Comment by kannanvijayan
5 days ago
I suppose you could have added root two as a fundamental as well. I suppose that's another problem with the irrationals: two irrationals that aren't linearly related by a rational are effectively two fundamentals from each others perspective.
It's a sad conclusion - though. Computation exists in the countable space. So there is no computationally representable symbolic model that can ever algebraically capture the reals.
The other thing that came to mind when you mentioned root-2 is a similar realization as with pi. That somehow a diagonal is not well defined in discrete terms with respect to two orthogonal vectors. So here once again, you have this weird impedance mismatch between orthogonality (a rotational concept) and diagonals (a linear concept).
I don't have the formalisms to explore these thoughts much further than this.. so it's hard to say whether this is just some trivial numerological-like observation or if there's something more to it. But it's kinda pleasant to think about sometimes.
Yeah, once I got to "all I need to do is add a root for every prime! And cube roots! And..." I realized this is a path of madness. ;)
It could be done symbolically, by generalizing from their rational representation:
To
Again this is tantalizingly close to being workable in Frink -- it supports 'dangling' (unevaluated) rational exponents on units, but not simple numbers.
The problem of course is that I'm trying to twist a (powerful!) calculator into something like a computer algebra system. I really should just use an actual CAS.
But like you say, I'd be happy if I could "just" have an exact representation of (if not the reals because that's impossible, then at least) any number I can describe in finite terms with normal math operators.
Cheers and good day