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In abstract algebra and
analysis Analysis is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics Mathematics (from Ancient Greek, Greek: ) includ ...
, the Archimedean property, named after the ancient Greek mathematician
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number ...

Archimedes
of
Syracuse Syracuse may refer to: Places Italy *Syracuse, Sicily, or spelled as ''Siracusa'' *Province of Syracuse United States *Syracuse, New York **East Syracuse, New York **North Syracuse, New York *Syracuse, Indiana *Syracuse, Kansas *Syracuse, Missou ...
, is a property held by some
algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s, 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 so that nx > y. It also means that the set of
natural numbers In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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 ''On the Sphere and Cylinder'' ( el, Περὶ σφαίρας καὶ κυλίνδρου) is a work that was published by Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek mathematician A mathematician is someone who uses an ex ...
''. The notion arose from the theory of magnitudes of Ancient Greece; it still plays an important role in modern mathematics such as
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in man ...
's axioms for geometry, and the theories of ordered groups,
ordered fieldIn mathematics, an ordered field is a field (mathematics), field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-comp ...
s, and local fields. An algebraic structure in which any two non-zero elements are ''comparable'', in the sense that neither of them is
infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not exist in the standard real number system, but do exist in many other number systems, such a ...
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 fieldIn mathematics, an ordered field is a field (mathematics), field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-comp ...
s, 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 Greece, ancient Greek geometer and physicist
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number ...

Archimedes
of
Syracuse Syracuse may refer to: Places Italy *Syracuse, Sicily, or spelled as ''Siracusa'' *Province of Syracuse United States *Syracuse, New York **East Syracuse, New York **North Syracuse, New York *Syracuse, Indiana *Syracuse, Kansas *Syracuse, Missou ...
. The Archimedean property appears in Book V of Euclid's Elements, 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's use of infinitesimals, Archimedes used infinitesimals in heuristic arguments, although he denied that those were finished mathematical proofs.


Definition for linearly ordered groups

Let and be Linearly_ordered_group#Definitions, positive elements of a linearly ordered group ''G''. Then is infinitesimal with respect to (or equivalently, is infinite with respect to ) if, for every natural number , the multiple is less than , that is, the following inequality holds: ::: \underbrace_ < y. \, 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
with a unit (1) — for example, a ring (mathematics), ring — a similar definition applies to . If is infinitesimal with respect to 1, then is an infinitesimal element. Likewise, if is infinite with respect to 1, 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 Embedding, embedded in any ordered field. That is, any ordered field has Characteristic (algebra), 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 r 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: ::: \forall\, \varepsilon \in K\big(\varepsilon > 0 \implies \exists\ n \in N : 1/n < \varepsilon\big).


Definition for normed fields

The qualifier "Archimedean" is also formulated in the theory of Valuation ring, 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 0 with the field element 0 and associates a positive real number , x, with each non-zero and satisfies , xy, =, x, , y, and , x+y, \le , x, +, y, . Then, is said to be Archimedean if for any non-zero there exists a natural number such that ::: , \underbrace_, > 1. \, 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, ::: , x+y, \le \max(, x, ,, y, ), respectively. 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. Monna.Monna, A. F., Over een lineare P-adisches ruimte, Indag. Math., 46 (1943), 74–84.


Examples and non-examples


Archimedean property of the real numbers

The field of the rational numbers can be assigned one of a number of absolute value functions, including the trivial function , x, =1, when , the more usual , x, = \sqrt, and the -adic absolute value functions. By Ostrowski's theorem, every non-trivial absolute value on the rational numbers is equivalent to either the usual absolute value or some -adic absolute value. 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. By this construction the field of real numbers is Archimedean both as an ordered field and as a normed field.Neal Koblitz, "p-adic Numbers, p-adic Analysis, and Zeta-Functions", Springer-Verlag,1977. On the other hand, the completions with respect to the other non-trivial absolute values give the fields of -adic numbers, where is a prime integer number (see below); since the -adic absolute values satisfy the ultrametric property, then the -adic number fields are non-Archimedean as normed fields (they cannot be made into ordered fields). 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. Denote by the set consisting of all positive infinitesimals. This set is bounded above by 1. Now proof by contradiction, assume for a contradiction that is nonempty. Then it has a least upper bound , which is also positive, so . Since is an upper bound of and is strictly larger than , is not a positive infinitesimal. That is, there is some natural number for which . On the other hand, is a positive infinitesimal, since by the definition of least upper bound there must be an infinitesimal between and , and if then is not infinitesimal. But , so is not infinitesimal, and this is a contradiction. This means that is empty after all: there are no positive, infinitesimal real numbers. The Archimedean property of real numbers holds also in constructive analysis, even though the least upper bound property may fail in that context.


Non-Archimedean ordered field

For an example of an
ordered fieldIn mathematics, an ordered field is a field (mathematics), field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-comp ...
that is not Archimedean, take the field of rational functions with real coefficients. (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.) To make this an ordered field, one must assign an ordering compatible with the addition and multiplication operations. Now if and only if ''f'' − ''g'' > 0, so we only have to say which rational functions are considered positive. Call the function positive if the leading coefficient of the numerator is positive. (One must check that this ordering is well defined and compatible with addition and multiplication.) By this definition, the rational function 1/''x'' is positive but less than the rational function 1. In fact, if is any natural number, then ''n''(1/''x'') = ''n''/''x'' is positive but still less than 1, no matter how big is. Therefore, 1/''x'' is an infinitesimal in this field. This example generalizes to other coefficients. Taking rational functions with rational instead of real coefficients produces a countable non-Archimedean ordered field. Taking the coefficients to be the rational functions in a different variable, say , produces an example with a different order type.


Non-Archimedean valued fields

The field of the rational numbers endowed with the p-adic metric and the p-adic number fields which are the completions, do not have the Archimedean property as fields with absolute values. All Archimedean valued fields are isometrically isomorphic to a subfield of the complex numbers with a power of the usual absolute value.Shell, Niel, Topological Fields and Near Valuations, Dekker, New York, 1990.


Equivalent definitions of Archimedean ordered field

Every linearly ordered field contains (an isomorphic copy of) the rationals as an ordered subfield, namely the subfield generated by the multiplicative unit 1 of , which in turn contains the integers as an ordered subgroup, which contains the natural numbers as an ordered monoid. The embedding of the rationals then gives a way of speaking about the rationals, integers, and natural numbers in . The following are equivalent characterizations of Archimedean fields in terms of these substructures. 1. The natural numbers are cofinal (mathematics), cofinal in . That is, every element of is less than some natural number. (This is not the case when there exist infinite elements.) Thus an Archimedean field is one whose natural numbers grow without bound. 2. Zero is the infimum in of the set . (If contained a positive infinitesimal it would be a lower bound for the set whence zero would not be the greatest lower bound.) 3. The set of elements of between the positive and negative rationals is non-open. This is because the set consists of all the infinitesimals, which is just the set when there are no nonzero infinitesimals, and otherwise is open, there being neither a least nor greatest nonzero infinitesimal. Observe that in both cases, the set of infinitesimals is closed. In the latter case, (i) every infinitesimal is less than every positive rational, (ii) there is neither a greatest infinitesimal nor a least positive rational, and (iii) there is nothing else in between. Consequently, any non-Archimedean ordered field is both incomplete and disconnected. 4. For any in the set of integers greater than has a least element. (If were a negative infinite quantity every integer would be greater than it.) 5. Every nonempty open interval of contains a rational. (If is a positive infinitesimal, the open interval contains infinitely many infinitesimals but not a single rational.) 6. The rationals are Dense set, dense in with respect to both sup and inf. (That is, every element of is the sup of some set of rationals, and the inf of some other set of rationals.) Thus an Archimedean field is any dense ordered extension of the rationals, in the sense of any ordered field that densely embeds its rational elements.


See also

* * *


Notes


References

* {{refend Field (mathematics) Ordered groups Real algebraic geometry