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. As such, it is a mathematical implementation of the historical concept of
adequality introduced by
Pierre de Fermat,
[Karin Usadi Katz and Mikhail G. Katz (2011) A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography. Foundations of Science.]
Se
arxiv
The authors refer to the Fermat-Robinson standard part. as well as
Leibniz'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
infinitesimals. 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 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
Limit
Given a sequence
(u_n), its limit is defined by
\lim_ u_n = \operatorname(u_H) where
H \in ^*\N \setminus \N is an infinite index. Here the limit is said to exist if the standard part is the same regardless of the infinite index chosen.
Continuity
A real function
f is continuous at a real point
x if and only if the composition
\operatorname\circ f is ''constant'' on the
halo
Halo, halos or haloes usually refer to:
* Halo (optical phenomenon)
* Halo (religious iconography), a ring of light around the image of a head
HALO, halo, halos or haloes may also refer to:
Arts and entertainment Video games
* Halo (franchise), ...
of
x. See
microcontinuity for more details.
See also
*
Adequality
*
Nonstandard calculus
Notes
References
*
H. Jerome Keisler. ''
Elementary Calculus: An Infinitesimal Approach''. First edition 1976; 2nd edition 1986. (This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~keisler/calc.html.)
*
Goldblatt, Robert. ''Lectures on the
hyperreals''. An introduction to nonstandard analysis.
Graduate Texts in Mathematics
Graduate Texts in Mathematics (GTM) ( ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standa ...
, 188. Springer-Verlag, New York, 1998.
*
Abraham Robinson. Non-standard analysis. Reprint of the second (1974) edition. With a foreword by
Wilhelmus A. J. Luxemburg
Wilhelmus Anthonius Josephus Luxemburg ( Delft, 11 April 1929 – 2 October 2018) was a Dutch American mathematician who was a professor of mathematics at the California Institute of Technology.
He received his B.A. from the University of Leide ...
. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1996. xx+293 pp.
{{Infinitesimals
Calculus
Nonstandard analysis
Real closed field