Infinitesimal number

In mathematics , a positive infinitesimal number is an object which, with regard to the order of the real numbers, is greater than zero , but smaller than any positive real number, no matter how small.

properties

Obviously there are no infinitesimals among the real numbers that meet this requirement, because such an infinite would have to fulfill the condition , since there is also a positive real number. In order to still be able to define such infinitesimals , either the above requirement must be weakened, or the real numbers must be embedded in a larger, ordered field , in which there is then space for such additional elements. The latter is the way in which algebraic infinitesimals are defined (Coste, Roy, Pollack), and also the way of non-standard analysis (NSA) (Robinson, Nelson). ${\displaystyle x\in \mathbb {R} }$ ${\displaystyle 0<x<{\tfrac {x}{2}}}$ ${\displaystyle {\tfrac {x}{2}}}$

An infinitesimal has the property that any arbitrary sum of finitely many (in the NSA: standard finitely many) terms of the amount of this number is less than 1: ${\displaystyle x\neq 0}$

${\displaystyle |x|+\dotsb +|x|<1}$ for any finite number of summands.

In this case is greater than any positive real (in the NSA: standard real) number. For the algebraic infinitesimals this means that the corresponding field extension is non- Archimedean . ${\displaystyle |1/x|}$

calculus

The first mathematician to use such numbers was arguably Archimedes , although he did not believe in their existence .

Newton and Leibniz use the infinitesimal numbers to develop their calculus of infinitesimal calculus (differential and integral calculus).

Typically they argued (actually only Newton, Leibniz uses monads , today roughly: broken or formal power series ):

To find the derivative of the function , we assume that it is infinitesimal. Then ${\displaystyle f'(x)}$ ${\displaystyle f\colon \mathbb {R} \ni x\mapsto x^{2}\in \mathbb {R} }$ ${\displaystyle \mathrm {d} x}$

${\displaystyle f'(x)={\frac {f(x+\mathrm {d} x)-f(x)}{\mathrm {d} x}}={\frac {x^{2}+2x\cdot \mathrm {d} x+\left(\mathrm {d} x\right)^{2}-x^{2}}{\mathrm {d} x}}=2x+\mathrm {d} x=2x,}$

because is infinitesimally small. ${\displaystyle \mathrm {d} x}$

Although this argument is intuitive and gives correct results, it is not mathematically exact: The basic problem is that it is initially considered non-zero (one divides by ), but in the last step it is considered equal to zero. The use of infinitesimal numbers was criticized by George Berkeley in his work: The analyst: or a discourse addressed to an infidel mathematician (1734). ${\displaystyle \mathrm {d} x}$ ${\displaystyle \mathrm {d} x}$

Historical advancement

Since then, the question of the infinitesimals has been closely linked to the question of the nature of the real numbers. It was not until the nineteenth century that Augustin Louis Cauchy , Karl Weierstrass , Richard Dedekind and others gave real analysis a mathematically strict formal form. They introduced limit value considerations that made the use of infinitesimal quantities superfluous.

Even so, the use of infinitesimal numbers was still considered useful for simplifying representations and calculations. Thus, if the property denotes to be infinitesimal, and accordingly the property of being infinite , can be defined: ${\displaystyle x\approx 0}$ ${\displaystyle N\approx \infty }$

• A (standard) result is a null sequence if for all applies: . ${\displaystyle (a_{n})}$ ${\displaystyle N\approx \infty }$ ${\displaystyle a_{N}\approx 0}$
• A (standard) function on a bounded interval is uniformly continuous if and only if for all , that applies from the following: . ${\displaystyle f}$ ${\displaystyle I}$ ${\displaystyle x,y\in I}$ ${\displaystyle x-y\approx 0}$ ${\displaystyle f(x)-f(y)\approx 0}$

In the 20th century, number range extensions of real numbers were found that contain infinitesimal numbers in a formally correct form. The best known are the hyper-real numbers and the surreal numbers .

In the nonstandard analysis by Abraham Robinson (1960), which contains the hyper-real numbers as a special case, infinitesimal numbers are legitimate quantities. In this analysis, the above-mentioned derivation of can be justified by a slight modification: we are talking about the standard part of the differential quotient and the standard part of is (if is a standard number; more details in the linked article). ${\displaystyle f\colon \mathbb {R} \ni x\mapsto x^{2}\in \mathbb {R} }$ ${\displaystyle 2x+\mathrm {d} x}$ ${\displaystyle 2x}$ ${\displaystyle x}$

swell

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