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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 ordered or normed 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).

The notion arose from the theory of magnitudes of Ancient Greece; it still plays an important role in modern mathematics such as David Hilbert's axioms for geometry, the theory of ordered groups, the one of ordered fields and the one of local fields.

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 ways of formulation. For example, in the context of the ordered fields, 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.

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Definition for linearly ordered groups

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:

 \underbrace{x+\cdots+x}_{n\text{ terms}} < y. \,

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.

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Ordered fields

An ordered field has some additional nice properties.

  • One may assume that the rational numbers are contained in the field.
  • 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.

In this setting, an ordered field K is Archimedean precisely when the following statement, called the axiom of Archimedes, holds:

Let x be any element of K. Then there exists a natural number n such that n > x.

Alternatively one can use the following characterization:

For any positive ε in K, there exists a natural number n, such that 1/n < ε.

Definition for normed fields

The qualifier "Archimedean" is also formulated in the theory of local fields as follows. Let F be a field endowed with an absolute value function, i.e. we have a way to associate a positive real number |x| for each x in F satisfying suitable compatibility relations. Then, F is said to be Archimedean if for any non-zero x in F there exists a natural number n such that

 |\underbrace{x+\cdots+x}_{n\text{ terms}}| > 1. \,

Examples and non-examples

Archimedean property of the real numbers

The field of the real numbers is Archimedean both as an ordered field and as a normed field. It is chiefly because the real numbers are obtained as the completion of the rational numbers, which themselves satisfy the axiom in both senses, with respect to an absolute value structure compatible with the ordering.

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.

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.

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. 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.

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.

Non-Archimedean local 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.

Equivalent definitions of Archimedean field

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. 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. 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.)

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 concept is named after the ancient Greek geometer and physicist Archimedes of Syracuse.

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"[2] or the Eudoxus axiom.

Archimedes used infinitesimals in heuristic arguments, although he denied that those were finished mathematical proofs.

Notes

  1. ^ Schechter 1997, §10.3
  2. ^ Knopp, Konrad (1951). Theory and Application of Infinite Series (English 2nd ed.). London and Glasgow: Blackie & Son, Ltd.. p. 7. 

References


The axiom of Archimedes can be stated in modern notation as follows:

Let x be any real number. Then there exists a natural number n such that n > x.

In field theory this statement is called the Axiom of Archimedes. The same name is also applied to similar statements about other fields or other systems of magnitudes; chiefly as one of David Hilbert's axioms for geometry.

In modern real analysis, it is not an axiom. It is rather a consequence of the completeness of the real numbers. For this reason it is often referred to as the Archimedean property of the reals instead.

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Formal statement and proof

Formally, the Archimedean property can be stated as follows:

Let c, ε be in R, the real numbers. Then the following two properties hold:

(i) For any positive c, there exists a natural number n such that n > c.

(ii) For any positive ε, there exists a natural number n, such that 1/n < ε.

The proof follows from the completeness of the real numbers:

First, observe that (i) and (ii) are equivalent since, if = 1, then (i) directly follows from (ii), and (ii) from (i).

We will prove (i) (thus also proving (ii)), using a proof by contradiction. Suppose there is a positive number c such that there is no natural number n greater than c. Clearly, nc for every n. This implies that the set of natural numbers, N, is bounded above by c. Thus, the completeness axiom of R asserts that N has a least upper bound, which we will call b.

Since b is the least upper bound of N, b − 1/2 is not an upper bound of N. Hence, we can choose an n in N such that n > b − 1/2. This implies that n + 1 > b − 1/2 + 1 > b. Therefore, n + 1 is a natural number that is larger than b. This contradicts b being an upper bound of N. This is a contradiction, which implies there is no upper bound for N.

Interpretation

In simple terms, the Archimedean Property can be thought of as either of the following two statements:

(1) Given any number, you can always pick an integer that is larger than the original number.

(2) Given any positive number, you can always pick an integer whose reciprocal is less than the original number.

These two statements correspond to (i) and (ii), respectively. To a mathematician, (1) and (2) indicate that the Archimedean property is capturing an important intuitive property of the real numbers.

Example usage

One of the most important uses of the Archimedean property in analysis is proving the important result:

\frac{1}{n} \rightarrow 0 as n \rightarrow \infty [1]

This is the statement that the sequence of unit fractions {1 / n} converges to 0.

Proof:

Let ε > 0. We need to find an N such that 1 / n < ε for all nN.

Using the Archimedean property, we can choose N such that 1 / N < ε. Thus 1 / n ≤ 1 / N < ε for all nN.

This statement is vital in establishing many of the properties of sequences of real numbers.

History

The first known statement of what is now called Archimedes Axiom is found in the writing of Eudoxus of Cnidus. The term itself was first used by the Austrian mathematician Otto Stolz in 1883.[2] An equivalent statement was also used by David Hilbert as one of his axioms of modern Euclidean geometry. See Hilbert's Axioms.

See also

References

  1. ^  Fitzpatrick, Patrick M. (2006) Advanced Calculus (2nd ed.). Belmont, CA: Thompson Brooks/Cole. ISBN 0-534-37603-7.
  2. ^  Eric W. Weisstein. "Archimedes' Axiom." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ArchimedesAxiom.html, Retrieved February 28, 2006.

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