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...
Suppose there is a repeating string in the sequence of digits in a and k0 is the first number that k0^2 if fully inside the repeating region. Then:
1. It is easy to show that (k+1)^2 can be only 1 digit longer than k^2.
2. Therefore, there is always k>k0, that the length of k^2 is a multiple of the length of the repeating string
3. If (k+1)^2 has the same length as k^2, then a repeating pattern is not possible. If (k+1)^2 is one digit longer than k^2, then (k+1)^2=10*k^2+m, where m<10, which again is not possible.
A similar proof can be offered for c.
Posted by Art M
on 2006-09-06 18:30:35