
Re: the binary tree
Posted:
Apr 17, 2009 1:39 AM


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.

