Hyperreal number

Summary

The hyperreal numbers form an ordered field ${}^*\R$ that contains the real numbers together with infinitesimal and infinite numbers. The usual construction is an ultrapower of $\R$: sequences of reals modulo agreement on a set in a nonprincipal ultrafilter on $\N$. By Łoś's theorem, ${}^*\R$ satisfies the same first-order statements as $\R$, which makes it the basic number system of nonstandard analysis. It is not Archimedean: an infinite hyperreal is larger than every natural number.