Date: Mar 31, 2013 9:07 PM
Author: fom
Subject: Re: Matheology § 224

On 3/31/2013 1:44 PM, Virgil wrote:
> In article
> <3d98da78-e43c-4550-812c-6436200744ec@vh9g2000pbb.googlegroups.com>,
> "Ross A. Finlayson" <ross.finlayson@gmail.com> wrote:
>

>>>> In a Complete Infinite Binary Tree, every binary rational path has only
>>>> finitely many left-child nodes or only finitely many right-child nodes,
>>>> whereas every other path has infinitely many of each.

>>>
>>> That is nonsense. 0.0101010101... has infinitely many of both sorts.
>>>
>>> Regards, WM

>>
>>
>> Well, you see Virgil has introduced a term in context the "binary
>> rational path"

>
> The standard definition of a binary rational is a rational whose
> denominator is a power of 2.
>
> In binary place value notation, they are the infinite strings starting
> at the binary point, then having onlybinary digits of 0 or 1, which end
> with either a string of infinitely many 0's or infinitely many 1's.
>
> Thus in a Complete Infinite Binary Tree they correspond to infinite
> paths with either only finitely many 1's or only finitely many 0's.
>



I will not disagree with your statement concerning "binary
rational path", but I did do a search and did not come up
with anything useful. That does not mean much since there
are far more pages with "binary" and "rational" used in a
context different from yours.

I did, however, find dyadic rationals.

http://en.wikipedia.org/wiki/Dyadic_rational

These "standard terms" are sometimes a pain.

What is important is that they are a dense subset
in spite of not being the entire class of rational
numbers.