Concept# Infinitesimal

Summary

In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the "infinity-th" item in a sequence.
Infinitesimals do not exist in the standard real number system, but they do exist in other number systems, such as the surreal number system and the hyperreal number system, which can be thought of as the real numbers augmented with both infinitesimal and infinite quantities; the augmentations are the reciprocals of one another.
Infinitesimal numbers were introduced in the development of calculus, in which the derivative was first conceived as a ratio of two infinitesimal quantities. This definition was not rigorously formalized. As calculus developed further, infinitesimals were replaced by limits, which can be calculated using the standard real numbers.
Infinitesimals regained popularity in the 20th ce

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The objective of this PhD thesis is the translation of, and the mathematical commentary on, a 16th-century Latin book. Its author, Diego Palomino is not well known. With a background in theology, he was a prior. In order to obtain his PhD at the University of Alcala (Madrid), he submitted a work, De mutations æris, in which he included a collection of what seems to be readings notes, entitled Fragmentum de inventionibus scientiarum. His readings have been drawn from various famous mathematicians of his time and ancient ones. The originality of his work relies mostly on his inventiveness and his style —which can be sarcastic— making the reading of it quite interesting and lively. His work consists for the main part in explaining some unclear demonstrations, or bringing new methods of solution ; he also innovates in solving pairs of indeterminate equations by providing the complete set of integral solutions. Before him, only one mathematician (Abu ̄ K ̄amil, at the end of the 10th century) did so, the other mathematicians restricting themselves to giving only one solution or a pair. Palomino did not hesitate in criticizing a well-established theory in ancient mathematics, namely Archimedes —although his critics seem to rely on a faulty edition. His book is entitled to have a significant place in the history of mathematics, for it both maintains the rigor of Greek classical mathematics and announces innovation, as did the 17th century, culminating with the discovery of the infinitesimal calculus.

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Diffusion-limited reactions are usually described within the Smoluchowski theory, which neglects interparticle interactions. We propose a simple way to incorporate excluded-volume effects building on simulations of hard sphere in the presence of a sink. For large values of the sink-to-particle size ratio Rs, the measured encounter rate is in good agreement with a simple generalization of the Smoluchowski equation at high densities. Reducing Rs, the encounter rate is substantially depressed and becomes even nonmonotonic for Rs≪1. Concurrently with the saturation of the rate, stationary density waves set in close to the sink. A mean-field analysis helps to shed light on the subtle link between such ordering and the slowing down of the encounter dynamics. Finally, we show how an infinitesimal amount of nonreacting impurities can equally slow down dramatically the reaction. © 2010 The American Physical Society