Monad (non-standard analysis)

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In non-standard analysis, a monad (also called halo^[1]) is the set of points infinitely close to a given point.

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

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

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

Notes

↑ Goldblatt, Robert (1998). Lectures on the Hyperreals. Berlin: Springer. ISBN 0-387-98464-X.

v t e Infinitesimals

History	Adequality Infinitesimal calculus Leibniz's notation Integral symbol Criticism of non-standard analysis The Analyst The Method of Mechanical Theorems Cavalieri's principle

Related branches	Non-standard analysis Non-standard calculus Internal set theory Synthetic differential geometry Constructive non-standard analysis

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

Scientists	Gottfried Wilhelm Leibniz Abraham Robinson Pierre de Fermat Augustin-Louis Cauchy Leonhard Euler

Physics Engineering	Infinitesimal strain theory Infinitesimal oscillations of a pendulum

Textbooks	Analyse des Infiniment Petits Elementary Calculus Cours d'Analyse

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