You are probably quite certain that the following numbers are irrational. But can you prove it?
a = 0.149162536496481...
b = 0.2357111317192329...
c = 0.1248163264128256...
Any number that has the following decimal representation: 0.a1a2a3..ak.. is irrational, if ak is an integer sequence so that 1<ak+1/ak<10. For a: (k+1)2/k2<=4, for c: 2(k+1)/2k=2. For b: Bertrand's postulate (http://en.wikipedia.org/wiki/Bertrand%27s_postulate) states, that for every n > 1 there is always at least one prime p such that n < p < 2n, therefore for primes pk+1/pk<2.
Edited on September 8, 2006, 2:57 pm
Edited on September 8, 2006, 2:59 pm
Edited on September 8, 2006, 3:01 pm
Edited on September 8, 2006, 5:16 pm
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Posted by Art M
on 2006-09-08 14:55:27 |
generalization
