Maximum run of one digit in decimal expansion of pi?
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From: Tim Little on 23 May 2010 21:27 On 2010-05-23, Rick Decker <rdecker(a)hamilton.edu> wrote: > A more restrictive form of this question is "for every positive > integer k, and every integer d, 0 <= d <= 9, is there a run of exactly k > d's in the decimal expansion of pi?" I'd guess that most mathematicians > would be agnostic on that question. Agnostic in the technical sense of having some slight hints of doubt perhaps, but strongly expecting the answer to be "yes". Not only that, but expecting such runs to appear infinitely many times for each k and d. - Tim
From: Peter Webb on 23 May 2010 22:59 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. __________________________________ With a tiny bit more work, we can easily provide a far stronger result, that there are an infinite number of Reals in which no digit repeats, and hence the longest run of every digit is one. Consider any Real - say pi or sqrt(2) - and write out its binary expansion. Let's say it is 0.1101... For each digit "0" substitute the digits "23". For each digit "1" substitute the digits "45". Consider this as a base 10 expansion So 0.1101... as binary -> 23.45452345 .... as decimal It is easy to see that: 1. The new number only contains the digits 2,3,4,5 and the maximum run of any of these digits is 1. 2. If the original binary number is irrational, so is the new number in base 10, and if the original binary number is rational, then so is the new number in base 10. (This doesn't even require the number to be irrational to create a Real with a maximum run of a single digit being one, but that was the original question in the context of pi being irrational) Peter Webb
From: Transfer Principle on 24 May 2010 13:38 On May 23, 12:44 pm, Shepherd Moon <shepherdm...(a)yahoo.com> wrote: > On May 23, 1:36 pm, spudnik <Space...(a)hotmail.com> wrote: > > on the wayside, > > 0.999.... does not = 1; > > it equals 1.000...., the "real"number, one; > > take a hop, a skip & a jump over Tony Robinson's bed > > (of coals). > I'm not sure I follow what you're saying. By "0.999.... does not = 1 > it equals 1.000...." are you implying that 1 <> 1.00? > Could you explain why 0.999... = 1.000... <> 1 if that is indeed what > you're claiming? It appears that spudnik is making a distinction often made by computer scientists. In many programming languages, there are integer variables and there are floating-point/real variables. An integer variable can contain the value 1, but not 1.0, and vice versa for real variables. As it turns out, standard set theories such as ZFC also make a similar distiction. In ZFC, naturals such as 1 are finite ordinals. In particular, the natural 1 equals the singleton {0}. But what about the real 0.999...? Reals are usually either Dedekind cuts or (equivalence classes of) Cauchy sequences. If we consider 0.999... as a D-cut, then it is the set of all rationals less than 0.999... -- in particular, it contains the rationals 0.9, 0.99, 0.999, and many others. This set is infinite -- countably infinite, to be sure, since Q is countable. If we used C-sequences instead, then 0.999... is an equivalence class of C-sequences. One such sequence is {0.9, 0.99, 0.999, ...}. It can be shown that there are uncountably many C-sequences in each equivalence class. In either case, the real 0.999... is an _infinite_ set, but the natural 1 is a _singleton_. By the Axiom of Extensionality, an infinite set cannot equal a singleton. Therefore 0.999... is not the same (set) as 1. Of course, mathematicians _identify_ the naturals with a subset of the reals all the time, but _identifying_ isn't the same as literal _equality_. We can write this as a formal proof: Theorem: The real 0.999... does not equal the natural 1. Proof: Case 1. D-cuts. The natural 1 has only one element, the empty set, but the elements of 0.999... are rationals. In particular, the rational 9/10, which is not equal to the empty set, is an element of 0.999..., so we have: (9/10)e0.999..., yet ~(9/10)e1 Thus by Extensionality, ~(0.999...=1). QED Case 1. Case 2. C-sequences. The natural 1 has only one element, the empty set, but the elements of 0.999... are C-sequences. In particular, the C-sequence {9/10, 99/100, ...}, which is not equal to the empty set, is an element of 0.999..., so we have: {9/10,99/100,...}e0.999..., yet ~{9/10,99/100,...}e1 Thus by Extensionality, ~(0.999...=1). QED Case 2. Notice that Megill's Metamath, since it is a _computer_-based prover, must find a way to distinguish between naturals and reals in order to prove 0.999...=1. In particular, the natural (finite ordinal) 1 is written as 1_o ("o" for ordinal), while the real 1 is written as 1_R. The number 1 without a subscript actually denotes the _complex_ number 1. The Metamath proof of 0.999... = 1 indeed proves that the _complex_ numbers are equal. So spudnik correctly points out that in both computer science and set theory, the real object 0.999... is not literally equal to the integer object 1. It definitely puts a new twist on the long-running debate about 0.999... and 1.
From: Aatu Koskensilta on 24 May 2010 13:51 Transfer Principle <lwalke3(a)lausd.net> writes: > So spudnik correctly points out that in both computer science and set > theory, the real object 0.999... is not literally equal to the integer > object 1. It definitely puts a new twist on the long-running debate > about 0.999... and 1. No it doesn't. The "debate" has nothing to do with the tedious technical details you mention. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: spudnik on 24 May 2010 13:55
I was taking monsieur Webb to task, for saying that *any* irrational number has a limited run of digits in *any* integral base ... unless it might be one that was (some how) specifically "constructed" to do that, with no other use for it. if you can't take the heat, get out of the frying-pan, or reduce the size of teh bed of coals to traverse. the problem of "0.999.... not equal to one" is, almost always, stated by the math-newbie in those terms; whereas, what is really meant, is if there is a non-infinitessimal difference between the two real numbers, cf. 1.000.... (and, this was all covered in Simon Stevin's booklet, _The Decimals_, way back in when ever .-) after that, comes the wonderful world of p-adics! thusNso: what consistency or species of feces would you like? I clikced on your webpage with the shrine to Hope or Athens, and your "Prime Gaps" page seems to be nothing, but a crude attempt at the sieve of Eristosthenes; congradulations -- you get every turd of that, tossed playfully back at you! are you literate in any language? (this is a big problem on the web, folks .-) > thus: > well, that is where the problem with assigning a particle > to a wave, a la de Broglie et al, comes. the assumption, > that causes folks to say "particle," is that because a quantum > of light is absorbed by one atom of siver dioxide (say, > in the photographic emulsion; or, other detector) --some how-- > that it must be that a rock of light hit the electronic orbital > (although > this is never specified, as to how it could be, and the whole problem > of EM is also hard to describe, and variously is). > > this is really all of a confusion from Newton's "geometrical optics," > that is, the "ray" of light, which is just one "normal" > to the wave (or Huyghens wavelet). > thus: > about your five "cloture" events, the real problem is that > "the Fed" was never properly ratified (and is unconstitutional > for that reason, if not directly; it is modeled > upon the Federal Reserve System > of England). of coursel the 527 cmtes. have essentially taken > over the TV advertizing on all national issues & candidates, > through an Act that was passed unaanimously in both houses. > > "Senate rules don't trump the Constitution" --http://GreaterVoice.org/60 > > thus: > I've been saying, for a while, that if "green" gasoline can > be made, and gasoline fuel cells, what is the problem > with Fossilized Fuels (TM), which ain't fossilized? ... anyway, > see "Green Freedom" in the article, > which is not quite what I was refering to! thusNso: Young's two-pinhole experiment was the thing that killed the corpuscle o'light, so that anomalous & highly particular (sik) set-ups with giant molecules are only of pecuniary interest ... er, what ever "pecuniary" means, you have to actually look at the "particulars" of the write-up with the fullerenes, to be able to say any thing of interest -- to make a hypothesis, beyond regurgtiating their delciious resultage. see, you did not even bother to deal with the whole idea of duality, that Pascal essntially created in projective geometry (cf., "two-column proofs" .-) another way to "check" your theory -- iff it is one -- would be to explain Snell's law, electromagnetically and/or with aether ... even if it is an "undefined element" of your axioms. > The 'particle' associated with the photon wave enters and exits a > single slit. The 'particle' occupies a very small region of the wave. thusNso: yeah, but you don't need the particle, at all; your theory says nothing, at all, either, at least til you attempt to make it do some thing. one way to "check" it, would be to explain permitivity & permeability with the theory, if it is a theory & not just typing & macro-ing practice. > The associated wave exits the both slits and creates interference > which alters the direction the particle travels. thusNso: being legally enjoined from using the googolplex, what's YUV? thusNso: complainant: there's a series of his books from Cambridge U. Press, paperback, very nice, but they want $45 for the one that I was peruzing at the bookstore! (I think, I'd bought one of these, before, for about $25, although I lost it.) the titles of these seem generally to be a list of three subjects, _This, That and Another Thing_, although also the usual format of several independent essays/chapters. r.i.p., MG. --Pi, the surfer's canonical value -- accept no other! http://wlym.com |