Pi must repeat

If Pi has infinite numbers following the decimal, at same point those numbers must fit into a sequence that has already been done, for whatever length of numbers.  Mathemetecians need only figure out how many numbers are repeated and for how long and in what places of the sequence.

>>65
If you accept that pi is transcendental, it isn't hard at all. The radicals are a subset of the algebraic numbers, which are (by definition) an algebraically closed field. If there was a polynomial with algebraic number coefficients which had pi as a root, then there are two possibilities:
(1) Pi is an algebraic number; contradiction, as pi is transcendental
(2) The algebraic numbers have a nontrivial algebraic extension; also a contradiction, as they are algebraically closed.

Of course, the hard part is the proof that pi is transcendental; if I recall correctly the idea is to show that i*pi (and therefore pi itself) must be transcendental because e is transcendental and e^(i*pi) = -1 is rational.