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Infinitesimal: Simple Definition

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In normal English, infinitesimal means “something that is extremely small”, but in mathematics it has an even stronger meaning. It is a quantity that is infinitely small; so small as to be non-measurable.


An infinitesimal is nonzero in size. In other words, it isn’t exactly zero. Despite it’s peculiarities, it still exhibits many of the properties of larger entities: properties such as angle or slope. These properties can be measured.

Infinitesimal Calculus

At the core of Calculus is the idea that, to really understand a curve, you have to understand what is happening at every instantaneous moment in time. That instant in time, when graphed on a curve, becomes an infinitely small interval—an infinitesimal. We call it a differential, and symbolize it as Δx. Summing up infinitely many infinitesimals gives us an integral.

Alternate Definition

The word has also, on occasion, been used to refer to functions which tend to zero. These are functions whose limits approach zero as the function approaches infinity.

History of Infinitesimal

The study of these infinitely small intervals is intrinsic to Calculus; in fact, Calculus has historically been known as ‘infinitesimal calculus’ or “the calculus of infinitesimals’. The word “calculus”, in that context, meant accounting or reckoning, and came from the name of a small counting pebble.

The study of infinitesimals began early; in fact, Archimedes, the Greek mathematician who lived from about 287 BC to 212 BC, gave the first logically rigorous definition of them. But they were not always well accepted. In fact, during the 17th century they were the subject of many political and religious controversies, and in 1632 there was actually a ban on infinitesimals issued by Roman clerics.

References

Keisler, Jerome. Foundations of Infinitesimal Calculus. Online Edition. Retrieved from https://www.math.wisc.edu/~keisler/foundations.html on December 6, 2018.
Stroyan, Kieth. A Brief Introduction to Infinitesiml Calculus. Retrieved from http://homepage.math.uiowa.edu/~stroyan/InfsmlCalculus/Lecture1/Lect1.pdf on December 6, 2018.

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Stephanie Glen. "Infinitesimal: Simple Definition" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/infinitesimal-definition/
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