Science Fair Project Encyclopedia
Archimedean field
In mathematics, an Archimedean field is an ordered field with the Archimedean property. Ultimately, the eponym is 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.
10-26-2009 08:16:03
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details