Maximum run of one digit in decimal expansion of pi?
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From: James Waldby on 23 May 2010 17:29 On Sun, 23 May 2010 12:44:51 -0700, "Shepherd Moon" wrote: > On May 23, 1:36 pm, "spudnik" <Space...(a)hotmail.com>" wrote: .... >> thusNso: >> the second part of the question is clearly trivial, and the first part >> seems to be its inverse, or what ever. >> > Could you explain why the second part of the question is trivial? .... The "spudnik" poster posted that text in another reply in a different thread (re Farey sequences) and apparently by mistake also pasted it into the post you quoted. That is, what "spudnik" posted is irrelevant. -- jiw
From: porky_pig_jr on 23 May 2010 17:39 > On May 23, 3:44 pm, Shepherd Moon <shepherdm...(a)yahoo.com> wrote: > I'm not sure I follow what you're saying. Don't worry about spudnik. He himself doesn't follow what he's saying either. Regarding the normal numbers, if I remember correctly, Billingsley (Probability and Measure) has a good discussion about those. May be somebody can jump in and confirm. I don't have that book at hand.
From: christian.bau on 23 May 2010 17:50 On May 23, 6:36 pm, spudnik <Space...(a)hotmail.com> wrote: > I don't see any neccesary resaon for *any* irrational number > to have a maximum run of any digit in what ever integral base; so, > rake one coal over yourself for propitiating such a silly idea! That's quite thoughtless of you. Take any irrational number. Express that irrational number in base 5, so all the digits are 0 to 4. Add 5 to every second digit and interpret this as a base ten number. There will be no repeating digits at all.
From: Pubkeybreaker on 23 May 2010 18:26 On May 23, 3:44 pm, Shepherd Moon <shepherdm...(a)yahoo.com> wrote: > On May 23, 1:36 pm, spudnik <Space...(a)hotmail.com> wrote: > > > I don't see any neccesary resaon for *any* irrational number > > to have a maximum run of any digit in what ever integral base; so, > > rake one coal over yourself for propitiating such a silly idea! > > Hello, > > I see no reason to get raked over the coals again. I asked my > questions in a courteous manner and even got some favorable responses > in terms of the value of the question. spudnik is a well known crank. Ignore him. And there are MANY irrational numbers which have a maximum run of some particular digit. Consider, e.g. .501001000100001000001....... This number is irrational. The digit 5 appears only once. The fact that he can't see such a thing indicates his level of mathemtical knowledge. > > Also, I never said anyone had to see a reason for any irrational > number to have a maximum run. I simply asked whether it is possible to > prove whether or not such a maximum run exists for pi, Actually, as others have indicated, we simply don't know. There is a conjecture that pi is normal, but there is no proof. We don't even know if it is provable.
From: Tim Little on 23 May 2010 21:21
On 2010-05-23, Shepherd Moon <shepherdmoon(a)yahoo.com> wrote: > So perhaps pi is closer to 0.625000... in that it has more than one > type of digit in it (all 10 decimal digits), yet may also have within > it an infinite run of one (or maybe all) of those 10 digits. Pi certainly does have all 10 digits in its decimal form, but there can be no infinite runs. Only rationals have infinite runs. There might be no limit to the size of finite runs, and this is actually expected of "nearly all" irrational numbers. It is not proven whether pi is one of those numbers. > So it seems to me, speaking as a naive layman, of course, that one > should always expect a run of any single digit in pi to be > interrupted by one of the other digits. Correct. > I suppose there is a third possibility - that each digit does indeed > have a maximum possible run in pi. But my math intuition, feeble as > it is, tells me that such a maximum run seems unlikely in an > irrational number. Yes, it has probability 0 in the usual measure. So numbers with a maximum bound for runs of each digit are "vanishingly rare". This isn't entirely a convincing proof that pi is normal is that sense, because pi has lots of properties that are "vanishingly rare". This holds for more than just maximum runs of each digit. If you take any set of sequences of digits whatsoever, almost every irrational number has arbitrarily long matching subsequences from every member of that set. Despite the general likelihood, it is remarkably difficult to prove this property for any given irrational not specifically constructed to have it (such as Champernowne's constant). - Tim |