For sqrt(2) I can tell you the order of magnitude and output as many digits as you want. I think that's plenty specific for this use case.

For zero I can not only do that, I can also count to it if you let me count both up and down, which seems like a very simple ask.

sqrt(2), and pretty much everything else you can think if, is computable- there's a program that can output rational numbers arbitrarily close.

BB(n) is not.