Archimedean field

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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 non-zero x in F there exists a natural number n such that

$|\underbrace{x+\ldots+x}_{n~ times}| > 1.$

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. Any complete Archimedean field is isomorphic (as an ordered field) to the field of real numbers.^{[citation needed]}

Archimedean fields are important in the axiomatic construction of real numbers.

Nonarchimedean fields with infinitesimal and infinitely large elements are also possible. An example using rational functions is presented at Archimedean property. Another example is the hyperreal numbers of nonstandard analysis.