Are there any examples of non-computable real numbers?

Is this true, that if we can describe any (real) number somehow, then it is computable?

For example, π is computable although it is irrational, i.e. endless decimal fraction. It was just a luck, that there are some simple periodic formulas to calcualte π. If it wasn’t than we were unable to calculate π ans it was non-computable.

If so, that we can’t provide any examples of non-computable numbers? Is that right?

The only thing that we can say is that these numbers are exist in many, but we can’t point to any of them. Right?

Answer

I haven’t thought this through, but it seems to me that if you let BB be the Busy Beaver function, then i=12BB(i)=21+26+221+2107+ ... 0.515625476837158203125000000000006 should be a noncomputable real number, since if you were able to compute it with sufficient precision you would be able to solve the halting problem.

Attribution
Source : Link , Question Author : Dims , Answer Author : r.e.s.

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