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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 century with Abraham Robinson's development of nonstandard analysis and the hyperreal numbers, which, after centuries of controversy, showed that a formal treatment of infinitesimal calculus was possible. Following this, mathematicians developed surreal numbers, a related formalization of infinite and infinitesimal numbers that include both hyperreal cardinal and ordinal numbers, which is the largest ordered field.
Vladimir Arnold wrote in 1990:
Nowadays, when teaching analysis, it is not very popular to talk about infinitesimal quantities. Consequently, present-day students are not fully in command of this language. Nevertheless, it is still necessary to have command of it.
The crucial insight for making infinitesimals feasible mathematical entities was that they could still retain certain properties such as angle or slope, even if these entities were infinitely small.
Infinitesimals are a basic ingredient in calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity.
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