Talk:Repeating decimal

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Request to simplify The case of 0.999...
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)


 * 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)


 * I was going to make the same suggestion. —Dominus (talk) 14:00, 8 May 2009 (UTC)


 * 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)

2607:FCC8:F4C2:7500:AC93:B28:BF40:52CA (talk) 19:55, 30 April 2018 (UTC)== Semi-cyclic number ==

The article uses the term "semi-cyclic" several times: Semi-cyclic number redirects to a section of this article. I did a quick Google search but couldn't find any useful references for this terminology. Can someone please give a definition of "semi-cyclic number" and a reliable source to show that this is not a neologism. Thank you. Gandalf61 (talk) 10:08, 8 May 2009 (UTC)
 * 1) "The number 076923 is thus semi-cyclic ; whereas the decimals of 1/11 and 1/3 are almost non-cyclic"
 * 2) "The following is an example of semi-cyclic number: $1/49$ = ..."
 * 3) "Thus $1/119$ has 96 repeating digits and is a semi-cyclic number"


 * 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)


 * 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)


 * 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)


 * 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)


 * 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)


 * 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)


 * 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 Avoid neologisms). I have therefore removed the term "semi-cyclic" from the article. Gandalf61 (talk) 11:47, 16 May 2009 (UTC)

How do I type a repeating decimal? 2607:FCC8:F4C2:7500:AC93:B28:BF40:52CA (talk) 19:55, 30 April 2018 (UTC) EvieSwan2405

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

Reaman numeral
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)


 * 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)


 * 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)


 * 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)

Semi-cyclic and non cyclic numbers
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 $p, q, r$ and etc are prime other than 2 and 5, and $k, l, m$ 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)

In English term, semi-cyclic numbers are not cyclic numbers but possess some aspect of cyclic behaviour. I believe this is quite a straight forward explanation. --Ling Kah Jai (talk) 02:48, 18 May 2009 (UTC)


 * 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)


 * 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)


 * 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)

Removal of Connection with Fermat's little theorem
Gandalf61, I can understand why you removed: 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)
 * Explanation of cyclic behavior,
 * Arithmetic proof of divisibility


 * 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)


 * Both articles do no mention relationship of FLT with cyclic numbers. --Ling Kah Jai (talk) 01:20, 19 May 2009 (UTC)


 * Since there is no response on this subject, I am putting this section back.--Ling Kah Jai (talk) 15:14, 9 September 2009 (UTC)


 * Under "Fractions with prime denominators", the article already says "The period of the repeating decimal of $1/p$ is equal to the order of 10 modulo p. If 10 is a primitive root modulo p, the period is equal to p&minus;1; if not, the period is a factor of p&minus;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)


 * 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)


 * 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)

Parasitic number
Arthur, what do you mean by unencyclopedic? I rather the section stay and the link at #See also's be deleted. I wish you can explain. --Ling Kah Jai (talk) 15:42, 10 September 2009 (UTC)
 * What can I say: "Wonders of ..." is one of the best WP:PEACOCK words I've ever seen. The links are arguable, but the section has no material which should be in any Wikipedia article which isn't already in cyclic number or parasitic number.  — Arthur Rubin  (talk) 15:49, 10 September 2009 (UTC)


 * If you do not agree with the heading, just change the heading. Wikipedia encourages links. What I am doing is adding a little bit introduction to the parasitic number which I think is proper. You act is Vandalism.


 * The links are not arguable. When I first have a longer, more detailed section for parasitic number in this article but named after something else, it was deleted by User:Gandalf61. He reminded me the existence of parasitic number. When I wrote the summary for parasitic number in this article, obviously User:Gandalf61 saw it as he deleted my the other sub-section but approved this section or otherwise he would have deleted it as well. You may ask him whether the link is appropriate before deleting it. You simply cannot accept a statement of truth that I wrote here. --Ling Kah Jai (talk) 17:37, 10 September 2009 (UTC)


 * The links are arguable, as they only relate indirectly through other articles. I was wrong to call them vandalism, if I did, but not wrong to call them inappropriate Wikilinks.  — Arthur Rubin  (talk) 18:26, 10 September 2009 (UTC)


 * Ling Kah Jai - for the record, I haven't expressed any view about your "Wonders of numbers: parasitic numbers" section, and I strongly object to your attempt to predict my opinions. Gandalf61 (talk) 20:29, 10 September 2009 (UTC)


 * I am pasting the section below for everyone consensus. The section is original saved I changed the section heading because of the comment. Let others have their comment. If Arthur does not allow other people to edit parasitic number, please say it loud and clear.

Parasitic numbers
The interesting cyclic behavior of repeating decimals in multiplication also leads to the construction of parasitic number. When a parasitic number is multiplied by n, not only it exhibits the cyclic behavior but the permutation is such that the last digit of the parasitic number now becomes the first digit of the multiple. For example, 102564 x 4 = 410156. Note that 102564 is the repeating digits of $4/39$ and 410156 the repeating digits of $16/39$.

--Ling Kah Jai (talk) 04:21, 11 September 2009 (UTC)


 * I am also pasting link to what I have amended for parasitic number:
 * http://en.wikipedia.org/w/index.php?title=Talk:Repeating_decimal&action=edit&section=24
 * http://en.wikipedia.org/w/index.php?title=Parasitic_number&oldid=312996712
 * Everybody can review and make a comment. I added section heading, wrote the new section 'simplified approach', reorganized the section 'general note' and added links. Arthur just undid everything. --Ling Kah Jai (talk) 04:32, 11 September 2009 (UTC)--Ling Kah Jai (talk) 04:35, 11 September 2009 (UTC)
 * Set aside the formula for the Ling Kah Jai (talk) 04:53, 11 September 2009 (UTC)


 * If no body object to the re-inclusion of the above, I will include it in one form or another.--Ling Kah Jai (talk) 04:53, 12 September 2009 (UTC)
 * I still don't see the need for a more complicated less mathematical alternative approach. — Arthur Rubin  (talk) 08:56, 12 September 2009 (UTC)
 * Did you read User:Ling Kah Jai/Other cyclic permutations? Still not convinced? Sigh! Where are other people's opinions? --Ling Kah Jai (talk) 02:44, 14 September 2009 (UTC)

Cyclic number
The {main|Repeating decimal} has been kept at cyclic number article for four months and has met many users' approval. In fact, the cyclic numbers were brought out and described in this article though without a proper heading long ago before other users created the article cyclic number. I think you have put in hard work for parasitic number and does not wish to see a simple approach proposed by other. What do I call that? --Ling Kah Jai (talk) 17:03, 10 September 2009 (UTC)--Ling Kah Jai (talk) 16:38, 10 September 2009 (UTC)
 * Wrong location for comment; please comment at Talk:Cyclic number, if at all. — Arthur Rubin  (talk) 17:29, 10 September 2009 (UTC)
 * Don't avoid my comment. You cannot answer my query then do not delete them! I accept changes but not deletion without reasons / pointing the mistakes and vandalism. --Ling Kah Jai (talk) 17:40, 10 September 2009 (UTC)


 * Stop your bogus accusations of other users of vandalism, it is considered very rude. Do yourself a favour and read WP:VAN to see what is and is not vandalism, in particular the paragraph starting with "Any good-faith effort to improve the encyclopedia, even if misguided or ill-considered, is not vandalism." — Emil J. 17:47, 10 September 2009 (UTC)


 * I doubt Arthur Rubin have read repeating decimal when he starts to delete links to the article. --Ling Kah Jai (talk) 18:05, 10 September 2009 (UTC) He followed me everywhere and delete my work giving pretext but not acceptable reasons. --Ling Kah Jai (talk) 18:07, 10 September 2009 (UTC)


 * It appears I didn't delete the links, although I intended to. You had just reordered the links, rather than adding them.  — Arthur Rubin  (talk) 23:07, 10 September 2009 (UTC)


 * I let you check for yourself whether you deleted the link: http://en.wikipedia.org/w/index.php?title=Repeating_decimal&diff=313143344&oldid=313011276
 * Look at the category of parasitic number. Cold category. Few people will realize there is such an article. Your intention is to keep this piece of article as an island. No links, less people access it, and you can keep people out from editing it. But no man is an island, so does knowledge! Knowledge is linked information. --Ling Kah Jai (talk) 06:40, 11 September 2009 (UTC)
 * OOPS, I only intended to delete the cryptography one. Fixed.  — Arthur Rubin  (talk) 07:18, 11 September 2009 (UTC)
 * You may claim to be an accident if you delete it once. But I remember you have deleted the link to parasitic number at least in another occasion. --Ling Kah Jai (talk) 09:02, 11 September 2009 (UTC)
 * I just had the statistics: repeating decimal and parasitic number were viewed respectively 4473 times and 324 times in the month of August 2009. Check it out at http://stats.grok.se/ --Ling Kah Jai (talk) 06:30, 12 September 2009 (UTC)--Ling Kah Jai (talk) 06:33, 12 September 2009 (UTC)

Other cyclic permutations
I am going to add this very long section, which is made possible by the method of simplified approach that was documented as parasitic number but was deleted by Arthur Rubin. This section is only complete if the short summary section, repeating decimal which I wrote but deleted by Arthur Rubin, is reverted back to this article. Please review whether I shall post it here or elsewhere: Other cyclic permutations Let Arthur Rubin see for himself: --Ling Kah Jai (talk) 11:25, 11 September 2009 (UTC)
 * if the so called direct method can do this or not; and
 * whether parasitic number belongs to a subtopic of repeating decimal.


 * The style in which you have written this proposed section at User:Ling Kah Jai/Other cyclic permutations is not encyclopedic and is very unclear. There are simples rule for finding numbers that give multiples of themselves when cyclically shifted right or left by any given number of places. If you want to include this information in Wikipedia you need to:
 * Make sure it is not already in an article.
 * Find a reliable source that describes the rules.
 * Write a section that clearly describes the rules, citing the source.
 * Add one or two well-chosen examples to illustrate the rules.
 * Instead, you seem to have found a large number of examples by trial and error, and you have documented these examples without mentioning the underlying pattern. Having fun with arithmetic is fine as a hobby, but the output is not notable or interesting, and does not belong in Wikipedia. Gandalf61 (talk) 12:06, 11 September 2009 (UTC)


 * Taken note and will improve to include the rules, to make it more apparent. The rule is simple to verify by arithmetic. --Ling Kah Jai (talk) 04:51, 12 September 2009 (UTC)


 * Gandalf61, User:Ling Kah Jai/Cyclic permutations is ready to be added to repeating decimal if you agree. --Ling Kah Jai (talk) 15:48, 14 September 2009 (UTC)


 * No, I do not agree. Look at what I said above. You haven't cited a source. You haven't explained the general rules (Arthur Rubin outlines the general rules at Talk:Parasitic number). You have far too many specific examples. Your style is still very unclear and difficult to understand. The whole section need to be made much shorter and much clearer. Gandalf61 (talk) 16:14, 14 September 2009 (UTC)
 * I will try. The arithmetic by Arthur Rubin on the left shift by double positions is correct but the solution is not necessary correct. Let him try out for n = 2, then he will know.--Ling Kah Jai (talk) 17:14, 14 September 2009 (UTC)


 * Only manage to add general rules and proof for 2 out of 3. Due to the length, I intend to start it as a new article and just leave a summary here.--Ling Kah Jai (talk) 18:00, 14 September 2009 (UTC)
 * I will add the rule for the last case.--Ling Kah Jai (talk) 18:29, 14 September 2009 (UTC)
 * There are no solutions for 2-digit cyclical left shift for n = 4, 10, 12, 20, 23, 25, 28, 34, 36, 40, 45, 50 and any numbers greater than 50. This cannot be predicted by Arthur Rubin's approach (giving wrong answers) but by the approach using repeating decimal.--Ling Kah Jai (talk) 01:02, 15 September 2009 (UTC)
 * Actually, it can be predicted; it just isn't immediately obvious. User:Arthur Rubin/Parasitic number produces:
 * n=4; d<24, and 32 | d
 * n=10; d<9, and 10 | d
 * Actually any n>=10 fails if you disallow leading zeros, as d<10 (to avoid carries in the multiply), and d>=10 to avoid leading 0s. But, continuing:
 * n=12: d<8, and 8 | d.
 * n=20: d<4, and 80 | d
 * n=23: actually, this one seems to work for d = 01, 02, or 03, with m = 6
 * 01 2987 * 23 = 2987 01
 * 02 5974 * 23 = 5974 02
 * 03 8961 * 23 = 8961 03
 * n=25: d<3, 25 | d
 * n=28: d<3, 4 | d
 * n=34, 36, 40, 50: d<2, 2 | d
 * n=40: d<2, 5 | d
 * n&ge;50: d<1 (!)
 * — Arthur Rubin (talk) 01:42, 15 September 2009 (UTC)
 * Ok! you know that each n value will work or not through individual inspection. Can your approach derive a pattern of n value that work or that do not work? My approach can!--Ling Kah Jai (talk) 02:01, 15 September 2009 (UTC)
 * It's true that, by looking more closely at the fractions (not exactly repeating decimals), one can show that the left shift calculation fails if:
 * $$10^k-n = 2^a 3^b 5^c$$, where a, b, and c are integers, and b&le;2.
 * On the other hand, your analysis doesn't directly eliminate k=2, n=51, m=42, does it? — Arthur Rubin  (talk) 02:04, 15 September 2009 (UTC)
 * My mistake for quoting n=23 to have no solution. My approach also provide solution.--Ling Kah Jai (talk) 02:08, 15 September 2009 (UTC)
 * My analysis does eliminate k=2, n=51, m=42 (I don't have to deduce what is m, which comes naturally) because it says that 1/49 x 51 > 1! Very simple.--Ling Kah Jai (talk) 02:12, 15 September 2009 (UTC)
 * (If $$10^k-n = 2^a 3^b 5^c,$$ then, regardless of the values of d and m, x will have the same digit repeated.)
 * As it stands, your approach cannot develop a general rule; your analysis for n=34 is not much different than mine. However, if you want a general rule, no solution exists if and only if:
 * $$GCD(10^\infty,10^k-n)\ge\frac{10^k}n-1$$
 * — Arthur Rubin (talk) 02:27, 15 September 2009 (UTC)
 * Again very simple and it has been listed in my text that X is represented by $1/F$ and if this fraction does not generate a repeating friction fraction decimal of with a period of 3 and above for k = 2 (2-digit shift), how could X be realistic? It is all in there with a little reasoning. That is the beauty - no lengthy formula!--Ling Kah Jai (talk) 02:36, 15 September 2009 (UTC)
 * On the other hand, the general rule that you are developing is copied from my concept(but incomplete / not entirely correct).--Ling Kah Jai (talk) 02:39, 15 September 2009 (UTC)--Ling Kah Jai (talk) 02:47, 15 September 2009 (UTC)
 * Sorry, it's from my concept, not yours. If you want details, I can run the GCD verification into my formulation, too.  — Arthur Rubin  (talk) 03:00, 15 September 2009 (UTC)

Then you have to work it on to make it correct. I wonder if you do not look at repeating decimal, how could you make it correct (rule for the exceptions) .--Ling Kah Jai (talk) 03:11, 15 September 2009 (UTC)
 * Very easily: If $$d \frac {10^m-1} b$$ is an integer, then $$GCD\left(10^\infty,b\right) | d$$.  Conversly, if  $$GCD\left(10^\infty,b\right) | d$$, then there is an m>0 such that $$d \frac {10^m-1} b$$ is an integer.   The latter follows from properies of powers of 10 modulo $$\frac b {GCD(b,d)}$$.  — Arthur Rubin  (talk) 04:06, 15 September 2009 (UTC)


 * Frankly speaking, I do't understand the $$|$$ symbol neither why do you have to use $$10^\infty$$ (what is your intention to achieve?) in $$GCD\left(10^\infty,b\right) | d$$.
 * I am giving you notice that I will create a new article based on my writing.--Ling Kah Jai (talk) 06:50, 15 September 2009 (UTC)

Theorem of repeating decimal
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)


 * 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)


 * 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)


 * I am not a Wikibooks expert (Wikisource seems inappropriate). But it's not the best.  — Arthur Rubin  (talk) 16:01, 23 September 2009 (UTC)

Phenomenon of repeating decimal
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)


 * 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)


 * 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)

Group theory
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)

Please also refer to proofs of Fermat's little theorem, somebody incorporated a proof based on group theory! --Ling Kah Jai (talk) 01:42, 14 October 2009 (UTC)

What is monid theory? I did a search on wikipedia. Nothing. --Ling Kah Jai (talk) 01:52, 14 October 2009 (UTC)


 * 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)


 * I am giving up. I cannot progress with you. --Ling Kah Jai (talk)01:18, 15 October 2009 (UTC)

Why does repetition begins where it does?
Section Repeating decimal tries to address this question, but it seems (to me) to miss the essential issue. Suppose 0&le;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)

The lede should be about rationals, not reals
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)

Beswick
Who is Beswick? Tkuvho (talk) 17:34, 1 January 2012 (UTC)


 * 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)
 * 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)
 * 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)


 * 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)

can a tail of nines be obtained by long division?
I added what seemed to be a non-controversial edit, to the effect that an unending tail of 9s cannot be obtained by long division. This was deleted twice, on the grounds that the editor "disagrees". Input would be appreciated. Tkuvho (talk) 17:55, 1 January 2012 (UTC)


 * This statement was added very shortly after I added a proof to 0.999... that used long division with reference. Tkuvho has been very active at that page.  If he didn't expect me to object, he must have missed the edit.  I had initially half expected this edit was a response to the proof I added.
 * Regardless, it is not a matter of my disagreement. Instead I provided a source, which uses the long division algorithm to produce an unending tail of nines.  It is difficult to state that something cannot be done when there is a source that does it.
 * Granted, in order to do this one must use a modified form of the division algorithm for integers, in which one chooses the remainder $$0<r\leq |b|$$, instead of $$0\leq r < |b|$$. But the long division algorithm, preformed with this modified form of the division algorithm certainly can produce an unending sequence of 9's. Thenub314 (talk) 19:24, 1 January 2012 (UTC)


 * If you feel Beswick's point of view is not notable, or otherwise inappropriate, perhaps a simple compromise would be to make no claims at all about what can or cannot be done with long division. Thenub314 (talk) 19:32, 1 January 2012 (UTC)


 * I think that Beswick's modification is not notable. I suggest we keep the remark about the impossibility of getting a tail of 9s through long division, and delete Beswick.  Tkuvho (talk) 19:34, 1 January 2012 (UTC)


 * I personally am not for contradiction a published article based on our opinion. Thenub314 (talk) 20:31, 1 January 2012 (UTC)


 * Agreed. We should go by Google Scholar's opinion, which in this case is negative.  Tkuvho (talk) 14:06, 2 January 2012 (UTC)


 * Google scholar list the paper, and another paper that cites it, and several papers that cite that. Google scholar also list around 80-ish papers by K Beswick on the subject of education.  I was a bit surprised by this (perhaps it is a bit inflated, google scholar is far from perfect). So I decided to switch over to google and find out who she was.  Her website is here.  It seems she hols an associate professor position.  She is an associate dean of research and the graduate research coordinator for her department.  Reading her achievements section she seems a fairly accomplished researcher on mathematics education.  Now not being in mathematics education, I do not know how many citations one should expect for any given paper. But overall I find your comment that her opinion is not notable not very convincing. Thenub314 (talk) 16:34, 2 January 2012 (UTC)
 * Don't put words in my mouth. I have no doubt Beswick is an accomplished and influential researcher.  This particular paper has had virtually no influence.  You mention the fact that the unique paper that cites Beswick's paper on .999 is in turn cited by several other authors.  How is this supposed to establish the notability of Beswick's paper??  Tkuvho (talk) 16:48, 2 January 2012 (UTC)
 * I had no intention of putting words in your mouth, but perhaps I simply misunderstood your position. As I said above I am content to just let the matter drop as long as we do not intend to contradict the article. Thenub314 (talk) 18:23, 2 January 2012 (UTC)
 * Which article do you claim we are contradicting, Beswick's? But she would be the first one to agree that one can't get a tail of 9s by using long division as it is taught to millions of children.  Tkuvho (talk) 11:18, 3 January 2012 (UTC)

Decimal Representation
The last paragraph of the lede (as of 8/29/2013) is:
 * "A decimal that is neither terminating nor repeating represents an irrational number (which cannot be expressed as a fraction of two integers), such as the square root of 2 or the number π. Conversely, an irrational number always has a non-repeating decimal representation."

I have a BUNCH of problems with this.
 * First, a decimal is a digit. I believe the author meant decimal representation.
 * Second, I know of NO decimal representations that are not finite. This seems to me to be definitional.
 * Third, isn't a repeating decimal a notation? Isn't it a augmentation of the decimal representation which allows an 'exact' decimal representation of numbers which would otherwise be truncated with lost accuracy?
 * Fourth, and I'm on shaky ground here, is it really true that every irrational number "always has a non-repeating decimal representation."? Are there not some irrational numbers which do NOT have a decimal representation?  We know they exist, but not their value (to one or more significant digits accuracy)?
 * Fifth, I wonder if there are rational numbers which the same objection applies? That is, they exist but can NOT be (yet, if ever) represented by a decimal number.  This, I have severe doubts about, but haven't the brains to establish the case one way or another.

We don't teach our kids that 0.5 "really" is shorthand for 0.500000... with the 0's continuing "forever". While mathematically a better way to think about it, perhaps, that's not what is always done. Is it? Two other issues, which I ask an editor's consideration:
 * series representation of any rational number X is  A + (1/10)^m * Σ (C ÷ (10^s)^n) (n=0 to ∞) where A is the non-recurring portion of the number, C is the recurring digits, s is the size of C (count of decimal digits in C) and m is the number of  places to the right of the decimal point that A "occupies".  The series section here seems unable or unwilling to handle the general case - but I have no references for this (if I got the math right).
 * Shouldn't this article have a section on different bases? For instance, 0.3 (base 10) has a repeating (non-truncating) decimal representation base 2; 0.3 = 0.010011(0011)72.172.10.35 (talk) 05:42, 29 August 2013 (UTC)


 * I reformatted your post so that your list of points is clearer.


 * On your first point, I agree that "decimal" should be "decimal representation". I have updated the article text.


 * On your second and third points, the standard definition certainly allows decimal representations to be infinite. If only finite representations were allowed then common fractions such as 1/3 and 1/7 would not have a decimal representation. Of course, we cannot write down the whole decimal representation if it is infinite - but that is a separate issue of notation, not the underlying representation.


 * On your fourth and fifth points, yes, it is true that every real number (no matter whether it is rational or irrational) has a decimal representation. This is because there is a simple algorithm that allows us in principle (given enough time, computing power etc.) to write down the decimal representation of any real number to any number of decimal places, no matter how large. How much of this decimal representation we actually know in practice is not relevant. For example, we can be sure that the decimal representation of pi has a 10100th digit, even though we may never know what this digit is. (A strict finitist would not agree with this, but they are a minority view among practicising mathematicians).


 * On your final point, representations in other bases are mentioned in the positional notation article. But note that these are not decimal representations because decimal implies base 10.


 * I hope this answers most of your questions. Gandalf61 (talk) 10:31, 29 August 2013 (UTC)

Period
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)
 * Yes. — Preceding unsigned comment added by 212.159.119.123 (talk) 14:52, 7 October 2016 (UTC)

Binary/decimal repeating
Hi. What is the relation between a length of repeating sequence of decimal and binary representation ? TIA --Adam majewski (talk) 18:12, 13 October 2013 (UTC)
 * Very complicated.
 * You don't say why you think the two numbers, two and ten, are so important. They happen to have come to your attention. — Preceding unsigned comment added by 212.159.119.123 (talk) 13:26, 14 October 2016 (UTC)

Dot notation in the UK
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)

Note
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)


 * ✅ Fixed. Bill Cherowitzo (talk) 15:35, 5 September 2016 (UTC)

Applications to cryptography formula
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)


 * 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)

Theorems determining the nature of the decimal expansions of rational numbers
the following are the Theorems determining the nature of the decimal expansions of rational numbers: THEOREM 1:let x be a rational number whose decimal expansion terminates. then, x can expressed in the form p/q, where p and q are co-primes, and the prime factorization of q is of the form 2^m × 5^n, where m, n are non-negative integers. THEOREM 2:let x= p/q be a rational number, such that the prime factorization of q is of the form 2^m × 5^n, where m,n are non-negative integers. then, x has a decimal expansion which terminates after k places of decimals, where k is the larger of m and n. THEOREM 3:let x= p/q be a rational number, such that the prime factorization of q is not of the form 2^m × 5^n, where m, n are non-negative integers, then, x has a decimal expansion which is no-terminating.

i request you to add these theorems to the article.Huzaifa abedeen (talk) 06:28, 30 September 2020 (UTC)Huzaifa abedeen


 * Theorems determining the nature of the decimal expansions of rational number.jpg

Huzaifa abedeen (talk) 08:46, 5 October 2020 (UTC)Huzaifa abedeen
 * Be wp:BOLD!
 * To use it as a source, you need to give the bibliographic data for the publication you show a picture of.--Nø (talk) 09:55, 5 October 2020 (UTC)

"not zero"
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)

Parentheses notation in Austria
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)