All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Numbers
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)

Comments: ( Back to comment list | You must be logged in to post comments.)
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
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (9)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2014 by Animus Pactum Consulting. All rights reserved. Privacy Information