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Topic: Proof the irrationality of pi
Replies: 8   Last Post: Sep 10, 2004 10:20 PM

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Yogi

Posts: 323
Registered: 12/6/04
Re: Proof the irrationality of pi
Posted: Sep 9, 2004 9:23 AM
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There is a very pretty theorem that says: If c is a positive real
number and there exist a function f(x) that is continuous on [0,c],
positive on (0,c) and such that f and it iterated anti-derivatives can
be taken to be integer valued at 0 and c, then c is irrational!

pi is a positive real number. The function f(x)= sin(x) is
continuous for all x and so on [0,pi] and is positive on (0, pi). All
anti-derivatives can be taken to be +/- sin(x) or +/- cos(x) (by
always choosing the constant of integration to be 0). Therefore
pi is irrational.




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