Editing Infinitesimal (section)

== Number systems that include infinitesimals ==

=== Formal series ===

==== 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.

==== The Levi-Civita field ====
The [[Levi-Civita field]] is similar to the Laurent series, but is algebraically closed. For example, the basic infinitesimal x has a square root. This field is rich enough to allow a significant amount of analysis to be done, but its elements can still be represented on a computer in the same sense that real numbers can be represented in floating-point.<ref>{{Cite web|url=http://www.uwec.edu/surepam/media/RS-Overview.pdf|title=Analysis on the Levi-Civita Field, a Brief Overview|last=Shamseddine|first=Khodr|archive-url=https://web.archive.org/web/20110608043202/http://www.uwec.edu/surepam/media/RS-Overview.pdf|archive-date=2011-06-08|url-status=dead}}</ref>

==== Transseries ====
The field of [[transseries]] is larger than the Levi-Civita field.<ref>{{Cite journal|last=Edgar|first=Gerald A.|date=2010|title=Transseries for Beginners|url=https://people.math.osu.edu/edgar.2/preprints/trans_begin/|journal=[[Real Analysis Exchange]]|volume=35|issue=2|pages=253–310|doi=10.14321/realanalexch.35.2.0253|arxiv=0801.4877|s2cid=14290638}}</ref> An example of a transseries is:

:<math>e^\sqrt{\ln\ln x}+\ln\ln x+\sum_{j=0}^\infty e^x x^{-j},</math>

where for purposes of ordering ''x'' is considered infinite.

=== Surreal numbers ===
Conway's [[surreal number]]s fall into category 2, except that the surreal numbers form a [[proper class]] and not a set.<ref name="Alling1985">{{citation | url = https://www.ams.org/journals/tran/1985-287-01/S0002-9947-1985-0766225-7/S0002-9947-1985-0766225-7.pdf | title = Conway's Field of surreal numbers | last = Alling | first = Norman | date = Jan 1985 | journal = Trans. Amer. Math. Soc. | volume = 287 | issue = 1 | pages = 365–386 | access-date = 2019-03-05 | doi=10.1090/s0002-9947-1985-0766225-7| doi-access = free }}</ref> They are a system designed to be as rich as possible in different sizes of numbers, but not necessarily for convenience in doing analysis, in the sense that every ordered field is a subfield of the surreal numbers.<ref name=bajnok>{{cite book|last=Bajnok|first=Béla|title=An Invitation to Abstract Mathematics|year=2013|publisher=Springer |isbn=9781461466369|quote=Theorem 24.29. The surreal number system is the largest ordered field|url=https://books.google.com/books?id=cNFzKnvxXoAC&q=%22surreal+numbers%22}}</ref> There is a natural extension of the exponential function to the surreal numbers.<ref name=G1986>{{cite book | last=Gonshor | first=Harry | title=An Introduction to the Theory of Surreal Numbers | year=1986 | publisher=Cambridge University Press | series=London Mathematical Society Lecture Note Series | volume=110 | isbn= 9780521312059 | doi=10.1017/CBO9780511629143 }}</ref>{{rp|at=ch. 10}}

=== Hyperreals ===
{{Main|Hyperreal number}}
The most widespread technique for handling infinitesimals is the hyperreals, developed by [[Abraham Robinson]] in the 1960s. They fall into category 3 above, having been designed that way so all of classical analysis can be carried over from the reals. This property of being able to carry over all relations in a natural way is known as the [[transfer principle]], proved by [[Jerzy Łoś]] in 1955. For example, the transcendental function sin has a natural counterpart *sin that takes a hyperreal input and gives a hyperreal output, and similarly the set of natural numbers <math>\mathbb{N}</math> has a natural counterpart <math>^*\mathbb{N}</math>, which contains both finite and infinite integers. A proposition such as <math>\forall n \in \mathbb{N}, \sin n\pi=0</math> carries over to the hyperreals as <math>\forall n \in {}^*\mathbb{N}, {}^*\!\!\sin n\pi=0</math> .

=== Superreals ===
{{Main|Superreal number}}
The [[superreal number]] system of Dales and Woodin is a generalization of the hyperreals. It is different from the super-real system defined by [[David Tall]].

=== Dual numbers ===
{{Main|Dual number}}
In [[linear algebra]], the [[dual number]]s extend the reals by adjoining one infinitesimal, the new element ε with the property ε<sup>2</sup> = 0 (that is, ε is [[nilpotent]]). Every dual number has the form ''z'' = ''a'' + ''b''ε with ''a'' and ''b'' being uniquely determined real numbers.

One application of dual numbers is [[automatic differentiation]]. This application can be generalized to polynomials in n variables, using the [[Exterior algebra]] of an n-dimensional vector space.

=== Smooth infinitesimal analysis ===
{{Main|Smooth infinitesimal analysis}}
[[Synthetic differential geometry]] or [[smooth infinitesimal analysis]] have roots in [[category theory]]. This approach departs from the classical logic used in conventional mathematics by denying the general applicability of the [[law of excluded middle]] – i.e., ''not'' (''a'' ≠ ''b'') does not have to mean ''a'' = ''b''. A ''nilsquare'' or ''[[nilpotent]]'' infinitesimal can then be defined. This is a number ''x'' where ''x''<sup>2</sup> = 0 is true, but ''x'' = 0 need not be true at the same time. Since the background logic is [[intuitionistic logic]], it is not immediately clear how to classify this system with regard to classes 1, 2, and 3. Intuitionistic analogues of these classes would have to be developed first.