In reply to Rick Wicklin.

Yes, I should have deduced that a pattern of digits can exist without repetition, and thereby still represent an irrational number. But it is irking me that numbers like pi and e might be infinite random sequences of digits, with no underlying structure or pattern...because at least intuitively that seems to imply that the natural phenomena they so beautifully represent are somehow intrinsically random as well. Silly idea, but maybe it's related to the probabilistic nature of quantum physics: If there WERE a pattern (like 0.1011011101111...) then we would KNOW the number even though we couldn't write it all out or represent it as a ratio. But if there is NO pattern or structure, then we can never really pin it down...like an intrinsic uncertainty. We can get closer and closer but never completely represent it. No..no, I'm wrong again, I'm always wrong, because pi and e are easily represented with tidy infinite sums. Aha, so infinite sums can therefore be random number generators?! I see I'm a little too existential, and not quantitative enough, about all this.