Well, an "actual" real number line would have a width (breadth) of zero and thus be entirely invisible.
And, if we take that the smallest actual "point" on a drawn line would be represented by the centers of the molecules or atoms of the ink/lead, then I would suspect that those locations are all at irrational distances from the end (transcendental in fact). So the number lines we know and love are actually irrational number lines with a finite number of points.:)
Obviously, number lines work, and we "see" the geometric version in our mind. Kids seem to see the geometric version pretty easily as well (it is the later formality that gives them problems). But if we drew them with dots or dashes that would probably play havoc with their senses.
I think the hardest aspect of the irrationals and rationals to portray in a tangible way is that between any two rationals there are an infinite number of rationals and an infinite number of "holes". Even though my son easily surmised (after learning fractions) that he can keep choosing a smaller and smaller number and get as close to zero as he wanted, I don't think he realized how empty that gap is. I get the distinct impression that he thinks he is filling the gap up.
If you want to see what this discussion looks like in baby talk, check out...
On Nov 25, 2013, at 1:05 PM, Joe Niederberger <firstname.lastname@example.org> wrote:
> To amplify a bit -- if we pick a real number (say in (0,1)) the odds we hit it with a randomly thrown dart is exactly zero. So it reasonable to assume that what goes for darts goes for light rays too. That randomly chosen real is "invisible". The real number line is almost, or completely, entirely invisible. Is the rational number line likewise? > > What do we really see then? > > Cheers, > Joe N