The Axiom of Archimedes usually takes the form that, given any positive real number x, there is a positive rational number less than x. In this form, the Axiom of Archimedes is involved in any demonstration that, given any real numbers x < y, there is a rational number r such that x < r < y. And this fact one depends upon whenever one uses decimals to approximate a number.
Yet the Axiom of Archimedes did not begin life as a statement about numbers; it began as a statement about magnitudes. In Aristotle’s Physics it takes the form:
“…for every finite magnitude is exhausted by means of any determinate quantity however small.”
I point out in the section that the Axiom of Archimedes does not come from a mathematical derivation, but is a basic observation about the world, specifically, about the nature of magnitudes such as length and weight.
As I delineate in Chapter 2, the central import of the Axiom of Archimedes is that magnitudes are measurable; that any positive magnitude Y can be measured by any other magnitude X of the same type: that for any finite precision requirement, there exists a rational number A, such that Y = AX. That is, within the precision context, Y is indistinguishable from AX.
Taking X and Y as concretely given magnitudes, I point out in the section that A is not unique, that for any prescribed precision standard there are multiple rational numbers A such that Y = AX. But suppose that X and Y arise in an abstract setting. Consider, for example, the relationship between the diagonal X and the edge Y of a square. In such cases, one wants a number A, not necessarily rational for which Y = AX regardless of precision context. In this example, one wants to show that A (in this case the square root of 2) is unique.
I argue in this case, using the Axiom of Archimedes, that A is unique, point out that existence of A requires more argument, and defer that argument to Chapter 4.
Finally, at the end of the section, I present the standard argument that, for any real numbers x < y, there exists a rational number r such that x < r < y.