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 and , there is an integer such that . It also means that the set of natural numbers is not bounded above. 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.
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.
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:
Because Archimedes credited it to Eudoxus of Cnidus it is also known as the "Theorem of Eudoxus" or the Eudoxus axiom.
Archimedes used infinitesimals in heuristic arguments, although he denied that those were finished mathematical proofs.
Definition for linearly ordered groups
Let and be positive elements of a linearly ordered group G. Then is infinitesimal with respect to (or equivalently, is infinite with respect to ) if, for any natural number , the multiple is less than , that is, the following inequality holds:
This definition can be extended to the entire group by taking absolute values.
The group is Archimedean if there is no pair such that is infinitesimal with respect to .
Additionally, if is an algebraic structure with a unit (1) — for example, a ring — a similar definition applies to . If is infinitesimal with respect to , then is an infinitesimal element. Likewise, if is infinite with respect to , then is an infinite element. The algebraic structure is Archimedean if it has no infinite elements and no infinitesimal elements.
Ordered fields
Ordered fields have some additional properties: The rational numbers are embedded in any ordered field. That is, any ordered field has characteristic zero. If is infinitesimal, then is infinite, and vice versa. Therefore, to verify that a field is Archimedean it is enough to check only that there are no infinitesimal elements, or to check that there are no infinite elements. If is infinitesimal and is a rational number, then is also infinitesimal. As a result, given a general element , the three numbers , , and are either all infinitesimal or all non-infinitesimal. In this setting, an ordered field is Archimedean precisely when the following statement, called the axiom of Archimedes, holds: "Let be any element of . Then there exists a natural number such that ." Alternatively one can use the following characterization:
Definition for normed fields
The qualifier "Archimedean" is also formulated in the theory of rank one valued fields and normed spaces over rank one valued fields as follows. Let be a field endowed with an absolute value function, i.e., a function which associates the real number with the field element 0 and associates a positive real number with each non-zero and satisfies and . Then, is said to be Archimedean if for any non-zero there exists a natural number such that
Similarly, a normed space is Archimedean if a sum of terms, each equal to a non-zero vector , has norm greater than one for sufficiently large . 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,
respectively. A field or normed space satisfying the ultrametric triangle inequality is called non-Archimedean.