals served as a mainstream tool of calculus for the next 150 years; unfortunately, mathematicians of the period never succeeded in formulating unambiguous rules for working with them and the idea was gradually abandoned in favour of the now-standard epsilon-delta method of Weierstrass. 

A rigorous theory of infinitesimals, known as nonstandard analysis, was developed by Robinson in the 1960s. Nelson and Hrbacek then separately provided axiomatic approaches.

The theory used here is based on research by Péraire and Hrbacek  and developed jointly with Lessmann and O'Donovan in order to be usable even at introductory level. Their book "Analysis with Ultrasmall Numbers" is published by CRC Press (see cover).

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An MAA review of the book can be found on the MAA website

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