Re: Representations of Pi, etc.

[tcmay@got.net (Timothy C. May)]

At 3:19 AM 1/5/96, jim bell wrote:

>But BTW, isn't it interesting, that news item from a few weeks ago, on an
>algorithm for determining individual bits in Pi, regardless of whether
>you've calculated all the previous ones.  Only problem is, it only works in
>hexadecimal (and, obviously, binary, etc, not decimal.
                                           ^^^^^^^^^^^

???

I didn't see this result you mention, but it surprises me. The part about
how it works in some bases, but not in decimal.

The "hand-waving" (motivational/informal) explanation for why I am
surprised is that "Nature doesn't care about bipeds with 10 digits vs.
bipeds, or whatever, with 2 digits or 16 digits." That is, results
applicable in base 16, hexadecimal, should be easily applicable in base 10.

And there is are interesting properties about the distribution of digits in
"random" numbers. Pi is of course not random by many definitions, but
shares certain important properties with random numbers. (Or sequences, if
you wish.) One of these is properties is that of _regularity_, the
frequency of digits. A regular number is one whose expansion has in the
limit the same frequency for all digits, and this is so in any base. Thus,
a regular number has an equal frequency (in the limit, blah blah) of 0s,
1s, 2s, 3s, etc. And switching to another base will not change this.

I recollect that pi has been proved to me regular, i.e., that pi has an
equal frequency of all digits, in the limit, in all bases.

(This is the sense in which we can argue that pi is "random." in the sense
that there are no correlations, no dependence of the n+1th digit on the nth
digit, and "no apparent order." Furthermore, there is no effective
compression of pi, except by some tricks, such as _naming_ it (a dictionary
compression, of sorts) or by specifying a program which computes it. Lots
of interesting issues about the real meaning of randomness and
compressability, about the "logical depth" of certain computations, etc. I
recommend "The Universal Turing Machine" (ed. by Haken, as I recall) for a
nice set of articles on these fascinating issues.)

In summary, I would be surprised to find that a method for calculating the
Nth digit of pi works for base N but not for base M (modulo some minor
efficiency factors related to machine architecture, etc.).

Any pointers to this result would be appreciated.

--Tim May

(By the way, randomness and regularity, real or only apparent, are some of
my favorite topics. Numbers which _appear_ to be regular, but which
actually aren't, are said to be "cryptoregular" (hidden regular). The
connection with cryptography is more than tangential: a text block or
number which _appears_ to be random or regular (the same frequency
definition applies to letters as well as digits) may be transformed by
application of a key to a nonrandom or nonregular thing. The connection
with entropy and randomness is right there, of course, and is left for the
interested folks to think about.)


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