In non-standard analysis, the standard part function is a function that provides a method of of translating calculations in the non-standard real numbers back to the real numbers. The standard part function was first defined by Abraham Robinson[citation needed] as a key ingredient in definitions of limits in non-standard analysis, which rigorously formalizes the calculations with infinitely small quantities called infinitesimals.
Definition
In nonstandard analysis deals primarily with the hyperreal line, is an extension of the real line that allows for infinitely small quantities. In the hyperreal line every real number has a collection of numbers (called a (monad) of hyperreals infinitely close to it. The standard part function associates to a finite hyperreal x, the unique standard real number x0 which infinitely close to it. In relationship is expressed symbolically by writting
- .
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.
Some authors develop the theory of non-standard analysis using Internal Set Theory. In this setting the standard part function "st" is not defined by an internal set [citation needed].
See also
References
- H. Jerome Keisler: Elementary Calculus: An Approach Using Infinitesimals. 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.)