Would a drawing of the rationals on a number line create the appearance of a line, nothing, or both?

Question

On a number line, the rationals are dense in R, meaning there are infinite rationals between rationals.

However, there are infinite undefined irrationals with a Lebesgue measure of 1.

So how do we describe the rationals on a number line? Would we get a "line", nothing, or both? Do points dense in R always create a line? Is there a paradox related to this?

Answer 1

You definitely do not get a line due to infinitely many missing points.

You also do not get nothing because you have infinitely many points in your set. what you get is a dense subset of the real line.

The intersection of every interval of real line with your set is infinite and it is dense in that interval.

You may call it a dust if you wish.