Infinitesimal
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In mathematics, an infinitesimal, or infinitely small number, is a number that is smaller in absolute value than any positive real number. A number x is an infinitesimal if and only if for every integer n, |nx| is less than 1, no matter how large n is. In that case, 1/x is larger in absolute value than any positive real number.
Nonzero infinitesimals are not members of the set of real numbers. In the cases of greatest interest, they are hyperreal numbers. Infinitesimal quantities are sometimes represented using the Greek letter epsilon (ε)[1], or by means of differentials (such as dx and dy).
When used as an adjective in the vernacular, "infinitesimal" means extremely small.
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[edit] History of the infinitesimal
The first mathematician to make use of infinitesimals was Archimedes (~250 BC)[2], although he did not believe in the existence of physical infinitesimals. See the article on how Archimedes used infinitesimals. The Archimedean property is the property of an ordered algebraic structure of having no nonzero infinitesimals.
In India from the 12th century until the 16th century, infinitesimals were discovered for use with differential calculus by Indian mathematician Bhaskara and various Keralese mathematicians.
When Newton and Leibniz developed calculus, they made use of infinitesimals. A typical argument might go:
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- To find the derivative f′(x) of the function f(x) = x2, let dx be an infinitesimal. Then,
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- since dx is infinitely small.
This argument, while intuitively appealing, and producing the correct result, is not mathematically rigorous. The use of infinitesimals was attacked as incorrect by Bishop Berkeley in his work The Analyst.[3] The fundamental problem is that dx is first treated as non-zero (because we divide by it), but later discarded as if it were zero.
It was not until the second half of the nineteenth century that the calculus was given a formal mathematical foundation by Karl Weierstrass and others using the notion of a limit. In the 20th century, it was found that infinitesimals could after all be treated rigorously. Neither formulation is right or wrong, and both give the same results if used correctly.
[edit] Modern uses of infinitesimals
Infinitesimals are legitimate quantities in the non-standard analysis of Abraham Robinson, which makes use of hyperreal numbers. In this theory, the above computation of the derivative of f(x) = x² can be justified with a minor modification: we have to talk about the standard part of the difference quotient, and the standard part of x + dx is x.
An alternative way of phrasing this argument uses the Leibniz notation where y = f(x) and dy/dx = f′(x), the slope of the tangent line or real linear function that best approximates f(x) at (x,y). Whatever dy is, it's a real multiple of the infinitesimal dx, only approximated by (2x + dx)dx. 2x + dx is a number, calculated to a "higher precision" than a standard real number, that rounds off to 2x as the nearest real number, the desired slope.
Alternatively, we can have synthetic differential geometry or smooth infinitesimal analysis with its roots in category theory. This approach departs dramatically from the classical logic used in conventional mathematics by denying the law of excluded middle--i.e., not (a ≠ b) does not have to mean a = b. A nilsquare or nilpotent infinitesimal can then be defined. This is a number x where x ² = 0 is true, but x ≠ 0 can also be true at the same time. With an infinitesimal such as this, algebraic proofs using infinitesimals are quite rigorous, including the one given above.
Yet another approach uses a syntactic extension of the language. Edward Nelson was the first to develop this approach. Here the idea is that there are standard and nonstandard reals, infinitesimal and unlimited reals.
[edit] References
- ^ See http://mathworld.wolfram.com/Epsilon.html
- ^ Archimedes, The Method of Mechanical Theorems, see the Archimedes palimpsest
- ^ George Berkeley, The Analyst; or a discourse addressed to an infidel mathematician
