TITLE: There's a real between any two rationals, a rational between any two reals, but more reals than rationals?
QUESTION [6 upvotes]: The following statements are all true:

Between any two rational numbers, there is a real number (for example, their average).
Between any two real numbers, there is a rational number (see this proof of that fact, for example).
There are strictly more real numbers than rational numbers.

While I accept each of these as true, the third statement seems troubling in light of the first two.  It seems like there should be some way to find a bijection between reals and rationals given the first two properties.
I understand that in-between each pair of rationals there are infinitely many reals (in fact, I think there's $2^{\aleph_0}$ of them), but given that this is true it seems like there should also be in turn a large number of rationals between all of those reals.
Is there a good conceptual or mathematical justification for why the third statement is tue given that the first two are as well?
Thanks!  This has been bothering me for quite some time.

REPLY [6 votes]: Maybe this helps: if two rationals agree to (say) 73 decimal places, then you can find reals between them by making your reals have the same first 73 decimals, then be careful with the 74th place, but then do anything you want with all the places after that (this isn't exactly right, if the 74th places differ by 1, but that's easy enough to fix). 
Now if you have two reals that agree to 73 places, you can find rationals between them by making your rationals have the same first 73 decimals, then be careful with the 74th, but then you have to make sure the rest of it is eventually periodic (or terminates). That restriction on the rationals, which wasn't there for the reals, may help you see why there are more reals (even between two rationals that agree to 73 places) than rationals.