Dr. Dobb's Python-URL! - weekly Python news and links (Mar 17)

[Chermside, Michael]

Robin Becker writes:
> I assume absolute normality excludes the case where one expresses the
> number in itself as a base or am I being more than usually stupid.

As far as I know, we're only talking about integer bases ( >= 2),
and non-integer numbers (the integer ones are CLEARLY not normal
since "x.000000..." has a lot more 0's than 1's, 2's, or 3's [1]),
so it doesn't come up.

> Also I suppose that being non-random implies finiteness (in some sense)
> so are we just talking 'symbol' count or information. After all there
> are very small symbolic representations of pi, but are they smaller in
> information content etc etc.

"Finite" in what sense? We're probably talking about numbers which
are finite in the sense that there's some integer larger than the
number (ie, no infinite quantities), but not finite in the sense
that they've got a finite number of digits. Since all we're really
interested in is the decimal expansion, we might as well presume the
number in question is in the range [0..1]. But that decimal goes on
infinitely, so you can pack lots of information into it (an "infinite"
amount if you like).

I don't think that the choice of symbols makes ANY difference in the
information content. I would say that "The ratio between a circle's
circumference and diameter", and "Pi", and the (infinite) string that 
I would get if I finished writing this out: "3.141592653589793..."
all have the same information content, despite being 55, 2, and an
infinite number of characters long (respectively).

But-one-sure-uses-a-lot-more-paper

-- Michael Chermside

[1] A really interesting argument can be made that it's got just as
  many 9's as 0's, since "x.00000..." = "(x-1).99999..." by most useful
  definitions. Does this mean that integers are normal in base 2?