Standard part function

In non-standard analysis, the standard part function "st" is a mathematical implementation of Pierre de Fermat's adequality. It is the key ingredient in Abraham Robinson's formalisation of Leibniz's infinitesimal definition (see ghosts of departed quantities) of the derivative as the ratio of two infinitesimals

${\displaystyle {\frac {dy}{dx}}}$,

see more at non-standard calculus.

Definition

The hyperreal line is an extension of the real line. Thus, every real number is accompanied by a cluster (monad) of hyperreals infinitely close, or adequal, to it. The standard part function associates to a finite hyperreal x, the standard real x₀ infinitely close to it, so that we can write

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

The standard part of any infinitesimal is 0. Thus if N is an infinite hypernatural, then (0.1)^N is infinitesimal, and st(0.1^N)=0.

The standard part function "st" is not an internal object.

See also

• Adequality
• Non-standard calculus

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.)