RE: Small probabilities

From: gordon brown <gbrown@euclid.colorado.edu>
Date: Mon Nov 07 2005 - 12:19:51 EST

A real number is rational if and only if its decimal expansion is a
repeating decimal, i.e. if from some point on it simply repeats some
segment forever. When you perform long division in which you divide by a
number n, there are only n possibilities for the remainder that you get
after each division step, and once you are to the place where your
quotient is beyond the decimal point, the calculations repeat themselves
each time you get the same remainder.

In your example, the spacing between the 1's is increasing as you go
along, and so no segment will keep repeating itself forever.

Gordon Brown
Department of Mathematics
University of Colorado
Boulder, CO 80309-0395

On Mon, 7 Nov 2005, Alexanian, Moorad wrote:

> 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 passť 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 12:22:02 2005

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