XPost: sci.math.num-analysis   
   From: david.brown@hesbynett.no   
      
   On 24/02/2026 07:08, Axel Reichert wrote:   
   > David Brown  writes:   
   >   
   >> Of course, confirming that the hexadecimal digits of pi are random   
   >> enough to pass such a test does not ensure that the decimal digits   
   >> would do so too.   
   >   
   > I was puzzled by the "Of course": To me, this is not intuitively clear.   
   > Is there any easy (not too technical) way to "see this"/make it   
   > plausible? My gut feeling (wrongly?) said that the base should not   
   > affect the randomness of a numerical pattern. "Of course" I am aware   
   > (and taught to dozens of numerical beginners) that, say, 0.5 in base 10   
   > has a non terminating representation in base 2, but "random" is neither   
   > representation.   
   >   
   > Pointers or simple counter-examples highly welcome!   
   >   
   > Axel   
      
   Numbers can be normal in some bases and not in others.  This is easy to   
   see if we pick related bases, such as base 2 and base 16.  For example,   
   let x be 1/3.  Then x is 0.0101010101... in base 2.  That is clearly   
   normal in base 2.  But in base 16, it is 0.55555555..., which is clearly   
   very far from normal.   
      
   I think (but I am not sure) that if a number is normal in base B, then   
   it will be normal in any other bases co-prime to B.  If the bases are   
   not co-prime, then things are not as clear (as shown by my simple example).   
      
   Almost all (in the technical mathematical sense) real numbers are normal   
   in all bases.  And lots of numbers (including pi) are believed to be   
   normal, and have been checked to various lengths in many bases.  But it   
   is extremely difficult to prove that any given number is normal, unless   
   it can be seen from its construction.   
      
   Being normal in a base is not sufficient to have the digits form a good   
   pseudo-random sequence, but it is a necessary condition for a uniform   
   distribution random sequence.   
      
   (I know you set the follow-ups to exclude comp.lang.c, since it is   
   off-topic in that group, but I added it again as I don't follow   
   sci.math.num-analysis, so I would not see any replies.  People have   
   different opinions on the pros and cons of limiting follow-up groups.   
   My opinion is that it is better to leave all groups there as long as   
   there are people from different groups in the discussion, even if it is   
   moving off-topic for a group, because limiting follow-up groups can lead   
   to fragmentation.  It is better for comp.lang.c to have one off-topic   
   thread than to have multiple threads that are part of the same   
   discussion but appear separately as groups are added and removed.  If   
   the discussion goes on for long, and becomes dominated by s.m.n regulars   
   and of no interest to c.l.c regulars, then it becomes time to move it   
   over to just that one group.  Others can have different opinions on such   
   matters.)   
      
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