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| In [[abstract algebra]] and [[analysis]], the '''Archimedean property''', named after the ancient Greek mathematician [[Archimedes]] of [[Syracuse, Italy|Syracuse]], is a property held by some ordered or normed [[group (algebra)|groups]], [[field (mathematics)|fields]], and other [[algebraic structure]]s. 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]]''.<ref>G. Fisher (1994) in P. Ehrlich(ed.), Real Numbers, Generalizations of the Reals, and Theories of continua, 107-145, Kluwer Academic</ref>
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| The notion arose from the theory of [[magnitude (mathematics)|magnitudes]] of Ancient Greece; it still plays an important role in modern mathematics such as [[David Hilbert]]'s [[Hilbert's axioms|axioms for geometry]], and the theories of [[linearly ordered group|ordered groups]], [[ordered field]]s, and [[local fields]]. | |
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| 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]].
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| This can be made precise in various contexts with slightly different ways of formulation. For example, in the context of [[ordered field]]s, one has the '''axiom of Archimedes''' which formulates this property, where the field of [[real number]]s is Archimedean, but that of [[rational functions]] in real coefficients is not.
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| == History and origin of the name of the Archimedean property ==
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| The concept is named after the [[ancient Greece|ancient Greek]] geometer and physicist [[Archimedes]] of [[Syracuse, Italy|Syracuse]].
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| The Archimedean property appears in Book V of [[Euclid's Elements|Euclid's ''Elements'']] as Definition 4:
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| {{Quote|Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another.}}
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| Because Archimedes credited it to [[Eudoxus of Cnidus]] it is also known as the "Theorem of Eudoxus"<ref name="Knopp1951">{{cite book|last=Knopp|first=Konrad|authorlink=Konrad Knopp|title=Theory and Application of Infinite Series|edition=English 2nd|page=7|year=1951|publisher=Blackie & Son, Ltd.|location=London and Glasgow|isbn=0-486-66165-2}}</ref> or the ''Eudoxus axiom''.
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| [[Archimedes's use of infinitesimals|Archimedes used infinitesimals]] in [[heuristic]] arguments, although he denied that those were finished [[mathematical proof]]s. | |
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| ==Definition for linearly ordered groups==
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| Let ''x'' and ''y'' be positive elements<!-- link has to be fixed --> of a [[linearly ordered group]] G. Then '''''x'' is infinitesimal with respect to ''y''''' (or equivalently, '''''y'' is infinite with respect to ''x''''') if, for every [[natural number]] ''n'', the multiple ''nx'' is less than ''y'', that is, the following inequality holds:
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| ::: <math> \underbrace{x+\cdots+x}_{n\text{ terms}} < y. \, </math>
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| The group G is '''Archimedean''' if there is no pair ''x'',''y'' such that ''x'' is infinitesimal with respect to ''y''.
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| Additionally, if ''K'' is an [[algebraic structure]] with a unit (1) — for example, a [[ring (mathematics)|ring]] — a similar definition applies to ''K''. If ''x'' is infinitesimal with respect to 1, then ''x'' is an '''infinitesimal element'''. Likewise, if ''y'' is infinite with respect to 1, then ''y'' is an '''infinite element'''. The algebraic structure ''K'' is Archimedean if it has no infinite elements and no infinitesimal elements.
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| ===Ordered fields===
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| An [[ordered field]] has some additional nice properties.
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| *One may assume that the rational numbers are contained in the field.
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| *If <var>x</var> is infinitesimal, then 1/<var>x</var> 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.
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| *If <var>x</var> is infinitesimal and <var>r</var> is a rational number, then {{math|<var>r</var> <var>x</var>}} is also infinitesimal. As a result, given a general element ''c'', the three numbers ''c''/2, ''c'', and 2''c'' are either all infinitesimal or all non-infinitesimal.
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| In this setting, an ordered field ''K'' is Archimedean precisely when the following statement, called the '''axiom of Archimedes''', holds:
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| : ''Let x be any element of K. Then there exists a natural number n such that n > x.''
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| Alternatively one can use the following characterization:
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| : For any positive ''ε'' in ''K'', there exists a natural number ''n'', such that 1/''n'' < ''ε''.
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| == 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 ''F'' 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 <math>|x|</math> with each non-zero <math> x\in F</math> and satisfies
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| <math>|xy|=|x| |y|</math> and <math>|x+y| \le |x|+|y|</math>. Then, ''F'' is said to be '''Archimedean''' if for any non-zero <math> x\in F</math> there exists a [[natural number]] ''n'' such that
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| ::: <math>|\underbrace{x+\cdots+x}_{n\text{ terms}}| > 1. \, </math>
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| Similarly, a normed space is Archimedean if a sum of <math> n </math> terms, each equal to a non-zero vector <math> x </math>, has norm greater than one for sufficiently large <math> n </math>. 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]],
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| ::: <math>|x+y| \le \max(|x|,|y|)</math>,
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| respectively. A field or normed space satisfying the ultrametric triangle inequality is called '''non-Archimedean'''.
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| The concept of a non-Archimedean normed linear space was introduced by A. F. Monna.<ref name=monna1>Monna, A. F., Over een lineare P-adisches ruimte, Indag. Math., 46 (1943), 74–84.</ref>
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| == Examples and non-examples == | |
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| ===Archimedean property of the real numbers===
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| The field of the rational numbers can be assigned one of a number of absolute value functions, including the trivial function <math>|x|=1,</math> when <math> x \neq 0</math>, the more usual <math>|x| = \sqrt{x^2}</math>, and the '''''p''-adic absolute value''' functions. One is Archimedean and the others non trivial are non--Archimedean ([[Ostrowski's theorem]]).{{clarify|date=September 2013}} The rational field is not complete with respect to non trivial absolute values. The completion with respect to the absolute value from the order is the field of real numbers; while the completions with respect to the others are the field of p--adic numbers, where p is a prime integer number (see below). By this construction the field of real numbers is Archimedean both as an ordered field and as a normed field. <ref>Neal Koblitz, "p-adic Numbers, p-adic Analysis, and Zeta-Functions", Springer-Verlag,1977.</ref>
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| <!-- "by axiom" side -->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 ''Z'' the set consisting of all positive infinitesimals. This set is bounded above by 1. Now [[proof by contradiction|assume for a contradiction]] that ''Z'' is nonempty. Then it has a [[least upper bound]] ''c'', which is also positive, so ''c''/2 < ''c'' < 2''c''. Since ''c'' is an [[upper bound]] of ''Z'' and 2''c'' is strictly larger than ''c'', 2''c'' is not a positive infinitesimal. That is, there is some natural number ''n'' for which 1/''n'' < 2''c''. On the other hand, ''c''/2 is a positive infinitesimal, since by the definition of least upper bound there must be an infinitesimal ''x'' between ''c''/2 and ''c'', and if 1/''k'' < ''c''/2 <= ''x'' then ''x'' is not infinitesimal. But 1/(4''n'') < ''c''/2, so ''c''/2 is not infinitesimal, and this is a contradiction. This means that ''Z'' is empty after all: there are no positive, infinitesimal real numbers.
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| One should note that the Archimedean property of real numbers holds also in [[constructive analysis]], even though the least upper bound property may fail in that context.
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| ===Non-Archimedean ordered field=== | |
| {{main|Non-Archimedean ordered field}}
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| For an example of an [[ordered field]] that is not Archimedean, take the field of [[rational function]]s 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 ''f'' > ''g'' 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 ''n'' is any natural number, then ''n''(1/''x'') = ''n''/''x'' is positive but still less than 1, no matter how big ''n'' is. Therefore, 1/''x'' is an infinitesimal in this field.
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| 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 ''y'', produces an example with a different [[order type]].
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| === Non-Archimedean valued fields ===
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| 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. <!-- Another example is the [[hyperreal numbers]] of [[nonstandard analysis]]. : (ed. I detest this, because the formal interpretation of the Axiom of Archimedes is indeed satisfied by hypernatural numbers in place of the "standard" natural numbers, which do not form a "hyperset" (or *-set, superset, whatever we call it) inside the system of the "hyperreal numbers".)--> All Archimedean valued fields are isometrically isomorphic to a subfield of the complex numbers with a power of the usual absolute value.<ref name=shell1>Shell, Niel, Topological Fields and Near Valuations, Dekker, New York, 1990. ISBN 0-8247-8412-X</ref> There is a non-trivial non-Archimedean valuation on every infinite field.
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| ===Equivalent definitions of Archimedean ordered field===
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| Every linearly ordered field ''K'' contains (an isomorphic copy of) the rationals as an ordered subfield, namely the subfield generated by the multiplicative unit 1 of ''K'', which in turn contains the integers as an ordered subgroup, which contains the natural numbers as an ordered [[monoid]]<!-- semigroup -->. The embedding of the rationals then gives a way of speaking about the rationals, integers, and natural numbers in ''K''. The following are equivalent characterizations of Archimedean fields in terms of these substructures.<ref name="Schechter">{{harvnb|Schechter|1997|loc=§10.3}}</ref>
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| 1. The natural numbers are [[cofinal (mathematics)|cofinal]] in ''K''. That is, every element of ''K'' 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.
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| 2. Zero is the [[infimum]] in ''K'' of the set {1/2, 1/3, 1/4, … }. (If ''K'' contained a positive infinitesimal it would be a lower bound for the set whence zero would not be the greatest lower bound.)
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| 3. The set of elements of ''K'' between the positive and negative rationals is closed. This is because the set consists of all the infinitesimals, which is just the closed set {0} when there are no nonzero infinitesimals, and otherwise is open, there being neither a least nor greatest nonzero infinitesimal. 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, a situation that points up both the incompleteness and disconnectedness of any non-Archimedean field.
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| 4. For any <var>x</var> in ''K'' the set of integers greater than <var>x</var> has a least element. (If <var>x</var> were a negative infinite quantity every integer would be greater than it.)
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| 5. Every nonempty open interval of ''K'' contains a rational. (If <var>x</var> is a positive infinitesimal, the open interval {{open-open|<var>x</var>, 2<var>x</var>}} contains infinitely many infinitesimals but not a single rational.)
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| 6. The rationals are [[Dense set|dense]] in ''K'' with respect to both sup and inf. (That is, every element of ''K'' 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.
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| ==Notes==
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| {{reflist}}
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| ==References==
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| {{refbegin}}
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| * {{Cite book|last=Schechter|first=Eric|authorlink=Eric Schechter|title=Handbook of Analysis and its Foundations|publisher=Academic Press|year=1997|isbn=0-12-622760-8|url=http://www.math.vanderbilt.edu/~schectex/ccc/|ref=harv|postscript=.}}
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| {{refend}}
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| [[Category:Field theory]]
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| [[Category:Ordered groups]]
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| [[Category:Real algebraic geometry]]
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