Standard part function: Difference between revisions

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In [[non-standard analysis]], the '''standard part function''' is a function from the limited (finite) [[Hyperreal number|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]].<ref>Karin Usadi Katz and [[Mikhail Katz|Mikhail G. Katz]] (2011) A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography. [[Foundations of Science]]. {{doi|10.1007/s10699-011-9223-1}} [http://www.springerlink.com/content/tj7j2810n8223p43/] See [http://arxiv.org/abs/1104.0375 arxiv]. The authors refer to the Fermat-Robinson standard part.</ref>

It can also be thought of as a mathematical implementation of [[Gottfried Wilhelm Leibniz|Leibniz]]'s [[Transcendental Law of Homogeneity]]. The standard part function was first defined by [[Abraham Robinson]] who used the notation <math>{}^{\circ}x</math> for the standard part of a hyperreal <math>x</math> (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 [[infinitesimal]]s. The standard part of ''x'' is sometimes referred to as its '''shadow'''.

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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''&prime;(''x'') exists, then

:<math>f'(x) = \operatorname{st}\left(\frac {f(x+h)-f(x)}h\right).</math>

==Notes==

{{Reflist}}

==See also==

*[[Adequality]]

*[[Non-standard calculus]]

==Notes==

{{Reflist}}

== 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.)

*[[Robert Goldblatt|Goldblatt, Robert]]. ''Lectures on the [[hyperreal number|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

*[[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

{{Infinitesimals}}