Re: [MLUG - DISCUSSION] fun math thing

[Stephen Montgomery-Smith <EMAIL:PROTECTED>]

Mike Miller wrote:
> On Tue, 30 May 2006, Mike Miller wrote:
>
> > > Do you mean irrational above instead of transcendental?
> >
> >
> > Yes!  I'm really losing my memory of high school math.
>
>
>
> When I was an HS senior I came up with this number:
>
> 0.12345678910111213141516...9899100101102103...99899910001001...
>
> It's just all the positive integers concatenated in order.  I guessed 
> that this was a transcendental number because it was irrational (it 
> never repeats) and I couldn't believe it might be the solution of a 
> polynomial equation, but I don't think I had a proof of that.
>
> I had a way of writing the number as a series, I think it was a double 
> sum.  After that I couldn't do much with it.  I always thought I'd bring 
> it out of retirement someday.
>
> Now, thanks to the web, I have discovered that this number was found by 
> others before me (and I'm not surprised, but I am very pleased).  It is 
> called Champernowne's number after David G. Champernowne, an English 
> mathematician.  He beat me to it by 42 years.  It is irrational, 
> transcendental and "normal" (a new one on me -- "normal" means all 10 
> digits appear with equal frequency) but not "absolutely normal."  It was 
> proved to be transcendental by Kurt Mahler in 1937 when he was an Asst. 
> Lecturer at Manchester U.
>
> I'm going to write to my old math teacher.  He really liked my number 
> and was even a bit jealous.  He was a very young guy and is still 
> teaching, but now at the college level.

I am guessing that "absolutely normal" means all b digits appear with 
equal frequency when written in base b, and this for every base 
b=2,3,... at the same time.

If you pick a real number between 0 and 1 using the uniform probability, 
then with probability 1 the number you pick will be absolutely normal. 
(You can prove this using the law of large numbers.)

However there is no naturally occuring number, like pi or e or sqrt(2), 
that is known to be absolutely normal.  (It is conjectured that most 
naturally occuring irrational numbers are normal, but I don't think that 
anyone is even close to providing a proof.)

Stephen

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