[John W. Burgeson]
| I pulled the following from Pickover's LISTSERV -- now I am really over
| my head!
| 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.
| 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.
It all depends on how you interpret the word "contain".
For any increasing sequence of natural numbers 0 < n_1 < n_2 <n_3 < ...,
there is a real number A with decimal expansion given by the rule
i-th decimal of A = n_i-th decimal of pi.
There are uncountably many such sequences, in fact, even uncountably
many such sequences with the property that 1 <= n_(i+1) - n_i <= 2 for
all i. The quote from Pickover's listserv deals with a very strict
subclass of sequences, namely those which are of the form n_i = N + i
for some number N.
-- Stein Arild StrÝmme <http://www.mi.uib.no/~stromme>
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