In non-standard analysis, the standard part function is a function from the limited (finite) hyperreals to the reals, which associates to every hyperreal, the unique real infinitely close to it. The standard part function was first defined by Abraham Robinson[citation needed] as a key ingredient in defining the concepts of the calculus such as the derivative and the integral in non-standard analysis, which rigorously formalizes the calculations with infinitesimals.
Definition
Nonstandard analysis deals primarily with the hyperreal line. The latter is an extension of the real line which contains, in addition to the reals, also 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. The 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.
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.)