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

Erik Max Francis max at alcyone.com
Wed Mar 19 02:14:50 CET 2003

"Greg Ewing (using news.cis.dfn.de)" wrote:

> Just out of curiosity, has any other irrational
> number (other than ones specially constructed to
> be so) been proved normal?

Nope, the only numbers known to be absolutely normal are the
artificially constructed ones like Champernowne or the Copeland-Erdos
constant (which is constructed similarly to the Champernowne constant,
except by concatenating primes instead of the positive numbers).  There
are non-artificial classes of numbers which are known to be b-normal
(where b is some number), e.g., normal in one particular base, but
absolute normality implies that it is b-normal for all b (e.g., absolute
normality means normality in all bases).  Some qualifications on
b-normality are known, like if a number is b^k-normal, then it's
b^m-normal also (for k, m integers); if x is b-normal, then so is q x +
r (q and r rational with q != 0), etc.  But general tests for normality
are an unsolved problem in mathematics.

> And what are the grounds for suspecting pi to
> be normal? Is it just "we don't have any particular
> reason to think it isn't", or is there more to
> it?

It's suspected that "most" irrational numbers are normal (in a
measure-theoretic sense, of course, since there are an uncountably
infinite number of irrationals).  Furthermore, actual tests of the
distribution of pi to however many billions of digits they're up to now
show no statistical deviation from normality.  As I pointed out earlier,
that isn't a proof, of course.

For more information, check out MathWorld:


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