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Re: Bizarritudes
Posted:
Jul 27, 2005 3:43 PM
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> > bischar <bisch_a_r@yahoo.fr> writes in article
> >
> <13842344.1122478023914.JavaMail.jakarta@nitrogen.math
>
> > forum.org> dated Wed, 27 Jul 2005 11:26:33 EDT:
> > >Would there be a function that would extract
> precise
> > digits and create a new number with them ?
> > >
> > >I mean a function that would make from pi
> > 3.04050205050909
> >
> >02080606030802090008040901090903050008000709040903070
>
> > 10406
> > >08020809060803080504010706 and
> > 0.1010906030807030304020403
> >
> >03070502080107060309070105020904040502000806000206000
>
> > 90802
> > >000402030201000 for instance.
> >
> > Anything which you can descibe in such a way that
> > f(x) is unambiguous
> > is a function. In the above case, you would
> define
> > f:R -> R^2, because
> > there are two different output values.
> >
> > The one caveat I see in your definition is that
> > f(.999999...) != f(1.0).
> > Unless you fix this, it's not exactly a function
> on
> > reals but on digits.
> >
> > >Or another function that would insert a zero
> between
> > all the digits composing the initial number ?
> >
> > Same problem with .99999...
> >
> > --Keith Lewis klewis {at} mitre.org
> > The above may not (yet) represent the opinions of
> my
> > employer.
>
> And what would a 'function on digits' look like ?
Bischar, I have the solution for all your problems:
1.- Consider the bijectve function f : R+ -> [0,1)
f( x )= (2/pi) arctan x
2. Consider the dynamical system x_(k+1)=frac(2x_k), which
will allow you to set out a binary codification of [0, 1) by means
of O^+(y) ( orbit of y = f ( x ) ).
3.- Now, for every x>=0 you have a "nice" expression for x.
The problem is, there will be your computer screen so big
for showing the results ?
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