What is infinitesimal?

Infinitesimal

Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. Students easily relate to the intuitive notion of an infinitesimal difference 1-"0.999...", where "0.999..." needs to be interpreted differently from its standard meaning as a real number. The insight with exploiting infinitesimals was that objects could still retain certain specific properties, such as angle or slope, even though these objects were quantitatively small. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series. It was originally introduced around 1670 by either Nicolaus Mercator or Gottfried Wilhelm Leibniz.

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Some articles on infinitesimal:

Infinitesimal - Logical Properties
... The method of constructing infinitesimals of the kind used in nonstandard analysis depends on the model and which collection of axioms are used ... We consider here systems where infinitesimals can be shown to exist ... This theorem is fundamental for the existence of infinitesimals as it proves that it is possible to formalise them ...
Levi-Civita Field
... a system of numbers containing infinite and infinitesimal quantities ... is the set of rational numbers, and is to be interpreted as a positive infinitesimal ... is equivalent to the assumption that is an infinitesimal ...
Elementary Calculus: An Infinitesimal Approach
... Elementary Calculus An Infinitesimal approach is a textbook by H ... The subtitle alludes to the infinitesimal numbers of the hyperreal number system of Abraham Robinson and is sometimes given as An approach using infinitesimals ...
Continuum (theory) - In Mathematics and Philosophy
... Bell isolates two distinct historical conceptions of infinitesimal, one by Leibniz and one by Nieuwentijdt, and argues that Leibniz's conception was implemented in Robinson's ... It is of interest to note that Leibnizian infinitesimals (differentials) are realized in nonstandard analysis, and nilsquare infinitesimals in smooth infinitesimal analysis" ...
Topological Derivative
... of a shape functional with respect to infinitesimal changes in its topology, such as adding an infinitesimal hole or crack ... the first-order term of the topological asymptotic expansion, dealing only with infinitesimal singular domain perturbations ...