On Thu, Dec 6, 2018 at 6:48 AM Bill Gosper <billgosper@gmail.com> wrote:
On 2018-12-05 16:44, Keith F. Lynch wrote:
The state of math is quite scandalous. For most numbers, we have no clue how to determine whether they're rational, algebraic, or transcendental. And, last I heard, nobody was able to prove *any* specific number to be normal,
In[108]:= N[ChampernowneNumber[], 42]
Out[108]= 0.123456789101112131415161718192021222324253
Mathematical constants treated as numeric by NumericQ and as constants by D.
ChampernowneNumber[b] is a *normal* transcendental real number whose base-b representation is obtained by concatenating base-b digits of consecutive integers.
It's normal to base 10. I think the claim is that nobody knows a specific number to be normal to every base. That said, the claim has to be restricted to computable numbers, since an algorithmically random real like the halting probability of a prefix-free universal Turing machine has to be normal to every base; if not, you could predict infinitely many (not necessarily contiguous) digits of it, which contradicts the definition of algorithmically random. -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com