Importance of Archimedean Property and Density of Rationals

[Mr Davis 97]

With regard to the real number system, what is the importance of the Archimedean property and the property that the rationals are dense in $\mathbb{R}$ (which is a consequence of the Archimedean property)?

Related to this, what is the most general structure for which the Archimedean property holds? Is it an ordered field? If so, then why do analysis courses go through the procedure of proving it using the least upper bound property, when it can be proved in a more general way, without reference to the peculiarities of the real number system?

 

[fresh_42]

What do you mean by Archimedean property? I assume you mean the Archimedean order of the reals.

With regard to the real number system, what is the importance of the Archimedean property

It is an inherited property of the natural numbers, so the importance is, that the natural numbers are still part of the reals and that we can do geometry. This is where the name comes from: Given an origin $P$ and a unit length $(PQ)$, then we can mark off this length as often as we reach above a given point $R$. You don't have this in finite fields.

and the property that the rationals are dense in $\mathbb{R}$ (which is a consequence of the Archimedean property)?

No. These are two different things. As said, the natural numbers, the integers and the rationals are also Archimedean ordered.

Related to this, what is the most general structure for which the Archimedean property holds?

For fields, all subfields of $\mathbb{R}$. In general, I'd say the additive half group $\mathbb{N}$.

Is it an ordered field? If so, then why do analysis courses go through the procedure of proving it using the least upper bound property, when it can be proved in a more general way, without reference to the peculiarities of the real number system?

This made me think that you might have meant something different, as the ordering and completeness are different properties. E.g. the complex numbers are complete, too, but don't have an Archimedean order.

 

[WWGD]

With regard to the real number system, what is the importance of the Archimedean property and the property that the rationals are dense in $\mathbb{R}$ (which is a consequence of the Archimedean property)?

Related to this, what is the most general structure for which the Archimedean property holds? Is it an ordered field? If so, then why do analysis courses go through the procedure of proving it using the least upper bound property, when it can be proved in a more general way, without reference to the peculiarities of the real number system?

The Reals can be given a model in which the Archimedean property does not hold -- the Nonstandard Reals -- but then you lose a lot of nice properties; the Standard Reals embed in it.

 

[mathwonk]

The first place I know of where the Archimedean property was used is in Euclid, Book V. He wanted to discuss similarity, or equality of ratios, for segments whose lengths were not commensurable. In order to do this he introduced the axiom now called Archimedean, and gave a limiting definition of equality of ratios for not necessarily commensurable segments. Namely two pairs of segments are in the same ratio, if and only if for every ratio of integers, their ratios are either both ≥ or both ≤ that ratio of integers. (I.e. a pair of segments are in a ratio greater than say 3/4 if when we divide the longer one into 4 equal parts, then 3 of those parts is less than the shorter one.)

I would say your observation that the axiom implies density of rationals is another manifestation of this usefulness of the axiom. I.e. just as Euclid did, we often make approximations with rationals in dealing with real numbers, and it is highly convenient. So perhaps the answer to your question is this: this axiom allows us to approximate real numbers arbitrarily well by rationals, and this is highly useful, e.g. in the first use of it by Euclid to define similarity of triangles whose sides are not in a commensurable ratio.

(There are other solutions to the problem of similarity that do not use density of rationals, e.g. one can define similarity of triangles by requiring that when placed vertex to vertex appropriately, they be inscribable in a triangle, i.e. that they be describable by a pair of intersecting lines in a circle. This is based on Euclid's Prop. III.35, but he seems not to have noticed it could be used as a substitute definition for similarity. This is interesting to me since both this proof and that of the basic similarity property for equiangular triangles, Prop. 4 in Book VI are based on area in Euclid.)

In analysis it is convenient to work with complete ordered fields. If completeness is defined by means of the least upper bound property, then all such fields are archimedean and isomorphic to the standard reals. If completeness is defined instead by the convergence of cauchy sequences, such fields can be non archimedean, and these also yield useful and interesting examples both in analysis and in geometry.

There is nothing mysterious about non archimedean orderings. E.g. we are used to polynomials which can be ordered in such a way that polynomials of higher degree are larger than those of lower degree. (Let a polynomial with real coefficients be called positive if its leading coefficient is positive. ) In particular no integer, nor any other constant, can be as great as the polynomial X. Thus 1/X is smaller than every fraction of form 1/n for a positive integer n.