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I've seen proofs that sqrt(2) and sqrt(3) can't possibly be expressed as a ratio of two integers. I imagine that a proof for e is out there somewhere.
Sqrt(2) exists in about the same way as unicorns, though it is much more useful for doing calculations.
I think the problem with what you're suggesting is that it just isn't possible to nail a irrational number down to one specific sequence of integers. They just don't settle down like that.
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