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In nonstandard analysis, a monad or also a halo is the set of points infinitesimally close to a given point.

Given a hyperreal number x in R, the monad of x is the set

${\displaystyle {\text{monad}}(x)=\{y\in \mathbb {R} ^{*}\mid x-y{\text{ is infinitesimal}}\}.}$

If x is finite (limited), the unique real number in the monad of x is called the standard part of x.

References

1. Goldblatt, Robert (1998). Lectures on the Hyperreals. Berlin: Springer. ISBN 0-387-98464-X.
2. Wood, Carol (4 Sep 2015). "The Infinitesimal Monad - Numberphile" (video). youtube.com. Numberphile. Retrieved 29 Dec 2022.
3. Keisler, Howard (19 June 2022). Foundations of Infinitesimal Calculus (PDF). Madison, Wisconsin, USA: University of Wisconsin Press. p. 2. Retrieved 19 December 2024.

v t e Infinitesimals
History	Adequality Leibniz's notation Integral symbol Criticism of nonstandard analysis The Analyst The Method of Mechanical Theorems Cavalieri's principle
Related branches	Nonstandard analysis Nonstandard calculus Internal set theory Synthetic differential geometry Smooth infinitesimal analysis Constructive nonstandard analysis Infinitesimal strain theory (physics)
Formalizations	Differentials Hyperreal numbers Dual numbers Surreal numbers
Individual concepts	Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set Law of continuity Overspill Microcontinuity Transcendental law of homogeneity
Mathematicians	Gottfried Wilhelm Leibniz Abraham Robinson Pierre de Fermat Augustin-Louis Cauchy Leonhard Euler
Textbooks	Analyse des Infiniment Petits Elementary Calculus Cours d'Analyse

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