Talk:Repeating decimal

message: This section is just too long winding and may be unnecessary. To find the value of 0.999..., all one need to ask is what is 1 - 0.999...
The answer is a digit 1 at an infinite decimal position. This equates it with zero! We only need two-line statement! I request to simplify this section for the sake of the general public. --Ling Kah Jai (talk) 05:07, 8 May 2009 (UTC)
replies: text: The topic of 0.999... is dealt with in much more detail in the article 0.999..., and is only slightly connected to repeating decimals. I would be happy to see this whole section removed, and to just have a link to 0.999... in the See also section. Gandalf61 (talk) 08:32, 8 May 2009 (UTC)

text: I was going to make the same suggestion.  —Dominus (talk) 14:00, 8 May 2009 (UTC)
replies: text: No objections, so I have removed the section and added a link to 0.999... in See also. Gandalf61 (talk) 11:52, 16 May 2009 (UTC)

text: Ok. The term is neologism for naming the headings of two sections. Cyclic numbers have been defined and thus it is easy to understand what are non-cyclic numbers. What do you suggest for the section heading if semi-cyclic is not used to show that "these numbers behave somewhere between cyclic and non-cyclic"? Thank you. --Ling Kah Jai (talk) 04:24, 11 May 2009 (UTC)
replies: text: I would have said that a "non-cyclic" number is any number that is not a cyclic number, but I think you are using that term in a different, more restricted, sense. What is not clear to me is the difference in your terminology between "semi-cyclic" and "non-cyclic" - what does "behave somewhere between cyclic and non-cyclic" mean ? If "1/11 and 1/3 are almost non-cyclic" then are they semi-cyclic ? Or are they non-cyclic ? If they are semi-cyclic, then can you give an example of a "non-cyclic" number in your terminology. Gandalf61 (talk) 06:27, 11 May 2009 (UTC)
replies: text:  I refer "non-cyclic" number to any number that does not possess any cyclic characteristic in multiplication. --Ling Kah Jai (talk) 10:52, 12 May 2009 (UTC)
replies: text: But what exactly do you mean by "semi-cyclic" ? You still have not explained this clearly. Your phrase "behave somewhere between cyclic and non-cyclic" is too vague. Let's take some examples. Which of the following numbers are "semi-cyclic" and which are "non-cyclic": 015873 ? 153846 ? 123456 ? 253968 ? Gandalf61 (talk) 11:52, 12 May 2009 (UTC)
replies: text: Number 015873 is derived from 1/63 and is semi-cyclic (relates with 5 others such as 10/63, 37/63, 55/63 etc). Number 153846 is from 2/13 and in the article I mentioned them to be semi-cyclic (relates with 5 other fractions). I don't know 123456 has any cyclic behavior. Number 253968 derived from 16/63 and relates with 5 other fractions (such as 34/63) having the same denominator so it is semi-cyclic.  I believe I have answered you the question. I rather let the terms be a bit vague as I am using "non-cyclic" and "semi-cyclic" as adjectives. If instead I define "non-cyclic numbers" and "semi-cyclic numbers", the definition certainly falls into the category of "new research". Then everybody has right to delete it. As it is how could people deprive my right to use these two adjectives? --Ling Kah Jai (talk) 13:58, 15 May 2009 (UTC)

text:  Every body can use an adjective like red without actual comparing the colour with certain colour chart or standard. Everybody can use the adjectives high, tall, fat as relative terms without defining them exactly. Do you think my statement is correct?  --Ling Kah Jai (talk) 14:23, 15 May 2009 (UTC)
replies: text: No, sorry, your argument does not hold. "Red", "high", "tall" and "fat" are commonly understood adjectives; they are part of everyday language and appear in a dictionary. You, however, have introduced an apparently technical, "semi-cyclic", which does not have a dictionary definition, and you cannot provide a reliable source to show that you are using that term in the correct sense. Since you admit that "semi-cyclic" is a neologism, and you are unable to give a precise definition of its meaning, it has no place in an encyclopedia article - it will only confuse readers and detract from the clarity of the article (see Wikipedia:Avoid neologisms). I have therefore removed the term "semi-cyclic" from the article. Gandalf61 (talk) 11:47, 16 May 2009 (UTC)
thread_title:  Request to simplify '''''The case of 0.999...''''' 

message: Gandalf61, I do not think that the heading non-cyclic numbers is appropriate when you refer them to include:
* numbers that have some cyclic behaviour (which I referred to as semi-cyclic); and
* usual numbers that do not have any cyclic behaviour.
This casual use of terms causes more confusion.--Ling Kah Jai (talk) 13:32, 16 May 2009 (UTC)
thread_title:  Non-cyclic number 

message: Gandalf61, except for the term and the section heading which I am thinking of changing, I intend to add back this section as the analysis and solution is much simpler than the version by the originator in the web. I am asking for your opinion since you deleted it. Thank you. --Ling Kah Jai (talk) 14:18, 16 May 2009 (UTC)
replies: text: Wikipedia is not a place to publish your own work - see WP:NOT. You will have to find a reliable source that uses the term "Reaman numeral" in the sense that you are using it, and derives the results that you are presenting. Without a reliable source, your section is original research, and does not belong in a Wikipedia article. And the puzzle page that is linked from Talk:Cyclic number is definitely not a reliable source. Gandalf61 (talk) 17:23, 16 May 2009 (UTC)
replies: text:  I meant I will use different term,e.g. "math puzzle" or something similar. My section is already a derivation! --Ling Kah Jai (talk) 02:52, 18 May 2009 (UTC)
replies: text: Wikipedia already has an article on these numbers - they are called parasitic numbers. A link to that article would be fine, since parasitic numbers can be derived from repeating decimnals, but there is no need to duplicate the whole explanation here. Gandalf61 (talk) 10:47, 18 May 2009 (UTC)
thread_title:  Reaman numeral 

message: Gandalf61, If I were to define the above terms, I will choose:
Semi-cyclic
Semi-cyclic numbers are represented by decimals derived from \frac{c}{p^k q^l r^m ... } excluding those classified as cyclic numbers.
Where  and etc are prime other than 2 and 5, and  and etc are positive integers or zeros (at least one of them shall be > 0).
Non-cyclic 
Non-cyclic numbers are any other numerals besides cyclic numbers and semi-cyclic numbers, e.g.,
*Terminating decimals, 
*Decimals derived from \frac{c}{2^a 5^b p^k q^l ...} excluding those classified as cyclic and semi-cyclic numbers.
This definition will then be in line with the article.
--Ling Kah Jai (talk) 01:39, 18 May 2009 (UTC)
replies: text: So then every string of digits is either cyclic or semi-cyclic - because every string of digits forms the repeating decimal part of some rational number. For example, 123456 is the repeating decimal part of 41152/333333. Gandalf61 (talk) 11:29, 18 May 2009 (UTC)
replies: text:  As you have pointed out, I will say yes, as the fractions 78187/333333, 115204/333333, 152041/333333 (and two others) can be expressed as decimals having the cyclic permutation of 123456. However, there are still non-cyclic numbers as defined above, such as terminating decimals and repeating decimals with preceding non-repeating digits - I use the full string of digits. --Ling Kah Jai (talk) 00:21, 19 May 2009 (UTC)

text:  I see what you mean, the issue can be resolved if the definition is based on full series of the decimal instead of the selected fixed digits only. --Ling Kah Jai (talk) 07:24, 19 May 2009 (UTC)
thread_title:  Semi-cyclic and non cyclic numbers 

message: Gandalf61, I can understand why you removed:
*Explanation of cyclic behavior,
*Arithmetic proof of divisibility
However I think the section  Connection with Fermat's little theorem shall be maintained because the connection may not be apparent to everybody. At one time somebody deleted my statement that the period is related to FLT. --Ling Kah Jai (talk) 04:32, 18 May 2009 (UTC)
replies: text: There is no need to duplicate information that the reader can get from the articles on Fermat's little theorem or cyclic number. There are links to both of those articles on the repeating decimal page - that is sufficient. Gandalf61 (talk) 11:36, 18 May 2009 (UTC)
replies: text:  Both articles do no mention relationship of FLT with cyclic numbers. --Ling Kah Jai (talk) 01:20, 19 May 2009 (UTC)

text:  Since there is no response on this subject, I am putting this section back.--Ling Kah Jai (talk) 15:14, 9 September 2009 (UTC)
replies: text: Under "Fractions with prime denominators", the article already says "The period of the repeating decimal of  is equal to the order of 10 modulo p. If 10 is a primitive root modulo p, the period is equal to p−1; if not, the period is a factor of p−1. This result can be derived from group theory or from Fermat's little theorem." Your section "Connection with Fermat's little theorem" is a series of examples that adds nothing to this, so I have removed it again. Gandalf61 (talk) 12:22, 10 September 2009 (UTC)
replies: text: My question: Is encyclopaedia meant for general public or mathematicians only? Do you think a illustration is better for public?--Ling Kah Jai (talk) 12:53, 10 September 2009 (UTC)
replies: text: Plainly stating your point of view is better than asking empty rhetorical questions. This article already contains plenty of examples of repeating decimals. If you want to add examples of Fermat's little theorem then these belong in the Fermat's little theorem article, not in this article. Gandalf61 (talk) 13:05, 10 September 2009 (UTC)
thread_title:  Removal of '''Connection with Fermat's little theorem''' 

message: I have completed an article with the above title currently kept at User:Ling_Kah_Jai/Theorem_of_repeating_decimal. I intend to:
*add this article to Wikipedia or Wikisource; and
*add a summary to Repeating decimal.
I was inspired by Gandalf61 and motivated by Arthur Rubin to write this article. In essence, the material is nothing new (original research) but merely re-organize information in a more easily understood manner. Any opinion against my putting up of this article in Wikipedia? --Ling Kah Jai (talk) 12:13, 23 September 2009 (UTC)
replies: text: Ling Kah Jai, I do not have the patience to find a nice way to say this, so I am just going to say it. Your proposed article is badly written, poorly organised, verbose and difficult to understand. More importantly, unless you can find a source that refers to this particular set of arithmetic facts as the "Theorem of repeating decimal" then by Wikipedia's definition it most definitely is original research and it does not belong on Wikipedia. Wikibooks may be a better home for your writing. Gandalf61 (talk) 12:35, 23 September 2009 (UTC)
replies: text: Gandalf61, is it really that bad that you cannot understand it? Can you suggest a way for me to improve it before I consider putting up to Wikisource or Wikibooks? --Ling Kah Jai (talk) 12:49, 23 September 2009 (UTC)
replies: text: I am not a Wikibooks expert (Wikisource seems inappropriate).  But it's not the best.  — Arthur Rubin  (talk) 16:01, 23 September 2009 (UTC)
thread_title:  Theorem of repeating decimal 

message: Gandalf61, like you say, these are mere arithmetic facts. If you are against the term, I have changed it to the new title as above or just can include it to Cyclic permutation of integer without any new title. Is this an acceptable article? If the writing / organization is bad but there is substance in it then perhaps other users can improve it. --Ling Kah Jai (talk) 08:07, 26 September 2009 (UTC)
replies: text: No. Why should you expect other editors to improve your bad writing and lack of clarity ? You have shown repeatedly that cannot work in the collaborative way that is fundamental to Wikipedia. You are creating articles that are unsourced and full of repetitive examples and trivial arithmetic. Your work is not suitable for Wikipedia. Please take it somewhere else. Gandalf61 (talk) 10:40, 26 September 2009 (UTC)
replies: text: I think that my writing has the clarity. On the other hand, it is not true that I did not work collaboratively. For discussion between us, how many times I respected your decision and how many times I did it my own way? --Ling Kah Jai (talk) 01:14, 28 September 2009 (UTC)
thread_title:  Phenomenon of repeating decimal 

message: Arthur Rubin, I don't remember who added in the group theory statement. It could be me or it could be others. Anyway, I support the statement.
For a multiplicative group modulo p (p is prime number other than 2 or 5): (Zp, *), a cyclic group / subgroup can be generated by 10. What is the order of the subgroup? At most (p -1). So do you think group theory can derive the answer? --Ling Kah Jai (talk) 01:32, 14 October 2009 (UTC)
replies: text: Wikipedia doesn't seem to have it; the relevant theory is the abstract theory of functions of one variable.  Much of it can be simulated by group theory.  — Arthur Rubin  (talk) 17:24, 14 October 2009 (UTC)
replies: text: I am giving up. I cannot progress with you. --Ling Kah Jai (talk)01:18, 15 October 2009 (UTC)
thread_title:  Group theory 

message: Section Repeating decimal#Reciprocals of integers not co-prime to 10 tries to address this question, but it seems (to me) to miss the essential issue. Suppose 0≤n<d are integers co-prime to each other and d is co-prime to 10 and
:{n \over d} = 0.q_1q_2q_3q_4\ldots \,
is the decimal expansion, then the remainders in the long division are defined by
:r_0 = n \,
:r_{k+1} = 10 \times r_k - q_{k+1} \times d \,.
One gets
:0 \leq r_k < d \,
:0 \leq q_k < 10 \,.
Now we know that for some distinct j and k we must have
:r_{j+1} = r_{k+1} \,
because there are only finitely many integers in [0,d). So we get
:10 \times r_j - q_{j+1} \times d = 10 \times r_k - q_{k+1} \times d \,
:10 \times (r_j - r_k) = (q_{j+1} - q_{k+1}) \times d \,.
Assume without loss of generality that
:r_j \geq r_k \,
(otherwise switch j and k), then
:(r_j - r_k) \geq 0 \,
:(q_{j+1} - q_{k+1}) \geq 0 \,.
Since 2 and 5 do not divide d, we get that 10 divides qj+1-qk+1 which, given that they lie in [0,10), is only possible if
:q_{j+1} = q_{k+1} \,
:10 \times (r_j - r_k) = (q_{j+1} - q_{k+1}) \times d = 0 \,
and thus
:r_j = r_k \,.
So we can work backwards inductively to show that all the remainders and digits separated by j-k decimal places must be the same, until we reach the decimal point where the definitions upon which the argument depends begin. JRSpriggs (talk) 08:51, 22 January 2011 (UTC)
thread_title:  Why does repetition begins where it does? 

message: Repeating decimal is a way of representing rational numbers.  The best proof of this is that repeating decimals have already appeared in the mathematical literature in the early 18th century.  The elegant theorem that a real number is rational if and only if it its decimal representation is eventually repeating is important, but there is no reason for the lede to start with the real numbers.  Tkuvho (talk) 14:11, 1 January 2012 (UTC)
thread_title:  The lede should be about rationals, not reals 

message: Who is Beswick?  Tkuvho (talk) 17:34, 1 January 2012 (UTC)
replies: text: The Beswick reference for "modified long division" is not covered in mathscinet.  I would suggest deleting it.  Tkuvho (talk) 17:38, 1 January 2012 (UTC)
replies: text: Beswick's procedure sounds pretty contrived as well as non-notable.  I have a simpler procedure for getting a tail of 9s: divide the number by 3 by using long division.  Then multiply back by 3, to get a tail of nines.  Tkuvho (talk) 17:51, 1 January 2012 (UTC)
replies: text:  Beswick is an educator and the article is a published in an education journal, which is why it is not on mathscinet. Thenub314 (talk) 19:26, 1 January 2012 (UTC)
replies: text: In the past 7 years, Beswick's text earned precisely one cite at Google Scholar.  I don't think it is notable.  It should be deleted from both pages.  Tkuvho (talk) 19:33, 1 January 2012 (UTC)
thread_title:  Beswick 

message: Hi. What means the period of the repeating decimal ? Is it a length of repeating sequence ? TIA --Adam majewski (talk) 17:25, 13 October 2013 (UTC)
replies: text: Yes.  — Preceding unsigned comment added by 212.159.119.123 (talk) 14:52, 7 October 2016 (UTC)
thread_title:  Period 

message: I was taught the dot notation at school in the UK by an Englishman who has probably never been to China. Why does Wikipaedia say it's a Chinese thing? 46.65.41.135 (talk) 16:53, 3 February 2014 (UTC)
thread_title:  Dot notation in the UK 

message: Under "Fractions with prime denominators", an equals sign ought to be congruence sign.  — Preceding unsigned comment added by 212.159.119.123 (talk) 11:30, 5 September 2016 (UTC)
replies: text: Fixed. Bill Cherowitzo  (talk) 15:35, 5 September 2016 (UTC)
thread_title:  Note 

message: Wikipedia is supposed to be accessible to the layperson.  I know that all math is not going to be that way, but usually when a variable is introduced we are given what those variables stand for.  What do the variables in the formula below stand for and can someone perhaps also give a simple example of its use here in the Talk section?  I think including what the variables stand for would be a nice thing to have in the main article.  An example may or may not be warranted in the main article, but I'd like to see that here, if at all possible.  Thanks.
Here's the Applications to Cryptography section as it appears currently:
Repeating decimals (also called decimal sequences) have found cryptographic and error-correction coding applications.[10] In these applications repeating decimals to base 2 are generally used which gives rise to binary sequences. The maximum length binary sequence for 1/p (when 2 is a primitive root of p) is given by:[11]
a ( i ) = 2 i     mod p     mod 2     {\displaystyle a(i)=2^{i}~{\bmod {p}}~{\bmod {2}}}  a(i)=2^{i}~{\bmod {p}}~{\bmod {2}}
These sequences of period p-1 have an autocorrelation function that has a negative peak of -1 for shift of (p-1)/2. The randomness of these sequences has been examined by diehard tests.[12]  — Preceding unsigned comment added by 66.104.142.198 (talk) 03:58, 9 October 2017 (UTC)
replies: text: You have a very valid point. I am familiar with the terms and the general area, but I still can't parse this so that it makes any sense. This section is being supported by two papers by the same author and an arXiv paper by someone else. These are primary sources and an unreliable source, so verifiability of this section is a problem. On those grounds I believe that the section could be removed and if some editor wanted to replace it they could be held accountable for increased clarity of exposition. (I realize that I am taking the easy way out by not tracking down and reading these papers myself, to find out what should be said about this, but an editor should not have to do that to improve an article.) --Bill Cherowitzo  (talk) 04:37, 9 October 2017 (UTC)
thread_title:  Applications to cryptography formula 

message: The article currently asserts:
:A repeating decimal or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero.
That's just not right.  I wouldn't be surprised if there's a source that says that somewhere, but it's still pretty silly.  There's nothing fundamentally different about repeating decimals whose digits are eventually all 0.  Separating out the "terminating" expansions is confusing and misleading.  All real numbers have infinitely long decimal expansions; it's just that sometimes it reaches a point where it repeats 0 forever.  We should really fix this.  I don't specifically know where to find sources. --Trovatore (talk) 04:31, 3 May 2023 (UTC)
thread_title:  "not zero" 

message: The article claims that the parentheses notation is used, amongst other countries, in Austria. I'm an Austrian native and I've never heard of this notation. In school and university we learned and used exclusively the vinculum notation. That claim seems to be a mistake. I'm happy to correct the article unless there are any objections or, the current authors prefer to make the modification themselves. DRappaport (talk) 08:00, 26 April 2024 (UTC)
thread_title:  Parentheses notation in Austria