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.

But that's the thing - each generation struggles with whether some new thing is a number. We're typically very inclusive, accepting imaginary numbers and even weirder things like surreal numbers, which we definitely can't count.

But as someone in this generation, I see a good argument for rejecting the big busy beaver numbers, which are provably outside of the realm of calculating with all the resources of our universe's runtime, from being fully accepted as numbers, any more than the first uninteresting number [0].

[0] https://en.wikipedia.org/wiki/Interesting_number_paradox

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.