Archimedean property
In
The notion arose from the theory of
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
History and origin of the name of the Archimedean property
The concept was named by
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]
Definition for linearly ordered groups
Let x and y 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 K 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
The concept of a non-Archimedean normed linear space was introduced by A. F. Monna.[4]
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 , when , the more usual , 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.[5] On the other hand, the completions with respect to the other non-trivial absolute values give the fields of p-adic numbers, where is a prime integer number (see below); since the -adic absolute values satisfy the
In the
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 field 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
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.[6]
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 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.[7]
- The natural numbers are 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.
- Zero is the infimumin 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.)
- 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.
- 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.)
- 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.)
- The rationals are 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
- 0.999... – Alternative decimal expansion of 1
- Archimedean ordered vector space – A binary relation on a vector space
- Construction of the real numbers
Notes
- ^ "Math 2050C Lecture" (PDF). cuhk.edu.hk. Retrieved 3 September 2023.
- ^ G. Fisher (1994) in P. Ehrlich(ed.), Real Numbers, Generalizations of the Reals, and Theories of continua, 107-145, Kluwer Academic
- ISBN 0-486-66165-2.
- MR 0015678.
- ^ Neal Koblitz, "p-adic Numbers, p-adic Analysis, and Zeta-Functions", Springer-Verlag,1977.
- ISBN 0-8247-8412-X
- ^ Schechter 1997, §10.3
References
- ISBN 0-12-622760-8. Archived from the originalon 7 March 2015. Retrieved 30 January 2009.