I pulled the following from Pickover's LISTSERV -- now I am really over
my head!
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What information can't pi contain in its digit stream?
Say that for example you have 3 atoms arranged in an equilateral triangle
formation. That means that at least one of the coordinates is equal to
the square root of 3, or 1.732etc. The three coordinates of the atoms
maybe could be (0,0), (1,0), and (1, 1.732..).
Because the square root of three is irrational, its expansion as a real
number will continue forever without a pattern. Thus, if pi is to
"contain" the sqrt(3) that means that at one point, the digits of pi
would have to revert to the digits of sqrt(3), and continue that way for
infinity. Which would mean that pi is not transcendental. Pi has been
proven to be transcendental, so an infinite expansiion of sqrt(3), or any
other root for that matter, is impossible.
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Ignoring the question of the proper definition of "transcendental," I
still don't see this follows. The infinite expansion of pi ought to be
able to contain a finite number of infinite expansions. Maybe.
John Burgeson (Burgy)
www.burgy.50megs.com
(science/theology, quantum mechanics, baseball, ethics,
humor, cars, God's intervention into natural causation, etc.)
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