In nonstandard analysis, the standard part function is a function from the limited (finite) hyperreal numbers is infinitesimal. As such, it is a mathematical implementation of the historical concept of adequality introduced by Pierre de Fermat, as well as Leibniz's transcendental law of homogeneity.

The standard part function was first defined by Abraham Robinson (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

, where the hyperreals are an ordered field, 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 x0 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

in the ultrapower construction, then

defines a Dedekind cut) 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 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

. The derivative is then defined as the standard part of the ratio:

Integral

is taken to be infinitesimal, exploiting a hyperfinite partition of the interval [a,b].

Limit

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

is constant on the halo. See microcontinuity for more details.

See also

• Adequality

• 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.