Intuitive explanation for how could there be "more" irrational numbers than rational? I've been told that the rational numbers from zero to one form a countable infinity, while the irrational ones form an uncountable infinity, which is in some sense "larger". But how could that be? There is always a rational between two irrationals, and always an irrational between two rationals, so it seems like it should be split pretty evenly. 

I think the key is what Henning mentioned, that is, 

There are more irrationals between any two different rationals than there are rationals between any two irrationals.

However, to gain some more intuition imagine binary expansion of a rational and irrational numbers, for example
\begin{align}
r_1 &= 0.00110100010\overline{0010101010} \\
i_1 &= 0.001101000100010101010 ??? \\
r_2 &= 0.00110100010\overline{0011111010} \\
i_2 &= 0.001101000100110101010 ???
\end{align}
The expansion of rationals are periodic, so every rational can be specified
with finite amount of information (bits). You can make it as long as you want, but it still will be finite. However, even if some irrationals can be specified using finite amount of information (e.g. $0.101001000100001000001\ldots$), there are infinitely many more of 
those that carry infinite amount of information (there are infinitely many more of those even between any two rationals). We cannot specify them$^\dagger$ (how would we?), but we know they exists and exactly those unspecifiable numbers make that there are more of irrationals than rationals. 
I hope this helps $\ddot\smile$

$^\dagger$ Here I'm assuming that we are using a language with finite or even countable number of symbols. On the other hand, for example, if we were to treat physical measures like "length of this stick" or "velocity of that bird" as specifications, and given that our world is continuous instead of discrete, then, almost surely, any such definition would be one of irrational number.
