Editing Infinitesimal (section)

==== Laurent series ====
An example from category 1 above is the field of [[Laurent series]] with a finite number of negative-power terms. For example, the Laurent series consisting only of the constant term 1 is identified with the real number&nbsp;1, and the series with only the linear term&nbsp;''x'' is thought of as the simplest infinitesimal, from which the other infinitesimals are constructed. Dictionary ordering is used, which is equivalent to considering higher powers of&nbsp;''x'' as negligible compared to lower powers. [[David O. Tall]]<ref>{{cite web |url=http://www.jonhoyle.com/MAAseaway/Infinitesimals.html |title=Infinitesimals in Modern Mathematics |publisher=Jonhoyle.com |access-date=2011-03-11 |url-status=dead |archive-url=https://web.archive.org/web/20110713115815/http://www.jonhoyle.com/MAAseaway/Infinitesimals.html |archive-date=2011-07-13 }}</ref> refers to this system as the super-reals, not to be confused with the [[superreal number]] system of Dales and Woodin. Since a [[Taylor series]] evaluated with a Laurent series as its argument is still a Laurent series, the system can be used to do calculus on transcendental functions if they are analytic. These infinitesimals have different first-order properties than the reals because, for example, the basic infinitesimal&nbsp;''x'' does not have a square root.