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, typically construed, states that given two positive numbers x and y, there is an integer n such that nx > y. It also means that the set of natural numbers is not bounded above.[1] Roughly speaking, it is the property of having no infinitely large or infinitely small elements. It was Otto Stolz who gave the axiom of Archimedes its name because it appears as Axiom V of Archimedes’ On the Sphere and Cylinder.[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. For example, a linearly ordered group that is Archimedean is an Archimedean group.

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.

Illustration of the Archimedean property.

1 History and origin of the name of the Archimedean property

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

The Archimedean property appears in Book V of Euclid’s Elements as Definition 4:

Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another.

Because Archimedes credited it to Eudoxus of Cnidus it is also known as the “Theorem of Eudoxus” or the Eudoxus axiom.[3]

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

2 Definition for linearly ordered groups

2.1 Ordered fields

3 Definition for normed fields

4 Examples and non-examples

4.1 Archimedean property of the real numbers

4.2 Non-Archimedean ordered field

4.3 Non-Archimedean valued fields

4.4 Equivalent definitions of Archimedean ordered field

5 See also