In
nonstandard analysis, the standard part function is a function from the limited (finite)
hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every such hyperreal
, the unique real
infinitely close to it, i.e.
is
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...
. As such, it is a mathematical implementation of the historical concept of
adequality introduced by
Pierre de Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
,
[Karin Usadi Katz and Mikhail G. Katz (2011) A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography. ]Foundations of Science
''Foundations of Science'' is a peer-reviewed interdisciplinary academic journal focussing on methodological and philosophical topics concerning the structure and the growth of science. It is the official journal of the Association for Found ...
.
Se
arxiv
The authors refer to the Fermat-Robinson standard part. as well as
Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
's
Transcendental law of homogeneity.
The standard part function was first defined by
Abraham Robinson who used the notation
for the standard part of a hyperreal
(see Robinson 1974). This concept plays a key role in defining the concepts of the calculus, such as continuity, the derivative, and the integral, in
nonstandard analysis. The latter theory is a rigorous formalization of calculations with
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...
s. The standard part of ''x'' is sometimes referred to as its shadow.
Definition
Nonstandard analysis deals primarily with the pair
, where the
hyperreals
are an
ordered field extension of the reals
, and contain infinitesimals, in addition to the reals. In the hyperreal line every real number has a collection of numbers (called a
monad, or halo) of hyperreals infinitely close to it. The standard part function associates to a
finite hyperreal ''x'', the unique standard real number ''x''
0 that is infinitely close to it. The relationship is expressed symbolically by writing
:
The standard part of any
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...
is 0. Thus if ''N'' is an infinite
hypernatural, then 1/''N'' is infinitesimal, and
If a hyperreal
is represented by a Cauchy sequence
in the
ultrapower construction, then
:
More generally, each finite
defines a
Dedekind cut on the subset
(via the total order on
) and the corresponding real number is the standard part of ''u''.
Not internal
The standard part function "st" is not defined by an
internal set. There are several ways of explaining this. Perhaps the simplest is that its domain L, which is the collection of limited (i.e. finite) hyperreals, is not an internal set. Namely, since L is bounded (by any infinite hypernatural, for instance), L would have to have a least upper bound if L were internal, but L doesn't have a least upper bound. Alternatively, the range of "st" is
, which is not internal; in fact every internal set in
that is a subset of
is necessarily ''finite'', see (Goldblatt, 1998).
Applications
All the traditional notions of calculus can be expressed in terms of the standard part function, as follows.
Derivative
The standard part function is used to define the derivative of a function ''f''. If ''f'' is a real function, and ''h'' is infinitesimal, and if ''f''′(''x'') exists, then
:
Alternatively, if
, one takes an infinitesimal increment
, and computes the corresponding
. One forms the ratio
. The derivative is then defined as the standard part of the ratio:
:
Integral
Given a function
on