Re: the binary tree
- From: Virgil <Virgil@xxxxxxxxx>
- Date: Thu, 16 Apr 2009 23:52:27 -0600
In article
<83a0328c-6ed4-4987-b5ae-72a44e213991@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Gvnaena Pura <tianran.chen@xxxxxxxxx> wrote:
On Apr 16, 11:27 pm, calvin <cri...@xxxxxxxxxxxxxx> wrote:
On Apr 16, 11:20 pm, Gvnaena Pura <tianran.c...@xxxxxxxxx> wrote:
On Apr 16, 10:58 pm, calvin <cri...@xxxxxxxxxxxxxx> wrote:
On Apr 16, 10:36 pm, Dave <dave_and_da...@xxxxxxxx> wrote:
On Apr 16, 9:05 pm, calvin <cri...@xxxxxxxxxxxxxx> 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.
Any "definition" of real numbers that depends on how one is allowed to
represent them symbolically is ultimately unsatisfactory.
.
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