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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)
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)
How do I type a repeating decimal?
2607:FCC8:F4C2:7500:AC93:B28:BF40:52CA (
talk)
19:55, 30 April 2018 (UTC) EvieSwan2405
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)
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)
Gandalf61, If I were to define the above terms, I will choose:
Semi-cyclic
Semi-cyclic numbers are represented by decimals derived from 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.,
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)
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)
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)
![]() | This ![]() It is of interest to the following WikiProjects: | ||||||||||
|
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)
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)
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#simplified approach but was deleted by Arthur Rubin. This section is only complete if the short summary section, repeating decimal#parasitic number 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)
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)
I have completed an article with the above title currently kept at User:Ling_Kah_Jai/Theorem_of_repeating_decimal. I intend to:
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)
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)
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)
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
is the decimal expansion, then the remainders in the long division are defined by
One gets
Now we know that for some distinct j and k we must have
because there are only finitely many integers in [0,d). So we get
Assume without loss of generality that
(otherwise switch j and k), then
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
and thus
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)
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)
Who is Beswick? Tkuvho ( talk) 17:34, 1 January 2012 (UTC)
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)
The last paragraph of the lede (as of 8/29/2013) is:
I have a BUNCH of problems with this.
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:
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)
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)
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)
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)
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)
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
Huzaifa abedeen ( talk) 08:46, 5 October 2020 (UTC)Huzaifa abedeen
The article currently asserts:
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)
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)
![]() | This article links to one or more target anchors that no longer exist.
Please help fix the broken anchors. You can remove this template after fixing the problems. |
Reporting errors |
Archives:
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)
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)
How do I type a repeating decimal?
2607:FCC8:F4C2:7500:AC93:B28:BF40:52CA (
talk)
19:55, 30 April 2018 (UTC) EvieSwan2405
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)
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)
Gandalf61, If I were to define the above terms, I will choose:
Semi-cyclic
Semi-cyclic numbers are represented by decimals derived from 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.,
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)
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)
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)
![]() | This ![]() It is of interest to the following WikiProjects: | ||||||||||
|
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)
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)
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#simplified approach but was deleted by Arthur Rubin. This section is only complete if the short summary section, repeating decimal#parasitic number 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)
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)
I have completed an article with the above title currently kept at User:Ling_Kah_Jai/Theorem_of_repeating_decimal. I intend to:
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)
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)
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)
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
is the decimal expansion, then the remainders in the long division are defined by
One gets
Now we know that for some distinct j and k we must have
because there are only finitely many integers in [0,d). So we get
Assume without loss of generality that
(otherwise switch j and k), then
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
and thus
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)
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)
Who is Beswick? Tkuvho ( talk) 17:34, 1 January 2012 (UTC)
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)
The last paragraph of the lede (as of 8/29/2013) is:
I have a BUNCH of problems with this.
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:
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)
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)
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)
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)
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)
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
Huzaifa abedeen ( talk) 08:46, 5 October 2020 (UTC)Huzaifa abedeen
The article currently asserts:
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)
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)