|
|
Re: Simple random number generator?
Posted:
Dec 1, 2012 8:56 PM
|
|
On 12/01/2012 07:21 PM, Christopher J. Henrich wrote:
> In article <pChMbSKOZRuQFw+m@invalid.uk.co.demon.merlyn.invalid>, Dr J
> R Stockton <reply1248@merlyn.demon.co.uk.invalid> wrote:
>
>> In sci.math message <50b6cabe$0$24749$607ed4bc@cv.net>, Wed, 28 Nov 2012
>> 21:38:34, Existential Angst <fitcat@optonline.net> posted:
>>
>>>
>>> So are the digits of pi random or not?
>>>
>>
>> The digits of pi are not random, because, unless the base is changed,
>> they are the same every time and can be defined relatively briefly, even
>> without assuming a definition of pi (which pi may formally lack).
>>
>> But any arbitrarily chosen sub-sequence of the digits of pi will, I
>> believe, pass an appropriate proportion of the usual tests for
>> randomness. Note that the full expansion of the digits of pi contains
>> as sub-sequences of a given length all possible digit strings of that
>> length, some of which will not look random to the untutored eye, such as
>> yours.
>>
>> A real mathematician could put that more precisely.
>
> If the digits of a number are uniformly distributed, so that in the
> (infinitely) long run 0's, 1's, 2's, etc. occur equally often, then the
> number is said to be "simply normal" (in base 10). If, for every k =
> 1,2, ..., all possible groups of k digits occur equally often in the
> long run, then the number is said to be "normal." (Reference: Hardy and
> Wright, /The/ /Theory/ /of/ /Numbers/ , Fourth ed., pp. 124-5.)
>
> Most of the irrational numbers that are interesting (such as sqrt(2) or
> pi) appear to be normal. Numbers that are not normal include all
> rational numbers, and some particular values of theta functions (for
> instance 0.1001000010000001 ... ).
>
> I think most mathematicians are of the *opinion* that pi is normal. But
> as far as I know, nobody has proved that it is, nor that it isn't. A
> frustrating situation. Part of the difficulty seems to be that the
> property of normality does not seem to be connected to other
> interesting properties of an irrational number.
>
Another question, this time on approximability by rational
numbers, can be asked for pi, for example.
Thm.: There are infinitely many pairs (m, n) of
positive integers such that
| pi - m/n | < 1/(n^2).
if 1/(n^2) is replaced by 1/(n^d), for a given real number
d >=2, one has the question:
Is it true that there are infinitely many pairs (m, n) of
positive integers such that
| pi - m/n | < 1/(n^d) ?
If d > 7.61, it is known to be false
Salikhov, 2008:
http://mathworld.wolfram.com/IrrationalityMeasure.html
--
http://www.youtube.com/watch?v=8IxeroqZSuo
|
|
