Archimedean property

In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields.

The property, as typically construed, states that given two positive numbers

It also means that the set of natural numbers is not bounded above.

[2] The notion arose from the theory of magnitudes of ancient Greece; it still plays an important role in modern mathematics such as David Hilbert's axioms for geometry, and the theories of ordered groups, ordered fields, and local fields.

An algebraic structure in which any two non-zero elements are comparable, in the sense that neither of them is infinitesimal with respect to the other, is said to be Archimedean.

A structure which has a pair of non-zero elements, one of which is infinitesimal with respect to the other, is said to be non-Archimedean.

This can be made precise in various contexts with slightly different formulations.

For example, in the context of ordered fields, one has the axiom of Archimedes which formulates this property, where the field of real numbers is Archimedean, but that of rational functions in real coefficients is not.

The concept was named by Otto Stolz (in the 1880s) after the ancient Greek geometer and physicist Archimedes of Syracuse.

[3] Archimedes used infinitesimals in heuristic arguments, although he denied that those were finished mathematical proofs.

Let x and y be positive elements of a linearly ordered group G. Then

This definition can be extended to the entire group by taking absolute values.

is an algebraic structure with a unit (1) — for example, a ring — a similar definition applies to

Ordered fields have some additional properties: In this setting, an ordered field K is Archimedean precisely when the following statement, called the axiom of Archimedes, holds: Alternatively one can use the following characterization:

The qualifier "Archimedean" is also formulated in the theory of rank one valued fields and normed spaces over rank one valued fields as follows.

with the field element 0 and associates a positive real number

Similarly, a normed space is Archimedean if a sum of

A field with an absolute value or a normed space is either Archimedean or satisfies the stronger condition, referred to as the ultrametric triangle inequality,

A field or normed space satisfying the ultrametric triangle inequality is called non-Archimedean.

The concept of a non-Archimedean normed linear space was introduced by A. F.

The rational field is not complete with respect to non-trivial absolute values; with respect to the trivial absolute value, the rational field is a discrete topological space, so complete.

The completion with respect to the usual absolute value (from the order) is the field of real numbers.

[5] On the other hand, the completions with respect to the other non-trivial absolute values give the fields of p-adic numbers, where

-adic absolute values satisfy the ultrametric property, then the

In the axiomatic theory of real numbers, the non-existence of nonzero infinitesimal real numbers is implied by the least upper bound property as follows.

is empty after all: there are no positive, infinitesimal real numbers.

(A rational function is any function that can be expressed as one polynomial divided by another polynomial; we will assume in what follows that this has been done in such a way that the leading coefficient of the denominator is positive.)

(One must check that this ordering is well defined and compatible with addition and multiplication.)

Taking the coefficients to be the rational functions in a different variable, say

The following are equivalent characterizations of Archimedean fields in terms of these substructures.