In
nonstandard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using (ε, δ)-definitio ...
, the standard part function is a function from the limited (finite)
hyperreal number
In mathematics, hyperreal numbers are an extension of the real numbers to include certain classes of infinite and infinitesimal numbers. A hyperreal number x is said to be finite if, and only if, , x, for some integer s 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 non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...
. As such, it is a mathematical implementation of the historical concept of
adequality
Adequality is a technique developed by Pierre de Fermat in his treatise ''Methodus ad disquirendam maximam et minimam'' (a Latin treatise circulated in France c. 1636 ) to calculate maxima and minima of functions, tangents to curves, area, center o ...
introduced by
Pierre de Fermat
Pierre de Fermat (; ; 17 August 1601 – 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 is recognized for his d ...
, as well as
Leibniz
Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to many ...
's
Transcendental law of homogeneity
In mathematics, the transcendental law of homogeneity (TLH) is a heuristic principle enunciated by Gottfried Wilhelm Leibniz most clearly in a 1710 text entitled ''Symbolismus memorabilis calculi algebraici et infinitesimalis in comparatione potent ...
.
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 history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using (ε, δ)-definitio ...
. The latter theory is a rigorous formalization of calculations with
infinitesimal
In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...
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
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Basic examples of ordered fields are the rational numbers and the real numbers, both with their standard ord ...
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 non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...
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
The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All fact ...
construction, then
:
More generally, each finite
defines a
Dedekind cut
In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind (but previously considered by Joseph Bertrand), are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of a set, ...
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''.
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 of
x. See
microcontinuity
In nonstandard analysis, a discipline within classical mathematics, microcontinuity (or ''S''-continuity) of an internal function ''f'' at a point ''a'' is defined as follows:
:for all ''x'' infinitely close to ''a'', the value ''f''(''x'') is in ...
for more details.
See also
*
Adequality
Adequality is a technique developed by Pierre de Fermat in his treatise ''Methodus ad disquirendam maximam et minimam'' (a Latin treatise circulated in France c. 1636 ) to calculate maxima and minima of functions, tangents to curves, area, center o ...
*
Nonstandard calculus
References
Further reading
*
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.)
*
Abraham Robinson. Non-standard analysis. Reprint of the second (1974) edition. With a foreword by
Wilhelmus A. J. Luxemburg. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1996. xx+293 pp.
{{Infinitesimals
Calculus
Nonstandard analysis
Real closed field