Date: Sep 12, 2012 6:54 PM
Author: Joe Niederberger
Subject: Computing pi (or not)
I intend this as a bran-teaser spinoff from the "Non-Euclidean Arithmetic" where the discussion veered
off into computable and non-computable real numbers and such.
So I take it most people here know there are algorithms that can compute the digits of pi. And some of those may know about "non-computable" real numbers. And some may know a little programming (I assure you this is mathematical content however.) Its easy to create an algorithm that in turn spits out all 1 place decimal numbers, then all 2-place, 3-place and goes on forever
(just like the algorithm that spits out pi digits.)
So, if we visualize this output as a graph theory tree, which has 10 branches down at every vertex, and a sequence of digits (with an implicit decimal point after the first) can we not say this algorithm is outputting the digits of pi? (It certainly is, there is a path down the tree which shows the digits of pi in sequence!)
Is it not outputting all the real numbers between 0 and 10? By tracing a suitable path down the tree we can find any real number we care to. (Yes, no, maybe?)
On the other hand, there is a hypothesis that the digits of pi are "normal" -- containing all possible 2-digit sequences (not only, but they all occur 1/100 of the time.) Likewise all possible 3-digit sequences, etc.
If true, then pi contains somewhere in its decimal expansion, sequentially, an encoding of the complete works of Shakespeare (any edition), complete encodings of the bible (any edition, any translation.) The sum total of human output all in one number, not just past knowledge, but all books yet to be written as well! [Imagine the enormity of the knowledge contained in the whole tree! Perhaps we should call it the "god tree (tm)".]
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