TITLE: Visual representation of the fact that there are more irrational than rational numbers.
QUESTION [12 upvotes]: Would anybody know of a visual or even (preferably) geometric representation of this?  
To make it more specific: Text, symbols and written numbers are predominantly used as labels, and and less to represent the (ir)rationals themselves, or relations between the two groups. Like, if you need an image or a picture for your answer, that's probably it.
For example:

It does not have to be a rigorous proof, just a visual approach—if possible.
At the same time, a (somewhat elementary) explanation of why it does not make sense to try to visualize this relation (geometrically) is just as welcome.

REPLY [9 votes]: If we make a rectangular grid with integer coordinates, it's possible to assign a unique angle to any rational number, using the definition $\tan \phi=y/x$ for $\phi \in (-\pi/2, \pi/2)$. 
For positive rationals it would look something like this:


It's intuitively clear to me that the lines corresponding to the rational numbers can't fill all the space here (the gaps are especially noticeable around the numbers with small denominators / numerators).

This can be considered an illustration to marty cohen's answer.