Whether or not a decimal expansion of a number looks random is not the
same question as whether or not it is transcendental. The Liouville number
.10100100000010...10... is transcendental where the number of zeros
between successive ones is 1!, then 2!, then 3!, etc., but it certainly
doesn't look random. On the other hand, the decimal expansions of such
irrational algebraic (i. e. nontranscendental) numbers as square roots of
non-square integers look as random as pi or e.
Gordon Brown
Department of Mathematics
University of Colorado
Boulder, CO 80309-0395
On Sun, 29 Jul 2001, D. F. Siemens, Jr. wrote:
> Burgy,
> Of course pi is not random at all. It is absolutely determined. But the
> sequence of integers to any base passes all known tests for randomness.
> Indeed, if a repeating decimal or other evident structure should appear,
> the number would not be transcendental. The conclusion is that all
> possible sequences of integers will be found if the computation is
> carried far enough. From something that George noted, this must be true
> of all transcendental numbers. But, so far as I know, only the decimal
> value of pi has been calculated to ridiculous lengths. But someone may
> have fooled around with e, the next best known, and I probably would not
> have heard.
>
> When I first encountered references to the nature of pi, a statement
> which must be either true or false, but which was not empirically known
> at the time, was 'There is a sequence of 3 3's in the decimal value of
> pi.' This was upped to 5 5's and 7 7's that I heard about. But I have no
> published evidence where the situation currently stands, though one
> should find 9 9's somewhere within about a billion integers. Finding the
> binary code for even a short program will likely take many more places. I
> know I'm not going to look for Windows' code in pi.
> Dave
>
> On Sun, 29 Jul 2001 19:05:20 -0600 John W Burgeson <burgytwo@juno.com>
> writes:
> > BTW, I think that pi is not really random. It just looks that way.
> > So
> > perhaps Microsoft's OS is really not in there after all, much less a
> > description of you or me.
> >
> > John Burgeson (Burgy)
> >
> > www.burgy.50megs.com
> > (science/theology, quantum mechanics, baseball, ethics,
> > humor, cars, God's intervention into natural causation,
> > etc.)
>
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