Archimedean field
From Wikipedia, the free encyclopedia
In mathematics, an Archimedean field is an ordered field with the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse.
In an ordered field F we can define the absolute value of an element x in F in the usual way by setting |x| = x for nonnegative x and −|x| = −x for negative x. Then, an Archimedean field F is one such that for any x in F there exists n in the natural numbers N for which |x| < n.
The real numbers form an Archimedean field. Moreover, it can be proved that any Archimedean field is isomorphic (as an ordered field) to a subfield of the real numbers.
Archimedean fields are important in the axiomatic construction of real numbers.
