e is 2.718281828459... Maybe Gauss put 1828 into this decimal expansion (twice in succession for emphasis!) to celebrate the year in which his differential geometry masterpiece “Disquisitiones generales circa superficies curva” was published. Jim Propp On Thursday, December 6, 2018, Adam P. Goucher <apgoucher@gmx.com> wrote:
It's not about unpredictability, it's about the stronger notion of incompressibility. You can show that Chaitin's constant is algorithmically incompressible*, which implies normal.
* proof: if not, you could create a program of N bits which knows how many programs (other than itself) of length <= N bits terminate, and then (by dovetailing) algorithmically determine *which* programs terminate. Then, it can decide to terminate if and only if it doesn't terminate.
Best wishes,
Adam P. Goucher
Sent: Thursday, December 06, 2018 at 8:33 PM From: "Andy Latto" <andy.latto@pobox.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Messages in pi
On Thu, Dec 6, 2018 at 10:02 AM Mike Stay <metaweta@gmail.com> wrote:
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.
I don't follow this. If I tell you that X has no 3s in it's base-10 expansion, which means it is not normal, how does this let you predict infinitely many digits (or even one digit!) of X?
Andy Latto
-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com
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