Dr. Dobb's Python-URL! - weekly Python news and links (Mar 17)
Erik Max Francis
max at alcyone.com
Wed Mar 19 23:23:29 CET 2003
mcherm at mcherm.com wrote:
> I agree that normality is usually considered a property of irrational
> numbers (which I think the original poster may NOT have realized) but
> if we DO extend it to the rational realm, then I had assumed that all
> rationals would be non-normal in all bases. But it turns out that it's
> not quite so easy, since some rational numbers have multiple valid
> decimal expansions.
But I think it's safe to say that when that happens, none of those valid
base-b expansions will be normal. They'll still be terminating or
repeating, which by definition means they can't be normal. It seems
like I'm just being anal, but normality is intimately tied with the
notion of irrationality. The definition of b-normality states that for
a decimal expansion in base b, every b-digit sequence of length k will
appear with the expected frequency. This can't possibly be the case for
an expansion which terminates or repeats, because even if the repeating
portion is very long, there must be some (in fact, an infinite number)
of sequences that don't appear anywhere at all in the expansion.
This can be seen by trying to construct such a number; the logical
result is to create Champernowne's constant in base b (C_b), which is
irrational. If it repeats, then some sequences _must_ be left out.
(Furthermore, there is some K such that no sequences of length k >= K
appear _anywhere_ in the expansion.)
> If you define normality by including all the
> of all the valid decimal expansions, then integers are normal in base
> 2. I doubt that's a USEFUL result, but it's certainly a curious one.
But in mathematics, when you're talking about something, even if it's
not unique, you don't consider all the different non-unique forms
simultaneously; you don't treat them as a superposition of states, you
treat each state individually.
Erik Max Francis / max at alcyone.com / http://www.alcyone.com/max/
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