Steiner ellipse

In non-standard 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. As such, it is a mathematical implementation of the historical concept of adequality introduced by Pierre de Fermat.

It can also be thought of as a mathematical implementation of Leibniz's Transcendental Law of Homogeneity. The standard part function was first defined by Abraham Robinson who used the notation ${}^{\circ }x$ for the standard part of a hyperreal $x$ (see Robinson 1974). This concept plays a key role in defining the concepts of the calculus, such as the derivative and the integral, in non-standard analysis. The latter theory is a rigorous formalisation of calculations with infinitesimals. The standard part of x is sometimes referred to as its shadow.

Definition

The standard part function "rounds off" a finite hyperreal to the nearest real number. The "infinitesimal microscope" is used to view an infinitesimal neighborhood of a standard real.

Nonstandard analysis deals primarily with the pair $\mathbb {R} \subset {}^{\ast }\mathbb {R}$, where the hyperreals ${}^{\ast }\mathbb {R}$ are an ordered field extension of the reals $\mathbb {R}$, 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 which is infinitely close to it. The relationship is expressed symbolically by writing

$\,\mathrm {st} (x)=x_{0}.$

The standard part of any infinitesimal is 0. Thus if N is an infinite hypernatural, then 1/N is infinitesimal, and st(1/N) = 0.

If a hyperreal $u$ is represented by a Cauchy sequence $\langle u_{n}:n\in \mathbb {N} \rangle$ in the ultrapower construction, then

${\text{st}}(u)=\lim _{n\to \infty }u_{n}.$

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 $\mathbb {R} \subset {}^{*}\mathbb {R}$ which is not internal; in fact every internal set in ${}^{\ast }\mathbb {R}$ which is a subset of $\mathbb {R}$ is necessarily finite, see (Goldblatt, 1998).

Applications

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

$f'(x)=\operatorname {st} \left({\frac {f(x+h)-f(x)}{h}}\right).$

Alternatively, if $y=f(x)$, one takes an infinitesimal increment $\Delta x$, and computes the corresponding $\Delta y=f(x+\Delta x)-f(x)$. One forms the ratio ${\frac {\Delta y}{\Delta x}}$. The derivative is then defined as the standard part of the ratio:

${\frac {dy}{dx}}={\text{st}}\left({\frac {\Delta y}{\Delta x}}\right)$.

See also

• Adequality
• Non-standard calculus

Notes

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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, 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. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1996. xx+293 pp. ISBN 0-691-04490-2