Archimedean property

Introduction

The Archimedean property is a trait shared by several algebraic structures, such as ordered or normed groups and fields, and is named after the ancient Greek mathematician Archimedes of Syracuse. The condition asserts that if two positive integers x and y are supplied, there is an integer n such that nx > y. It also implies that the set of natural numbers above is unbounded.  It is the property of not having indefinitely big or endlessly small constituents, to put it another way. Because it occurs as Axiom V of Archimedes’ On the Sphere and Cylinder, it was Otto Stolz who gave the axiom its name.

The idea came from Ancient Greece’s theory of magnitudes, and it’s still used in modern mathematics in places like David Hilbert’s axioms for geometry and the theories of ordered groups, ordered fields, and local fields. Archimedean structure is an algebraic structure in which any two non-zero components are similar in the sense that neither is infinitesimal with regard to the other. Non-Archimedean structures have a pair of non-zero components, one of which is infinitesimal in comparison to the other. A linearly ordered Archimedean group, for example, is an Archimedean group.

With slightly varied phrasing, this may be made more precise in many scenarios. The axiom of Archimedes, for example, formulates this quality in the context of ordered fields, where the field of real numbers is Archimedean but the field of rational functions in real coefficients is not.

History and origin of the name of the Archimedean property

The Archimedean property’s name has a long and illustrious history. Otto Stolz (in the 1880s) named the notion after Archimedes of Syracuse, an ancient Greek geometer and scientist. The Archimedean property is defined as follows in Book V of Euclid’s Elements:

 Magnitudes are stated to have a ratio to one another that can surpass one another when multiplied. 

It’s also known as the “Theorem of Eudoxus” or the Eudoxus Axiom since Archimedes attributed it to Eudoxus of Cnidus. Infinitesimals were utilised by Archimedes in heuristic arguments, although he rejected that they were complete mathematical proofs.

Examples and non-examples

Archimedean property of the real numbers

A number of absolute value functions, IxI = 1 when x ≠ 0 including the trivial and p-adic absolute value functions, can be given to the field of rational numbers. Every non-trivial absolute value on the rational numbers is identical to either the ordinary absolute value or some p-adic absolute value, according to Ostrowski’s theorem. The rational field is not complete in terms of non-trivial absolute values, but it is a discrete topological space in terms of trivial absolute values, hence it is complete. The field of real numbers is the completion with regard to the typical absolute value (from the order). The field of real numbers is Archimedean both as an ordered and as a normed field as a result of this design. The completions with respect to the other non-trivial absolute values, on the other hand, yield the fields of p-adic numbers (see below); because the p-adic absolute values fulfil the ultrametric condition, the p-adic number fields are non-Archimedean as normed fields (they cannot be made into ordered fields).

 Even if the least upper bound condition may fail in constructive analysis, the Archimedean property of real numbers holds.

Non-Archimedean ordered field

Consider the field of rational functions with real coefficients as an example of a non-Archimedean ordered field. (A rational function is any function that can be represented as one polynomial divided by another polynomial; we’ll assume in the rest of this section that the denominator’s leading coefficient is positive.) To make this an ordered field, you must choose an ordering that is consistent with addition and multiplication. Now, f > g is true only if and only if f g > 0, thus we just need to specify which rational functions are positive. If the numerator’s leading coefficient is positive, the function is called positive. (It’s important to double-check that the ordering is clear and compatible with addition and multiplication.) The rational function 1/x is positive but smaller than the rational function 1 according to this definition. In reality, if n is any natural integer, then n(1/x) = n/x is positive but less than 1, regardless of the size of n. As a result, in this field, 1/x is infinitesimal.

His example may be applied to a variety of coefficients. A countable non-Archimedean ordered field is obtained by using rational functions with rational rather than real coefficients. Changing the coefficients to rational functions in a different variable, such as y, results in a different order kind of example.

Non-Archimedean valued fields

The p-adic number fields that are completions and the field of rational numbers supplied with the p-adic metric do not have the Archimedean property as fields with absolute values. Isometrically isomorphic to a subfield of the complex numbers with a power of the typical absolute value are all Archimedean valued fields.

Conclusion

The Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a characteristic shared by various algebraic structures, such as ordered or normed groups and fields. If two positive numbers x and y are given, the condition says that there is an integer n such that nx > y. It also indicates that the above-mentioned collection of natural numbers is infinite. To put it another way, it is the characteristic of not having infinitely large or infinitely small parts.