Standard part function: Difference between revisions

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[[Image:Gottfried Wilhelm von Leibniz.jpg|thumb|200px|right|''[[Gottfried Wilhelm Leibniz]]'' Inventor of infinitesimal calculus]]

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 of the derivative as the ratio of two infinitesimals
:<math>\tfracfrac{dy}{dx}</math>,
see more at [[non-standard calculus]].

==Definition==

The standard part function associates to a [[finite]] [[hyperreal number|hyperreal]] ''x'', the standard real ''x₀'' infinitely close to it, so that we can write

:<math>\,\mathrm{st}(x)=x_0</math>.

The existence of the standard part function is a consequence of the [[completeness of the reals]] or the fact that [[finite]] [[closed interval]]s of the reals are [[compact]].