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

[Michael Hudson]

Tim Peters <tim.one at comcast.net> writes:

> [Tim]
> > Also, AFAIK, no absolutely normal number is known (constructed or
> > not).  This is curious because the set of non-normal reals has
> > measure 0 ("almost all" reals are absolutely normal).
> 
> [Michael Hudson]
> > <hand-waving>
> > But for a number to be "known", it must be comptuable in some sense
> > and there are only countable many Turing machines...
> > </hand-waving>
> 
> That's an excellent argument for why we can't show an example of an
> uncomputable real, but really no more convincing wrt normality than wrt
> irrationality.

Well, I guess.  But rationality seems a much more tractable notion
than normality -- it seems "finite" in some sense.  I'm not sure
rationality is that good an example, as having irrationals is somehow
the point of the reals...  Transcendence would be a better example,
but I'm an algebraist -- what's a transcendent number, again? <wink>.

> After all, we do have examples of numbers normal wrt specific bases,
> and there's no obvious way in which absolute normality "should be"
> intractable.

> > Why does anyone care about normality, anyway?
> 
> I think because attempts to define what random means lead naturally
> to normality.  If, e.g., pi is shown *not* to be normal, then it
> would be very hard to believe it's random in any reasonable sense.
> If it is shown to be absolutely normal, I expect reasonable people
> would disagree about whether that's strong enough to conclude it's
> random, though.

Except that pi clearly *isn't* random... but this leads very quickly
into the "the smallest integer than can't be described in less than
100 characters" type paradox, and I don't care about them, either :-)

Cheers,
M.

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