R Hansen says: >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).
Really? You think physical distances are truly all irrational? And then what distance could ever be demarcated as 1?
R Hansen says: >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".
What I see with my mind's eye is not holes but rather that the minds concept of mathematical "line" and "point" and the relationship between them is just not similar to pebbles laid in a row. And yet a lot of common talk in mathematics is always leaning on that conception. As if the line is made of of a bunch of points (or points and holes).
In my mind at least line and point are distinct species with a curious but easily visualized game that can be played - I can always zoom in to the space between any two points on a line, and find a line segment between them. That line segment is similar in certain respects to any other segment, and so I can play the game over and over. The only thing "tangible" is this notion of a never ending process.
Its also quite "tangible" (in the imaginary sense of "tangible! What??) to view incommensurable lengths in a similar way: imagine (or image ;-) multiple copies of two segments of two incommensurable lengths being laid end to end -- and the frustration that they never, ever line up on the cracks.