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. 

See Cantor's diagonal argument. There are infinitely more reals between each "lattice point" of a rational. This is the basis for having multiple levels of infinity.
