Number r is:

1.Rational, if the function

is periodic

2.Irrational, when HS_r[x] is not a periodic function.]]>

And there are various proofs for irrational numbers.

But there are also "Open Questions", see Wikipedia Article (near end), so the best you could do (using current knowledge!) would be:

disc(x) = 1 when rational

disc(x) = -1 when irrational

disc(x) = 0 when unknown

Now, a computer program could run through thousands of decimal places looking for a repeating pattern and never discover it, but that would not prove anything - it could still be the ratio of two very large numbers!

]]>Please give me some advice.

I think there will exist such.]]>