On 25 Mrz., 15:48, "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote: > 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 > rational > > 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.