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Re: Computing pi (or not)
Posted:
Sep 13, 2012 10:31 AM


Joe Niederberger wrote (in part):
http://mathforum.org/kb/message.jspa?messageID=7889637
> 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 2digit > sequences (not only, but they all occur 1/100 of the time.) > Likewise all possible 3digit 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)".]
The tree "paradox" is a standard technique/idea in set theory and logic, and a good starting point for those interested would be to google "infinite binary tree" and "binary tree" AND "real numbers".
http://www.google.com/search?q=%22infinite+binary+tree%22
http://www.google.com/search?q=%22binary+tree%22+%22real+numbers%22
The messages in pi idea showed up in Carl Sagan's book "Contact" (but not in the movie made from the book), something I've posted a fair amount about in the past, for example these two posts:
sci.math: "Contact"/pi [23 October 2000] http://mathforum.org/kb/message.jspa?messageID=272526
mathteach: Pi & contact [24 April 2001] http://mathforum.org/kb/message.jspa?messageID=1480429
  begin technical math aside 
Incidentally, for pi to have this property, a much weaker hypothesis than "pi is a normal number" suffices.
Almost all real numbers, in the sense of Lebesgue measure, are normal (i.e. the set of nonnormal numbers has Lebesgue measure zero), but the opposite is true in the case of Baire category (almost all real numbers, in the sense of Baire category, are NOT normal). Thus, while the set of normal numbers is really big in the sense of Lebesgue measure, it's also really small in the sense of Baire category.
On the other hand, the property of having all finite strings of decimals appearing in a real number's decimal expansion holds for real numbers that form a much larger set of real numbers. Indeed, this set of real numbers is really big in the sense of Lebesgue measure AND really big in the sense of Baire category. In fact, the distinction is even more extreme than this, as I indicate in the post below.
sci.math: Omnitranscental numbers [19 February 2003] http://mathforum.org/kb/message.jspa?messageID=459843
  end technical math aside 
Dave L. Renfro
 End of Forwarded Message



