From: ksbrown@seanet.com (Kevin Brown)
Newsgroups: sci.math
Subject: Re: Can e & pi be easily related?
Date: Mon, 22 Jun 1998 04:15:26 GMT

On 21 Jun 1998 jarweinst@aol.com (JarWeinst) wrote:
> I don't even think we know whether e*pi is irrational,
> although all our intuitions say [it is]... Indeed, showing that 
> e*pi is rational would revolutionize computations for both
> numbers..

It's known that e * pi is not rational, but we don't whether
e + pi is rational, and we certainly don't know if both
are transcendental, although we do know that at least one 
of them must be.

By the way, here's an interesting little formula involving
the product of pi and e (which will be utterly unintelligible
if you're reading this with something other than a fixed 
pitch font, and maybe even regardless):
    _______
   / pi e             1      1        1
  / -----    =   1 + --- + ----- + ------- + ....
 /    2              1*3   1*3*5   1*3*5*7


               +  1
                 ------------------
                        1
                  1 + ----------------
                            2
                      1 + --------------
                                  3
                            1 + -----------
                                       4
                                 1 + --------
                                           5
                                      1 + ------
 
                                           1 + ...


I probably don't even need to say who came up with this - which 
raises an interesting question: Is any other mathematician's work 
so instantly recongizable?  For the record, Ramanujan posed the
above formula as question 541 in the Journal of Indian Math.

On an unrelated point, does anyone know where the first "3"
appears in the base-pi representation of e?  I think the leading
digits are something like

            e = 2.2021201002111...(3)?...         base pi

where the kth digit is the largest integer d such that d/pi^k is less
than or equal to the current remainder.  Similarly the representation
of pi in the base e is

          pi  =  10.101002020002111...     base e

but of course here there can never be any digits other than 0,1, 
or 2.
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