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These are no fractions, are they? (Posted on 2006-09-05) Difficulty: 3 of 5
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...

See The Solution Submitted by JLo    
Rating: 2.5000 (2 votes)

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proof for a | Comment 8 of 15 |
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
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