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Re: the binary tree
Posted:
Apr 16, 2009 11:20 PM
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On Apr 16, 10:58 pm, calvin <cri...@windstream.net> wrote:
> On Apr 16, 10:36 pm, Dave <dave_and_da...@Juno.com> wrote:
>
> > On Apr 16, 9:05 pm, calvin <cri...@windstream.net> wrote:
>
> > > Can one be haunted by a mathematical concept?
> > > The binary tree answers yes for me. That I can
> > > construct, in a countable number of steps, a fully
> > > comprehensible representation of every real number
> > > between zero and one, is astonishing every time
> > > I think of it.
>
> > > What else in math is so easy to do and has equally
> > > breathtaking results, I wonder.
>
> > I was with you until you said "every real number." Are you sure you
> > don't mean "any real number"?
>
> All real numbers between zero and one. The construction
> I have in mind is the simple one of starting with a
> binary point at the topmost node, and then going from
> left to right below it to the two nodes at that level,
> and then going from left to right for the four nodes
> at the next level, and so on. Of course it helps
> visualization to halve the distance down to each level
> so that the possible paths from node to node are seen
> approaching individual points on the unit line at the
> bottom, but my main point is that this construction
> requires a countable number of steps to lay out the
> uncountable number of paths to the real numbers between
> zero and one.
Is this more astonishing than, say, one can write down any
real number between 0 and 1 in a countable number of digits,
even though for each digit, one has only 10 choices.
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