Talk:Normal number

message: "No rational number is normal to any base, since the digit sequences of rational numbers are eventually periodic."
Is this right?  Obviously rationals are periodic, but what about 2/3 in base 2 (0.1010101...), or 1234567890/9999999999 in base 10 (0.12345678901234567...)?
:I think it is right. Based on the definition, it isn't only single digits that have to appear equally, it's all sequences of digits. So for 2/3 to be normal to base 2, you need to show that 111010101100001101, for instance, occurs as if 2/3 were a random  string of digits, when of course it does not occur at all. And similarly with your other example. Eric119 20:53, Mar 26, 2004 (UTC)
:I believe there is an inconsistency with the claim at the end of the article that "not a single irrational algebraic number has ever been proven normal in any base", since we know that Copeland-Erdős constant is normal.
:Oh wait, it said irrational ALGEBRAIC number - and the Copeland-Erdős constant looks pretty transcendental to me. (Michael Currie)
::I think the conjecture that every irrational algebraic number is normal is not due to Bailey and Crandal as asserted but goes back to Borel:
É Borel, Sur les chiffres décimaux de $\sqrt 2$ et divers problèmes de probabilités en chaîne, C. R. Acad. Sci. Paris 230 (1950) 591--593. Réédité dans : Œuvres d'É. Borel  vol. 2, Éditions du CNRS, Paris, 1972, pp. 1203--1204. [J.-P. Allouche]
:::Has the Copeland-Erdős constant been proven transcendental?  If not, perhaps the statement should be "not a single number has been proven to be both normal in any base and algebraic", which is a bit different from "not a single irrational algebraic number has ever been proven normal in any base".  There's also a bit of an ambiguity, in that the latter can be interpreted as "there is no base in which a single irrational algebraic number has ever been proven normal", or "not a single irrational algebraic number has ever been proven normal in every base".  Also, the word "irractional" is a bit redundant, as the statement that no rational number has been proven normal in every base is rather trivial.Flarity 20:27, 31 May 2006 (UTC)
thread_title: Rational numbers and normality

message: The current text says "The set of non-normal numbers is known to be uncountable; this follows easily by simply omitting one digit from each real number." This seems like it could be worded better unless I am missing something. When I hear "omitting a digit" I think of dropping, say, the 15th digit of a number so that the formerly 14th and 16th digits are now adjacent. But I don't immeditely see how that would demonstrate that non-normal numbers are uncountable because it doesn't seem like it would produce non-normal numbers.
If, instead, we drop all instances of a particular numberal in the decimal expansion (all "9"'s for instance), we clearly have a non-normal number for each Real. Even looking at it that way, the result doesn't seem to immediately follow because there will be many duplicates. For instance imagine pi with 3 9's inserted immediately after the decimal point, and pi with 4 9's inserted immediately after the decimal point. The "drop all 9's" procedure would turn both of those into pi. Given that the procedure results in a proper subset, how do we know that the subset isn't countable?
I think a clearer explanation would be helpful, or at least a pointer to another reference.
: It seems to me that it would be more easily provable to have a reversible transformation. e.g. "write a digit 0 after every digit 1", "write the number in octal and interpret the result as decimal" or "write in binary-coded decimal and interpret as hexadecimal" or similar. --62.173.111.114 15:16, 18 November 2005 (UTC)
replies: text:  Consider all numbers which only have the digits 1 and 7. None is normal, and there are clearly uncountably many. The current proof should be read "consider all numbers which avoid the digit 5. None is normal, and there are uncountably many". 192.16.204.80 07:31, 9 February 2006 (UTC)
replies: text: Good point. I changed the formulation. -- Jitse Niesen (talk) 17:52, 9 February 2006 (UTC)
replies: text: Since the the non-normal numbers are uncountable, why does the text say "While a general proof can be given that almost all numbers are normal"? ReytheRed (talk) 20:05, 16 June 2011 (UTC)
replies: text: This is a technical use of the phrase almost all — in context, it means that the set of exceptions has Lebesgue measure zero.  Is that not explained? --Trovatore (talk) 20:16, 16 June 2011 (UTC)
replies: text: It isn't really clear, the page for Almost all has two definitions, one which applies to normal numbers, one that does not. ReytheRed (talk) 22:19, 16 June 2011 (UTC)
replies: text: I've added an in-line explanation; feel free to tweak it. --Trovatore (talk) 22:28, 16 June 2011 (UTC)
thread_title:  Demonstration of uncountability of non-normal numbers 

message: A while back I nominated Lexicon (mathematics) on Wikipedia:Requested articles/mathematics, as there is an intriguing link on Mathworld Real Numbers: that almost all real numbers are lexicons, meaning that they do not obey the law of large numbers. I now think its refering to Normal numbers. Can anyone hear shed some light on the issue? --Salix alba (talk) 15:12, 7 March 2006 (UTC)
replies: text: Looks like a lexicon seems to be describing a normal number in base 2.  Looks that way to me anyway.  I've never heard of a lexicon in these terms before.  I suggest we merge the article here. Fieari 05:34, 11 February 2007 (UTC)
replies: text: A disjunctive sequence is a sequence in which every finite string appears, and as the article on normal numbers explains, these are not the same as normal sequences. According to the Wikipedia description of a lexicon, it is a real number whose base-k expansion, for every k \geq 2, is disjunctive. I have never heard of a lexicon, but if that is indeed the definition, then it is not difficult to construct a lexicon that is not normal in base 2. Fix a standard integer pairing function (j,k). Define the binary sequence S = s_1 0^{f(1)} s_2 0^{f(2)} \ldots  (and call the real number is represents r), where, for each i, f(i) is chosen large enough that S is not normal, and, if i = (j,k), then s_i is chosen to be the lexicographically first binary string that causes the jth k-ary string to appear in the base-k expansion of r. Then r is a lexicon but is not normal base 2. I will remove the suggestion to merge the articles. Pexatus 08:05, 11 February 2007 (UTC)

text: I went ahead and did the merge anyway. They're not the same, but they're so closely related that there's not much more to say about disjunctive sequences and lexicons than what we already say here. And the pre-merge lexicon article was not good: it didn't say anything about the connection to normal numbers, and gave credit to a paper in 1998 for discovering the concept and proving that almost every number is a lexicon, despite the existence of a much older literature on normal numbers. The easiest way to fix the problem seemed to be to add a little more here about the "lexicon" terminology and redirect the other article here. —David Eppstein (talk) 19:03, 23 April 2008 (UTC)
thread_title:  Lexicon (mathematics) 

message: If so amazingly few numbers have ever been shown to have this property, why of all the words that could have been chosen was "normal" used?  They seem pretty "abnormal" to me.  JackofOz 16:29, 19 May 2006 (UTC)
replies: text: Almost all numbers have the property. Few individual and interesting numbers have been shown to be normal, but 'random' sequences will have it. Charles Matthews 18:56, 19 May 2006 (UTC)
thread_title: Why "normal"?

message: "No rational number is normal to any base, since the digit sequences of rational numbers are eventually periodic."
Is this true? 123456789/9999999999 is definitely normal in base 10. --Hirak 99 10:10, 2 May 2007 (UTC)
replies: text: It's simply normal in base 10, but not simply normal in base 100, so not normal in base 10.  — Arthur Rubin |  (talk) 16:02, 2 May 2007 (UTC)
replies: text: Right, I got the definition wrong. Thanks a lot for clarifying. Please feel free to delete this section if you want, now that it's cleared up. --Hirak 99 19:43, 2 May 2007 (UTC)
thread_title:  Rationals 

message: From a recent post in sci.math:
 From:  "Dave L. Renfro" <renfr...@cmich.edu>
 Newsgroups: sci.math
 Subject: Re: Randomness of digits within pi
 Date: Fri, 09 Nov 2007 10:59:22 -0800
 
   "While a general proof can be given that "almost all"
   numbers are normal, this proof is not constructive ..."
 
 If anyone here edits these pages, you might want
 to revisit this statement. First of all, "this proof"
 seems inappropriate to me, since no specific
 proof had been singled out prior to this. Second,
 I strongly doubt the standard proof that one can find
 in Niven's "Irrational Numbers" is not constructive,
 or at least can't easily be reworded so that it is.
 In fact, the result itself is due to Borel, who
 had very strong constructivists leanings.
 
 Dave L. Renfro  —Preceding unsigned comment added by Loadmaster (talk • contribs) 19:26, 12 November 2007 (UTC)
thread_title:  General proof for almost all numbers 

message: Agafonov showed an early connection between finite-state machines and normal sequences: every subsequence selected from a normal sequence by a regular language is also normal. In other words, if one runs a finite-state machine on a normal sequence, where the finite-state machine outputs the digit it just read whenever it enters an accepting state, then the sequence it outputs will be normal. (Agafonov 1968)
:Am I missing something? Take the following finite automaton:
::state 1: input 0 -> go to 2, input 1 -> go to 1, initial, not accepting
::state 2: input 0 -> go to 2, input 1 -> go to 1, accepting
:Obviously, the machine will only output zeroes (and since the input sequence is normal, it will output an infinite number of zeroes), which is obviously not a normal sequence in base 2. - Mike Rosoft (talk) 22:43, 30 December 2007 (UTC)
replies: text: Hmm, you're right, this looks wrong as stated. Maybe it's the digit the machine is about to read -- that would make sense. --Trovatore (talk) 02:16, 31 December 2007 (UTC)
replies: text: Good catch. Pexatus (talk) 07:51, 15 February 2008 (UTC)
replies: text: I assume this has been fixed? --Lendorien (talk) 14:24, 24 March 2009 (UTC)
thread_title: Connection to finite-state machines

message: I'm sorry to put my request in here, but since I'm not so familiar with the cite-template I can't put it on the page on my own:
Could anyone, who is better in wiki-markup than me, please add this to the refferences?
http://crd.lbl.gov/~dhbailey/dhbpapers/bcnormal.pdf (The authors are on the title-page)
(I think this article might be interesting)  —Preceding unsigned comment added by 80.108.153.97 (talk) 16:24, 2 January 2008 (UTC)
thread_title:  Further reading 

message: The article states that normal numbers have an even distribution of digits in "every base", but perhaps this should read "every rational base"? I mean, for any normal number x, it is exactly 10 in base x, right? 219.78.90.231 (talk) 16:19, 8 March 2008 (UTC)
thread_title:  "Every Base"? 

message: I changed
:A normal number in base r is normal in base s if and only if log r / log s is a rational number. (Schmidt 1960)
to
:A normal number in base r is normal in base s if log r / log s is a rational number. (Schmidt 1960)
The original wording couldn't be right: Chaitin's constant is 2-normal and 3-normal, but log 2/log 3 is irrational.  Can anyone look this up and see what the original has?
CRGreathouse (t | c) 03:17, 28 July 2008 (UTC)
thread_title:  Mistake re: Schmidt 1960? 

message: The last example of the properties and examples section is nonsensical. 
I cannot extract a formulation of Theorem 1.1 of the cited paper which resembles what it says here. Maybe the correct statement is taken from a later version of the same paper? It obviously does need to be corrected. Kluto (talk) 11:52, 29 October 2008 (UTC)
thread_title:  Bailey & Misiurewicz hot spots 

message: What does # mean in a mathematical expression? Is this standard at all? 24.28.82.87 (talk) 03:25, 31 May 2009 (UTC)
replies: text: It depends on the context, I'm sure. I've sometimes seen it used to denote the cardinality of a set; that is, #S is the same quantity more commonly written as |S|. But I don't see a # in this article; where did you find it? —David Eppstein (talk) 04:21, 31 May 2009 (UTC)
replies: text: It's at the very bottom of the "Properties and examples" section, the reference says "# denotes count", so thats a definition, but it seems to me that that bullet point in the section requires some additional explanation. Maybe if I knew a little more about normal numbers, that equation would be immediately meaningful to me, but I don't, and it isn't. 24.28.82.87 (talk) 20:45, 31 May 2009 (UTC)
replies: text: Ok, I've added an explanation. —David Eppstein (talk) 21:09, 31 May 2009 (UTC)
replies: text: Well, what are x and y? Out of context, this equation means nothing to me. Maybe I'm just stupid though, who knows. 24.28.82.87 (talk) 23:31, 31 May 2009 (UTC)
replies: text: I'm pretty sure that x is the number whose normality is in question. But I don't know what y is, and looking at the given reference I didn't see anything that would count as an explanation: the only statement I can see that has similar nested lim inf and lim sup is Theorem 3.5, but the rest of the expression is different. Also, the condition that it's normal when the nested lim is infinite is exactly backwards from the condition in Theorem 3.5. I've just removed the expression altogether. If it was important, someone else can re-add this material in a more understandable way. —David Eppstein (talk) 23:50, 31 May 2009 (UTC)
thread_title:  # 

message: I don't want to come across as hardheaded on this usage — it's not familiar to me, but this is not really my field, so I don't really know how much workers in the field use the phrase absolutely normal number.  All I can say is that, in my limited experience, I haven't seen it much.
But I do think that we need to use it (as opposed to mention it) either always or never, at least in this one article.  When there are some occurrences of normal and others of absolutely normal, the casual reader can be forgiven for thinking that these are two different notions, and that the second one is stronger.  --Trovatore (talk) 23:24, 4 March 2010 (UTC)
thread_title:  absolutely normal number 

message: The term "b-dense" is used three times, with no link to a definition. Furthermore, googling the term returns nothing relevant at first glance. The term is used with no introduction whatsoever.
Furthermore, the entire b-dense paragraph (starting "A disjunctive sequence is a sequence...") seems to just go off on a random tangential diversion from the original train of thought. Why not just delete that paragraph in its entirety? It doesn't seem to contribute anything.  —Preceding unsigned comment added by 207.22.18.83 (talk) 00:21, 16 April 2010 (UTC)
thread_title:  b-dense and such 

message: Does a "normal number" contain every possible number of finite length?  If so that answer should probably be put in here.  Pi is often used as an example by pirates as a number that contains copywritten data, but I want to know what mathematicians have to say about that.
Citizen Premier (talk) 03:53, 24 November 2010 (UTC)
replies: text: Yes, it's a fairly easy consequence of the definition that the base-b representation of a normal number contains every possible finite string of base-b digits.  Maybe worth mentioning, not sure.  Would be nice to find it in a source; otherwise it's technically original research, though not the sort that anyone's likely to challenge. --Trovatore (talk) 04:23, 24 November 2010 (UTC)
thread_title:  I don't get anything on this page, I just came here to answer one question... 

message: The article says:
:only very few specific numbers have been shown to be normal
Surely this is not correct, since it is quite easy to construct many normal numbers by constructions analogous to the one that produces Champernowne's constant. I suspect what was meant here something like: very few numbers with any unrelated interesting properties have been shown to be normal.  —Mark Dominus (talk) 16:51, 16 September 2012 (UTC)
replies: text: The article defines "normal" to mean normal in every base. Is Champernowne's constant really normal in every base, or only in base 10? The article seems to suggest it is known only that it is normal in base 10. —Bkell (talk) 18:41, 16 September 2012 (UTC)
replies: text:  It seems to me that this is a problem throughout the whole article.  When the term absolutely normal is used in this article, it always means "normal in every base".  When the term normal in base b  is used, it always means "normal in base b".  But when the term normal is used, it seems to go back and forth.  It would be nice for each instance to be verified. — Lawrence King (talk) 01:51, 15 March 2015 (UTC)
thread_title:  "Very few specific numbers have been shown to be normal" 

message: Can we please not have this template here?  It hasn't been used here before.  Let's treat this like a WP:RETAIN instance. --Trovatore (talk) 18:35, 1 October 2012 (UTC)
replies: text: I reverted your edit except its only part which was substantiated: removal of . I know that this was rude, but discarding a legitimate improvement, namely , by your edit, was silly. If you want to know, why I behave rude and do attack (you), then ask on my user_talk, please. Incnis Mrsi (talk) 18:44, 1 October 2012 (UTC)
thread_title:  {{pi}} 

message: The paragraph Liouville numbers are abnormal in base b, in which base the number was constructed. was removed as "remove incorrect statement".  That's not quite correct, although one has to be careful about exactly what one means by Liouville number.  If we use the definition of a Liouville constant in base b as \sum_n b^{-n!}, it is visibly not normal: indeed, the sequence of digits 111 never occurs.  However, taking the more general definition of a number of infinite irrationality exponent, then it is indeed possible for a Liouville number to be normal: see for example   Deltahedron (talk) 18:05, 16 January 2013 (UTC)
thread_title:  Liouville numbers 

message: This statement seems really implausible. Isn't 1/3 = .010101.... simply normal in base 2 but not rich? It doesn't contain '11', for example.
Every normal number is certainly rich in the corresponding base, but simple normality doesn't seem strong enough.  — Preceding unsigned comment added by 138.16.16.185 (talk) 23:15, 16 March 2013 (UTC)
replies: text: I don't have access to the reference at the moment, but I think this comment is correct.  Deltahedron (talk) 07:37, 17 March 2013 (UTC)
thread_title:  "Every simply normal number is rich in the corresponding base" 

message: The article seems to say that Becher and Figueira give a proof of the existence of computable absolutely normal numbers without providing a means to calculate any such number, but looking at the linked paper they seem to give an explicit algorithm for calculating the number? Unless the article merely means to say that they don’t go on to specify the output of the algorithm, which is perhaps not a very practical one in terms of computation time. If the latter is what is meant it should be rephrased to say so more clearly.136.25.104.172 (talk) 02:56, 28 July 2021 (UTC)
thread_title:  Unclear statement regarding existence of computable examples 

message: OK, this is very much a POV thing, but surely, surely, surely, it MUST be mentioned somewhere about the incident of "Engraved Pi" on the second last page of  Carl Sagan's fabulous epic work "Contact"?
If you havent read the book, then yeah, that sentence will make no sense. Solution: Read the book!  — Preceding unsigned comment added by 2001:8003:E422:3C01:753A:CACA:A4C6:DA18 (talk) 12:01, 27 December 2021 (UTC)
thread_title:  Crying out for one example "In Fiction". 

message: I'm not sure what it even means to have a base for n < 2 or for non-integer bases.  How would one even define base 1 or base -23 or base 7/16 or base pi?
Maybe I'm missing something, but simply saying "A number is said to be normal  if it is normal in all bases." would seem to be sufficient. Mr. Swordfish (talk) 02:04, 28 June 2023 (UTC)
replies: text: See for example unary numeral system and non-integer base of numeration wow, that's an incredibly awkward title for the latter article --Trovatore (talk) 02:28, 28 June 2023 (UTC)
thread_title:  A number is said to be normal ... if it is normal in all integer bases greater than or equal to 2 

message: 666 3ékhjàß fiican bampvb Xasan ā@ 105.27.124.166 (talk) 19:06, 24 September 2023 (UTC)
thread_title:  Muqdisho 666 

message: There is Normal number (computing), and both seem important enough that none should take precedence and Normal number be rather a disambiguation page. Musaran (talk) 20:06, 21 October 2023 (UTC)
thread_title:  Rename to "Normal number (mathematics)"?