Standard part function

In non-standard analysis, the standard part function "st" is the key ingredient in Abraham Robinson's resolution of the paradox of Leibniz's definition (see Ghosts of departed quantities) of the derivative as the ratio of two infinitesimals

Gottfried Wilhelm Leibniz Inventor of infinitesimal calculus

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

see more at non-standard calculus.

Definition

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 existence of the standard part function is a consequence of the completeness of the reals or the fact that finite closed intervals of the reals are compact.

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

See also

• 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.)