From: Alexanian, Moorad <alexanian@uncw.edu>

Date: Mon Nov 07 2005 - 10:49:36 EST

Date: Mon Nov 07 2005 - 10:49:36 EST

The sum of the series with terms 1/10^(n(n+1)/2) from n=1 to infinity is the number 0.101001000100001000001 ... I am not sure if that helps to establish whether the number is rational or irrational. The partial sum is indeed a rational number, however, the infinite series is a limiting process and so one does not know if the limit gives rise to a rational or irrational number. Perhaps some mathematician can answer that question.

Moorad

-----Original Message-----

From: asa-owner@lists.calvin.edu [mailto:asa-owner@lists.calvin.edu] On Behalf Of mrb22667@kansas.net

Sent: Sunday, November 06, 2005 8:49 PM

To: asa@calvin.edu

Subject: Re: Small probabilities

*> > ...Conclusion: It is a virtual certainty of probability that the entire
*

*> > coded works of Shakespeare as well as everything else are to be found
*

*> > somewhere in pi. Okay - maybe I'm the only one fascinated by this,
*

*> > since the magnitudes of digits required to find anything interesting
*

*> > would dwarf the paltry (relatively tiny) trillions of digits now known.
*

*> >
*

*> > So if the universe was infinite in either time or space, any
*

*> > improbability becomes probable - hence the evolutionary attachment to
*

*> > steady state theories with their infinite time spans or infinite #s of
*

*> > 'parallel' universes in which so called 'randomness' can do its tricks.
*

*> > Not that I really buy any of this (or its negation either) - there are
*

*> > too many assumptions in it for either creationists or their antagonists
*

*> > to wield it effectively regarding probabilities. The conclusions
*

*> > displayed in the eventual creationist or evolutionist pedagogy are, IMO,
*

*> > more a reflection of the carefully chosen assumptions than a proof (or
*

*> > even an evidence) of a reality.
*

*> >
*

*> > --merv
*

*> >
*

*> > by the way - I'm sure my suggestion that anything can be found in
*

*> > irrational #s must be passé discussion among mathematicians, & yet I
*

*> > don't remember ever hearing such a thing proposed. Does it sound
*

*> > logical? I think there could be a formal proof done on this.
*

*> >
*

*>
*

*> Although I wouldn't be surprised if any given finite string of digits
*

*> exists somewhere in pi, this is not true of just any irrational number.
*

*> Although we usually demonstrate the existence of transcendental numbers,
*

*> i.e. those that are not roots of any polynomial with integer coefficients,
*

*> by proving that the set of algebraic numbers is countable but the set of
*

*> real numbers is uncountable, there is a constructive proof of this fact
*

*> due to Liouville in 1851. The sum of the numbers 10^-j! (ten to the minus
*

*> j factorial) from j=1 to infinity is an example. The digits of its decimal
*

*> expansion are all either 0 or 1.
*

*>
*

*> Gordon Brown
*

*> Department of Mathematics
*

*> University of Colorado
*

*> Boulder, CO 80309-0395
*

*>
*

*>
*

Thanks -- I'm familiar with transcendental numbers, but I'm still trying to

understand the class you referred to as algebraic numbers. I googled that

phrase, and got oogled by things I'm still trying to wrap my mind around. I

once encountered the question of whether or not 0.101001000100001000001 ...

would be an irrational number, and until now, I had forgotten about it. Could I

take it correctly from your reference to the 'Liouville proof' that the above

would indeed be irrational? It doesn't technically have the standard

repetition, right? If it was rational, I'd love to see the fraction. It's been

too long since I studied all the types of infinite series in calculus.

--merv

Received on Mon Nov 7 10:52:02 2005

*
This archive was generated by hypermail 2.1.8
: Mon Nov 07 2005 - 10:52:02 EST
*