Infinitesimal
In mathematics, a positive infinitesimal is an object that is with respect to the order of the real numbers greater than zero, but smaller than any small positive real number.
Properties
Obviously, there are among the real numbers no infinitesimals, which meet this requirement, for such x should satisfy 0
An infinitesimal x not equal to 0 has the property that any sum of finitely many ( in the NSA: Default finitely many ) elements of the amount of that number is less than 1:
In this case, | 1 / x | is larger than any positive real ( in the NSA: standard real) number.
This implies that the algebraic infinitesimal that the associated field extension is non- Archimedean.
The first mathematician who made use of such numbers is Archimedes, although he did not believe in their existence.
Newton and Leibniz use the Infinitesimalzahlen to develop their calculations of calculus ( differential and integral calculus ).
Typically, they argued (actually, Newton, Leibniz used monads, today in about: broken or formal power series ) as:
To find the derivative f '(x ) of the function f ( x ) = x ² to determine, we assume that dx is infinitesimal.
Then
Because dx is infinitesimally small.
Although this argument provides intuitive and evident to correct results, it is not mathematically exact: The basic problem is that dx is initially considered not equal to zero ( we divide by dx) and later it is considered as if it were equal to zero.
The use of Infinitesimalzahlen was criticized by George Berkeley in his work: The analyst: or a discourse Addressed to at infidel mathematician (1734 ).
The question of the infinitesimal has since been closely linked to the question of the nature of real numbers.
Only in the nineteenth century was given by Augustin Louis Cauchy, Karl Weierstrass, Richard Dedekind and others of real analysis, a mathematically rigorous formal form.
They led a limit considerations that made unnecessary the use of infinitesimal quantities.
Nevertheless, the use of Infinitesimalzahlen was still considered useful for the simplification of representations and computations.
Thus, if the property referred to be infinitesimal, and accordingly, the property of being infinitely are defined:
In the 20th century speed range extensions of the real numbers were found containing infinitesimal numbers in formally correct form.
The best known are the hyper- real numbers and the surreal numbers.
In nonstandardanalysis Abraham Robinson (1960), containing the hyper real numbers as a special case, are legitimate Infinitesimalzahlen sizes.
In this analysis can be x ² justified by a slight modification of the above-mentioned derivative of f (x ) =: We are talking about the standard part of the derivative and the standard part of 2x dx is 2x (where x is a standard number; further details in the linked article ).
Calculus
Historical development
Swell