value of pi and 22/7

[Rotwang]

On 17/03/2011 18:49, Ian Kelly wrote:
> On Thu, Mar 17, 2011 at 11:36 AM, Jeffrey Gaynor<jgaynor at ncsa.uiuc.edu>  wrote:
>> There are fun math questions, for instance, is there a run of a million 1's someplace in the decimal expansion of pi? Maybe so, but we just don't know, since we've only computed the first trillion or so digits.
>
> Since pi is irrational I would be surprised if there isn't one
> eventually.  Out of my own curiosity, do you know what the longest
> known string of repeating digits in pi is?

Note that Liouville's constant, the number sum_{j = 1}^\infty 10^{-j!} 
is easily seen to be irrational (and is also transcendental), but no 
string of a million 1's, or any digit other than 0 and 1, appears in its 
decimal expansion. The relevant concept is that of a normal number, 
which is one whose digits "look random":

http://en.wikipedia.org/wiki/Normal_number

Pi has not been proved to be normal but it is suspected on purely 
statistical grounds that it is, since almost all real numbers are normal 
(in the sense that the set of non-normal real numbers has Lebesgue 
measure 0).