 My Math Forum Proove: Irrational numbers have unique distribution
 User Name Remember Me? Password

 Number Theory Number Theory Math Forum

September 10th, 2010, 02:42 PM   #1
Member

Joined: Aug 2010

Posts: 49
Thanks: 0

Proove: Irrational numbers have unique distribution

Quote:
 I have a question about irrational numbers.
damn, there are irrational numbers, which do not fulfill this: Let us take a random sequence of 0 and 1 and use this for the a_i... I thought it can be proven that sqrt(2) has equally distriputed a_i but this mean nothing, if the proof is right, at all..

I was just fascinated by the distribution of the numbers "we know", like e or so...

Any help appreciated!

Best regards,
Jens September 10th, 2010, 05:24 PM #2 Global Moderator   Joined: May 2007 Posts: 6,754 Thanks: 695 Re: Proove: Irrational numbers have unique distribution I am not sure what you are trying to say. There is a concept call "normal numbers", which are numbers whose expansion (binary, decimal, or any other base) is statistically random. There is a theorem (as I recall from a long time ago) which states that almost all numbers (in a measure theory sense) are normal. September 10th, 2010, 05:52 PM   #3
Member

Joined: Aug 2010

Posts: 49
Thanks: 0

Re: Proove: Irrational numbers have unique distribution

Quote:
 Originally Posted by mathman I am not sure what you are trying to say.
Understandable, sorry, since I forgot the following term:
Quote:
 There is a concept call "normal numbers", which are numbers whose expansion (binary, decimal, or any other base) is statistically random.
This is what I mean. My counterexample showed, that not every real number is normal.
Quote:
 There is a theorem (as I recall from a long time ago) which states that almost all numbers (in a measure theory sense) are normal.
This is at least where my mathematics ends. I am not a mathematician (but physiscist), I was simply very confused by "finding out" that for example e seems to be "normal", taking the number (in base 10) and applying statistics. I heard measure theory only in the context Lebesque integrals... So: Is there any number where it is proven that it is normal? Is it prooven that the most numbers are normal (seems so if there is a theorem)? So, if I would proove that sqrt(2) is normal, is this something interesting at all?

Thanks,
Jens September 10th, 2010, 08:38 PM   #4
Global Moderator

Joined: Nov 2006
From: UTC -5

Posts: 16,046
Thanks: 938

Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms
Re: Proove: Irrational numbers have unique distribution

Quote:
 Originally Posted by Jensel So: Is there any number where it is proven that it is normal?
Yes. 0.12345678910111213... and 0.23571113171923... have both been proven normal -- the concatenations of the natural numbers and the primes, respectively.

Quote:
 Originally Posted by Jensel Is it prooven that the most numbers are normal (seems so if there is a theorem)?
Yes. "Almost all" (in a particular mathematical sense) numbers are normal.

Quote:
 Originally Posted by Jensel So, if I would proove that sqrt(2) is normal, is this something interesting at all?
That would be very interesting -- and probably extremely hard. Tags distribution, irrational, numbers, proove, unique Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Mighty Mouse Jr Algebra 1 October 16th, 2010 07:46 PM kiechle Advanced Statistics 5 September 29th, 2009 07:12 AM PseudoCode Number Theory 2 March 6th, 2009 11:03 PM Torres Number Theory 4 December 22nd, 2008 03:33 PM kiechle Applied Math 1 December 31st, 1969 04:00 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top

Copyright © 2019 My Math Forum. All rights reserved.      