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In abstract algebra, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some groups, fields, and other algebraic structures. Roughly speaking, it is the property of having no infinitely large or infinitely small elements (i.e. no nontrivial infinitesimals). This can be made precise in various contexts, for example, for fields with an absolute value, where the ordered field of real numbers is Archimedean, but the field of p-adic numbers with the p-adic absolute value is non-Archimedean.
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 called Archimedean. A structure which has a pair of non-zero elements, one of which is infinitesimal with respect to the other, is called non-Archimedean. For example, a linearly ordered group that is Archimedean is an Archimedean group, and a field with a non-Archimedean absolute value is a non-Archimedean field.
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Definition
Let x and y be positive elements 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:
The group G is Archimedean if there is no pair x,y such that x is infinitesimal with respect to y.
Additionally, if K is an algebraic structure with a unit (1) — for example, a 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.
For fields
In an ordered field, some additional nice properties apply.
- If x is infinitesimal, then 1/x 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 x is infinitesimal and r is a rational number, then rx is also infinitesimal. As a result, given a general element c, the three numbers c/2, c, and 2c are either all infinitesimal or all non-infinitesimal.
Archimedean property of the real numbers
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 assume by contradiction that Z is nonempty. Then it has a least upper bound c, which is also positive, so c/2 < c < 2c. Since c is an upper bound of Z and 2c is strictly larger than c, 2c must be strictly larger than every positive infinitesimal. In particular, 2c cannot itself be an infinitesimal, for then 2c would have to be greater than itself. Moreover since c is the least upper bound of Z, c/2 must be infinitesimal. But 2c and c/2 cannot have different types by the above result, so there is a contradiction. The conclusion follows that Z is empty after all: there are no positive, infinitesimal real numbers.
It is interesting to 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.
Example of a 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 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. Write the rational function in the form of a polynomial plus a remainder over the denominator, where the degree of the remainder is less than the degree of the denominator (using the Euclidean algorithm for polynomials). Call the function positive if either (1) the leading coefficient of the polynomial part is positive, or (2) the polynomial part is zero and the leading coefficient of the remainder is positive. (One must check that this ordering is well defined and compatible with the addition and multiplication operations.) 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.
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.
Equivalent definitions of Archimedean field
Every linearly ordered field K embeds (an isomorphic copy of) the rationals as an ordered subfield, namely the subfield generated by the multiplicative unit 1 of K, which in turn embeds the integers as an ordered subgroup, which embeds 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 K. The following are equivalent characterizations of Archimedean fields in terms of these substructures.[1]
1. The natural numbers are 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.
2. The infimum in K of the set {1/2, 1/3, 1/4, …} exists and is zero. (If K contained a positive infinitesimal it would be a lower bound on the set whence zero could not be the greatest lower bound.)
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.
4. For any x in K the set of integers greater than x has a least element. (If x were a negative infinite quantity every integer would be greater than it.)
5. Every nonempty open interval of K contains a rational. (If x is a positive infinitesimal, the open interval (x,2x) contains infinitely many infinitesimals but not a single rational.)
6. The rationals are 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.
History and origin of the name of the Archimedean property
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.
The concept is named after the ancient Greek geometer and physicist Archimedes of Syracuse. Archimedes stated that for any two line segments, laying the shorter end-to-end only a finite number of times will always suffice to create a segment exceeding the longer of the two in length.[citation needed] If we take the shorter line segment to have length x, then any (larger) positive real number y defines a longer line segment, so we recognise Archimedes' claim as the Archimedean property of real numbers. Nonetheless, Archimedes used infinitesimals in heuristic arguments, although he denied that those were finished mathematical proofs.
Because Archimedes credited it to Eudoxus of Cnidus it is also known as the "Theorem of Eudoxus"[2] or the Eudoxus axiom.
Notes
- ^ Schechter 1997, §10.3
- ^ Knopp, Konrad (1951). Theory and Application of Infinite Series (English 2nd ed.). London and Glasgow: Blackie & Son, Ltd.. p. 7.
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