Re: Small probabilities

From: gordon brown <gbrown@euclid.colorado.edu>
Date: Sun Nov 06 2005 - 21:04:03 EST

Merv,

Rational numbers are a subset of algebraic numbers since they always
satisfy a linear equation ax+b=0, where a and b are integers. Any number
that is not algebraic is transcendental by definition. Thus transcendental
numbers are irrational, but many irrational numbers are not
transcendental. For example, the square root of 2 is irrational but not
transcendental. Even though transcendental numbers are infinitely more
common (since algebraic numbers occupy zero space on the real line), it is
exceedingly difficult to give examples of them and prove that they are
transcendental. Pi and e are examples of transcendental numbers.

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

On Sun, 6 Nov 2005 mrb22667@kansas.net wrote:

>
> > > ...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 Sun Nov 6 21:04:50 2005

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