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September 10th, 2010, 02:42 PM   #1
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Proove: Irrational numbers have unique distribution

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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
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September 10th, 2010, 05:24 PM   #2
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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.
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September 10th, 2010, 05:52 PM   #3
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Re: Proove: Irrational numbers have unique distribution

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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
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September 10th, 2010, 08:38 PM   #4
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Re: Proove: Irrational numbers have unique distribution

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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.

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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.

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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.
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