Date: Mar 25, 2013 11:09 AM
Subject: Re: Matheology § 224
On 25 Mrz., 15:48, "Ross A. Finlayson" <ross.finlay...@gmail.com>
> On Mar 25, 4:45 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> > Does the Binary Tree that contains all rational paths also
> > contain all irrational paths?
> Given that, for R[0,1]:
> a) each irrational has a unique infinite expansion as path
That is the question. If so, why has never anybody written it using
digits or bits?
> b) each initial segment of the expansion is the initial segment of a
> c) every rational's path is in the tree
That is the question too. Why has never anybody written the complete
decimal- or binary expansion of a periodic rational?
> d) the union of finite initial segments of the expansion as tree
> contains the expansion as path
> e) thus each irrational's expansion is a path in the tree of rationals
> then, yes, that appears to be so.
I agree with your conclusion but not with the premises.
Remember: Never has anybody written an infinite sequence other than by
using the symbolic method: "1/9" or "1/pi" or "1/(SUM 1/n!)". These
however are only names to identify or formulas to construct infinite
paths - not paths that belong to the Binary Tree.