Real numbers vs. the real number line

Question

I don't know how to formulate this question precisely, so let me explain where I am coming from, noting that I know little about nonEuclidean geometry.

I was thinking about how to explain how complicated the real numbers are to someone without much mathematical training, and started out by thinking of how it's natural in the classroom to say something like "The real numbers look like this ...", then draw a line on the chalkboard with some dots to indicate that it extends indefinitely in both directions. Then one could discuss how much the notion of the real number line abstracts from the thing on the chalkboard. For example, it's hard to infer completeness by zooming in on chalk that looks less and less like a line the more you magnify.

But that made me wonder why it seems to be so natural to identify the real numbers, which I guess is intrinsically an algebraic object (up to isomorphism, the unique complete ordered field) with a geometric object (which is, I suppose, a one dimensional Euclidean space, though the chalkboard might encourage someone to think of it as a one dimensional object in two dimensional space).

Wiki, for example, seems to take this identification for granted:

In mathematics, the real line, or real number line is the line whose points are the real numbers.

So my questions are along the following lines. First, why is it so natural to make this identification? To what extent is it adding additional (e.g. metric) structure, so that it should be thought of an embedding of an algebraic object in a geometric one? Second, are other identifications with geometric objects, e.g. lines in nonEuclidean geometries, in some sense less natural? For example, does one get a less desirable metric structure?

Answer 1

It's a bit problematic to justifying thinking of real numbers as a line in Euclidean space or some manifold as I feel that when imagining Euclidean space or manifolds etc, this assumption is implicitly used. I guess you can do all of linear algebra in an abstract way and then note that Euclidean space is a particularly nice example of a vector space and then see how the real line fits in there, but that just moves the question.

As for why the line is natural, it's quite reasonable to imagine the integers as an infinite set of evenly spaced points. They should be evenly spaced, I guess so that it's easy to imagine addition and multiplication: shifting points along and scaling the points. Once you have a line of integers, you immediately get a notion of distance between them. (Maybe the desirability of this property comes from actually measuring things in real life, you want the process of measuring to be invariant under where you put the origin and how big your unit length is). Thinking of proper fractions as fractions of the distance between the integers you can fill in all the rational numbers, because you have a clear notion of multiplying fractions by integers by repeated addition. The fractions also inherent the same notion of distance. So I guess that's the justification for viewing the rational numbers as a line, and this makes the field of rational numbers a metric space once you consider the metric of the absolute value of the difference.

Now the real numbers are the completion of the rational numbers with the difference norm, which is an abstract concept, but it's quite clear where any irrational number would sit on the line (just get better and better approximations) and I think this is way of coming to the conclusion that the real numbers form the real line, I suppose the real line implicitly implies the choice of the norm of the difference metric. To know that the reals can be visualised in this way, all you need to do is prove that they are ordered.

To partially answer the second question, there are other metrics that you can put on the rational numbers, see for example the p-adic numbers, and the resulting completions have nice visualizations, but are not isomorphic to the real numbers (when fixing the rationals at least).

Answer 2

This maybe could come from Euclid himself, in the Elements, on which he defines the basic geometric formes, points and lines.

He defines the line as "a cluster of infinite points" (inside the quotes because I don't remember the exact term he used). This could be problematic because the naturals, the integers and the rationals could all be thought as such, especially the rationals since there exists a sequence of rationals tending to any real number.

But the naturalness maybe comes from our previous experience and/or natural senses such as vision. Maybe this is the way that we think about numbers that are so complete that you can never say which number is the next (so does the rationals in this sense).

The most mind boggling thing for me, however, is that we keep finding new kinds of numbers inside the real numbers: such as the transcendentals. The one I get specially amazed is the non computable ones, they exist but we can't compute them or write them down, even more, they have full measure. So we just work with the negligible minority of numbers on the real line. (There is some time since I read about it, if you want I can tell you the reference)

And for the first question: probably you could show the complexity of the real numbers by deriving it with them, collaboratively. And make them do the most work. You could also tell the history of numbers to them as you go along.