From: ags@seaman.cc.purdue.edu (Dave Seaman)
Subject: Re: Completely Normal Numbers
Date: 14 Oct 1999 00:18:39 -0500
Newsgroups: sci.math
Keywords: Copland-Erdos constant
In article <3804FC8C.7D5A@kodak.com>, Jonathan Hoyle wrote:
>>> 0.12345678910111213141516... is completely normal.
>
>I don't think so, since if we convert to it hex, we get it being
>something less recognizable: 0.1F9ADD3... which doesn't appear all that
>"normal".
A better choice would be the number
0.23571113171923...
obtained by concatenating the primes. Copeland and Erdos proved in 1945
that this number is normal (in all bases), and it is therefore called the
Copland-Erdos constant.
--
Dave Seaman dseaman@purdue.edu
Pennsylvania Supreme Court Denies Fair Trial for Mumia Abu-Jamal
==============================================================================
From: Fred Galvin
Subject: Re: Completely Normal Numbers
Date: Thu, 14 Oct 1999 01:15:43 -0500
Newsgroups: sci.math
On 14 Oct 1999, Dave Seaman wrote:
> A better choice would be the number
>
> 0.23571113171923...
>
> obtained by concatenating the primes. Copeland and Erdos proved in 1945
> that this number is normal (in all bases), and it is therefore called the
> Copland-Erdos constant.
Which is it, Copeland or Copland? I'm not sure I understand what you're
saying. Let C(b) be the number you get by concatenating the primes in base
b. Are you saying that the number C(10) is the Cop(e)land/Erdos constant,
and has the property that, if you convert this number to any other base,
it will be normal in the new base? Is this a special property of the base
10, or does it work for other bases as well? Is C(2), say, normal in base
10? Is there an infinite sequence of Cop(e)land/Erdos constants, one for
each base?
I wonder why Klee and Wagon weren't aware of this 1945 result when they
wrote their (1991) book _Old and New Unsolved Problems in Plane Geometry
and Number Theory_; on p. 251 they say, "an explicit example of a number
that is normal in all bases is still lacking."
I hate to say this, but I'm starting to suspect that you may have
misstated the result. Maybe they proved that, for each base b, the number
C(b), obtained by concatenating the primes in base b, is normal in base b?
But then it wouldn't be a constant, as it would depend on the base.
All right, you've got me confused (not hard to do); please clear this up.
==============================================================================
From: Fred Galvin
Subject: Re: Completely Normal Numbers
Date: Thu, 14 Oct 1999 11:11:31 -0500
Newsgroups: sci.math
On 14 Oct 1999, Dave Seaman wrote:
> In article ,
> Fred Galvin wrote:
> >On 14 Oct 1999, Dave Seaman wrote:
> >
> >> A better choice would be the number
> >>
> >> 0.23571113171923...
> >>
> >> obtained by concatenating the primes. Copeland and Erdos proved in 1945
> >> that this number is normal (in all bases), and it is therefore called the
> >> Copland-Erdos constant.
>
> >Which is it, Copeland or Copland?
>
> It's Copeland. I got the information from the CRC Encyclopedia of
> Mathematics. The constant I gave originally is called Champernowne's
> constant, which is stated to be normal in base 10. The Copeland-Erdos
> constant is stated to be normal.
>
> My copy of the encyclopedia is at home and temporarily out of reach,
> but an on-line version is available at Eric's Treasure Trove of
> Mathematics . Chapters
> "C" (Champernowne's constant and Copeland-Erdos constant) and "N"
> (normal number) are available today, Oct. 14.
>
> >I'm not sure I understand what you're
> >saying. Let C(b) be the number you get by concatenating the primes in base
> >b. Are you saying that the number C(10) is the Cop(e)land/Erdos constant,
> >and has the property that, if you convert this number to any other base,
> >it will be normal in the new base? Is this a special property of the base
> >10, or does it work for other bases as well? Is C(2), say, normal in base
> >10? Is there an infinite sequence of Cop(e)land/Erdos constants, one for
> >each base?
>
> I believe that is what the brief encyclopedia entry is saying. I did not
> see any discussion of repeating the construction in other bases.
>
> >I wonder why Klee and Wagon weren't aware of this 1945 result when they
> >wrote their (1991) book _Old and New Unsolved Problems in Plane Geometry
> >and Number Theory_; on p. 251 they say, "an explicit example of a number
> >that is normal in all bases is still lacking."
>
> >I hate to say this, but I'm starting to suspect that you may have
> >misstated the result. Maybe they proved that, for each base b, the number
> >C(b), obtained by concatenating the primes in base b, is normal in base b?
> >But then it wouldn't be a constant, as it would depend on the base.
>
> >All right, you've got me confused (not hard to do); please clear this up.
>
> Perhaps you should direct your questions to Eric Weisstein.
Hardy & Wright quote the Erdos-Copeland result (4th edition p. 128), and
it's as I suspected: for each base, a number proven normal in that base
alone. They give the original reference as Bull. Amer. Math. Soc. 52
(1946), 857-860.