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

mcherm at mcherm.com mcherm at mcherm.com
Wed Mar 19 22:47:22 CET 2003

Michael Chermside writes:
> [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?

Erik Max Francis writes:
> No, because either it's got an infinite number of zeroes or an infinite
> number of ones.  
> Neither expansion is normal (they contain exactly none
> of the other digit)
> and you can't include both simultaneously. 
Why not?
> Furthermore, normality or non-normality can only be properties of normal
> numbers, and both of these expansions terminate or repeat, meaning that
> they are rational, not irrational.  So therefore the normal/non-normal
> qualification can't apply at all.

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. If you define normality by including all the digits
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.

-- Michael Chermside

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