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Infinitesimals (ε) and infinities (ω) on the hyperreal number line (ε = 1/ω) In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the "infinity-eth" item in a sequence.
Gottfried Wilhelm Leibniz argued that idealized numbers containing infinitesimals be introduced. The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using limits rather than ...
Keisler's Elementary Calculus: An Infinitesimal Approach defines continuity on page 125 in terms of infinitesimals, to the exclusion of epsilon, delta methods. The derivative is defined on page 45 using infinitesimals rather than an epsilon-delta approach. The integral is defined on page 183 in terms of infinitesimals.
The foundations of differential and integral calculus had been laid. In Cauchy's Cours d'Analyse, we find a broad range of foundational approaches, including a definition of continuity in terms of infinitesimals, and a (somewhat imprecise) prototype of an (ε, δ)-definition of limit in the definition of differentiation. [39]
In traditional approaches to calculus, differentials (e.g. dx, dy, dt, etc.) are interpreted as infinitesimals. There are several methods of defining infinitesimals rigorously, but it is sufficient to say that an infinitesimal number is smaller in absolute value than any positive real number, just as an infinitely large number is larger than ...
The word calculus is a Latin word, meaning originally "small pebble"; as such pebbles were used for calculation, the meaning of the word has evolved and today usually means a method of computation. Meanwhile, calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the study of continuous change.
Smooth infinitesimal analysis is a modern reformulation of the calculus in terms of infinitesimals. Based on the ideas of F. W. Lawvere and employing the methods of category theory, it views all functions as being continuous and incapable of being expressed in terms of discrete entities.
Instead, Cauchy formulated calculus in terms of geometric ideas and infinitesimals. Thus, his definition of continuity required an infinitesimal change in x to correspond to an infinitesimal change in y. He also introduced the concept of the Cauchy sequence, and started the formal theory of complex analysis.