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Re: the binary tree
Posted:
Apr 17, 2009 1:39 AM
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On Apr 16, 11:27 pm, calvin <cri...@windstream.net> wrote:
> On Apr 16, 11:20 pm, Gvnaena Pura <tianran.c...@gmail.com> wrote:
>
>
>
> > 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.
>
> You can have as many choices as you like for each 'digit'
> simply by choosing the proper base. I like having the
> least number of choices and therefore using base 2.
Okay, should have used a more sarcastic tone. This is how elementary
school textbook define real numbers, except they use base 10. So
why is this surprising.
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