I understand that there is an irrational number between any (and every) two rationals. And I can see that there is a rational number between any (and every) two algebraic irrational numbers. But I'm having a hard time convincing myself that there must be a rational between any (and every) two transcendental numbers. Can this be proven, and is the proof accessible? It seems to me that if every two transcendental numbers are separated by a rational, then there has to be as many of one as of the other. But the rationals and algebraics are countable, and the transcendentals are uncountable. This is where I run into trouble. Now, I can see that for any two specifically named transcendentals there must be a rational between them. But since the transcendentals are uncountable, there must be transcendentals that cannot be so easily named (except perhaps to say that one must be very, very near the other). I've no idea how they prove specific numbers are transcendental, but I gather it's hard. Is it impossible in the general case? And I know I was playing fast and loose with the description "very, very close" but the specification of a distance [tex]\delta[/tex] makes the second point separated by a nameable distance from the first, and I'm thinking of something much, much closer. Is there a way to express this more formally? Thanks in advance for any (and every) responses.