Editing Differential (mathematics) (section)

=== Nonstandard analysis ===

The final approach to infinitesimals again involves extending the real numbers, but in a less drastic way. In the [[nonstandard analysis]] approach there are no nilpotent infinitesimals, only invertible ones, which may be viewed as the [[multiplicative inverse|reciprocal]]s of infinitely large numbers.<ref name="nonstd"/> Such extensions of the real numbers may be constructed explicitly using equivalence classes of sequences of [[real number]]s, so that, for example, the sequence (1,&nbsp;1/2,&nbsp;1/3,&nbsp;...,&nbsp;1/''n'',&nbsp;...) represents an infinitesimal. The [[first-order logic]] of this new set of [[hyperreal number]]s is the same as the logic for the usual real numbers, but the [[completeness axiom]] (which involves [[second-order logic]]) does not hold. Nevertheless, this suffices to develop an elementary and quite intuitive approach to calculus using infinitesimals, see [[transfer principle]].