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Such as number modulo 3 := sum of the digits (decimal base) Example: 62837 mod 3 = 6+2+8+3+7 mod 3 = 26 mod 3 = 2+6 mod 3 = 8 mod 3 = 2
Another: number modulo 7 := number lest the last digit - 2 * last digit (decimal base) Example: 62837 mod 7 = 6283-14 mod 7 = 6269 mod 7 = 626-18 mod 7 = 608 mod 7 = 60-16 mod 7 = 44 mod 7 = 2
Ïnteresting would be an algorithm for numbers modulo 31; with that you could calculate in your head certain check digits to forge a social security number in situ...
As I mentioned in the discussion page for modulo operation, I made an edit to that article today that clarifies that computers do not constrain a and n to be integers. Rather, the evaluation of a mod n in computers is carried out such that quotient q is an integer and remainder r is allowed to be a non-integer.
I am unsure of the ramifications of this on the remainder article. It seems that it is possible to define "remainder" more generally, not just in the context of pure integer division / the division algorithm, so I think this should be perhaps mentioned here. Please do this, if you can, or correct me if I'm wrong. Thanks - mjb 04:44, 30 Jan 2005 (UTC)
Thank you for your edits. However, you are not mathematically correct.
The word "remainder" means "leftover", "residue". As such, the remainder of division of 5 by 3 is not 5, is not 8, it is 2, and only 2. As such, your addition of the section "Other definitions" in the remainder article is not mathematically correct.
The concept of remainder is well-defined in Mathematics. The only possible ambiguity is when you talk about negative numbers, there you have a choice in sign. But that is all. Please take a look at Division algorithm and Euclidean algorithm.
I think you confuse remainder with modular arithmetic.
You are right that the remainder can be defined for real numbers, I will add a section for this. But, still the same basic truth holds, the remainder is always smaller in absolute value than the quotient.
Tomorrow I will go back to remainder and correct things. Please do not take it personally. Oleg Alexandrov | talk 16:02, 30 Jan 2005 (UTC)
When you say remainder for reals does not make sense since reals can be divided without remainder, I think you are assuming too much. It is true that reals can be divided without remainder, but this is only if you require that the quotient also be real. The modulo operation in computing (at least according to my observations) finds the remainder, given that q is limited to the set of integers — even though a, n, and r can be real. Thus there is often a remainder, which of course must be real. I do not see this as violating any principles. qn+r=a is still true and, I assume, provable, for the given sets of values allowed for each variable. Yet for some reason it is offensive to you. I don't understand why. The modulo operation merely "solves for r", given real or integer a and n and certain bounds, one of which is q being {…,−1,0,1…}. These equations, taken with their bounds, are statements of fact; one is no more valuable than the other. Mathematicians use one set of bounds, computer scientists another. This may not always be the case in the future. - mjb 00:24, 31 Jan 2005 (UTC)
To say "it can be proved" is correct English. See for example this Britannica article. For some reason, it sounds more natural to my ear than "it can be proven". That's why I will change this in remainder. Oleg Alexandrov 05:07, 10 Mar 2005 (UTC)
Can we discuss this before getting carried away with the edits today? — mjb 22:51, 11 May 2005 (UTC)
Actually, I probably would have removed them all if I hadn't been in a bit of a hurry. But I'm not going to be adamant about this point. Michael Hardy 02:49, 12 May 2005 (UTC)
−
instead of a hyphen to represent minus or negative.Are there other guides we can refer to?
Regarding
or  
)… the pros are that it
The cons are that:
Is the formatting bug something you can live with? Is there a workaround?
I think it is reasonable to assume that standard algebraic notation is best for formulas, so two variables q and d being multiplied are probably best written as qd when presented in a formula. That is, when citing a formula, don't use ×
.
However, when explaining, for a general audience (not just mathematicians), the formula or giving examples of its application, I feel it is prudent to use the multiplication sign — "×", coded as ×
. So, I think these examples should remain as-is (although I added no-break spaces here):
Do others agree with this convention?
Wikipedia seems to be pretty good at dealing with unescaped "<" characters, but since the MediaWiki markup also uses HTML tags, I think it would be ideal to use <
when we want to represent the less-than sign, just like in HTML and XML.
Does anyone know of a style guide that addresses this issue and makes recommendations?
Grammar question:
Is either one acceptable? Does one sound better than the other? — mjb 08:56, 9 August 2005 (UTC)
I just added a paragraph to the article. There seem to be no consensus among programming languages implementations regarding the choice for the remainder when negative integers are involved.
In the article today −42 = 9×(−5) + 3
seems to be saying -42 divided by -5 has a quotient of 9 with a remainder 3.
This may be true, based on the article definition, but I'm not sure we should accept that. I think most people would accept that ±42 divided by ±5 would give some kind of 8 as a quotient, not a 9. Additionally, the article definition of remainder does not allow the ambiguity introduced in the section with the word ambiguity. Isn't THAT ambiguous? Something needs to be changed! I vote we find an actual source for the definition of remainder. 75.17.12.230 07:58, 8 September 2007 (UTC)
I've seen the phrase "divides into" in print before, as in "a does not divide into b", referring to the fact the remainder of a/b is nonzero. Alternatively, "divides evenly by" is used sometimes to the same effect.
Could there be a place for this information either in this article or in Division (mathematics)#Notation or Multiple (mathematics)? —Preceding unsigned comment added by Kostmo ( talk • contribs) 05:12, 15 February 2009 (UTC)
There are some other definitions of remainder that should be considered in this article:
I have been adding article links and references for these subjects. RockMagnetist ( talk) 18:58, 29 January 2013 (UTC)
I've written and edited middle school maths textbooks for students on two continents and taught post-high school maths on a third, and I've yet to encounter residue in a mathematical sense. Where is this used? Thanks. -- Unicorn Tapestry {say} 14:48, 18 August 2013 (UTC)
above is incorrect,misleading and needs to be edited by an expert. naming the "quotient" quotient or q is incorrect,or at least the link is incorrect, because in this modulo theory the quotient should be an integer and in the link provided it does not necessarily has to be an integer.for example. 7/2=3,5 . Here 7=a , 2=d . So to make modulo work quotient wich is defined in the link is 3,5 and by modulo theory it has to be an integer, number 3 and reminder has to be number 1 — Preceding unsigned comment added by 91.124.154.226 ( talk • contribs) 18:19, 25 September 2013
The starting paragraph
does not seem quite right since the remainder could be zero: 8/2 is an integer, 8 divided by 2 has remainder 0.
This early definition contradicts what is later explained in the page:
and
🥺 2409:4042:269F:45F7:8CC8:F7B7:B350:614E ( talk) 09:11, 19 April 2022 (UTC)
This article is rated C-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
This article was selected as the article for improvement on 22 April 2013 for a period of one week. |
Such as number modulo 3 := sum of the digits (decimal base) Example: 62837 mod 3 = 6+2+8+3+7 mod 3 = 26 mod 3 = 2+6 mod 3 = 8 mod 3 = 2
Another: number modulo 7 := number lest the last digit - 2 * last digit (decimal base) Example: 62837 mod 7 = 6283-14 mod 7 = 6269 mod 7 = 626-18 mod 7 = 608 mod 7 = 60-16 mod 7 = 44 mod 7 = 2
Ïnteresting would be an algorithm for numbers modulo 31; with that you could calculate in your head certain check digits to forge a social security number in situ...
As I mentioned in the discussion page for modulo operation, I made an edit to that article today that clarifies that computers do not constrain a and n to be integers. Rather, the evaluation of a mod n in computers is carried out such that quotient q is an integer and remainder r is allowed to be a non-integer.
I am unsure of the ramifications of this on the remainder article. It seems that it is possible to define "remainder" more generally, not just in the context of pure integer division / the division algorithm, so I think this should be perhaps mentioned here. Please do this, if you can, or correct me if I'm wrong. Thanks - mjb 04:44, 30 Jan 2005 (UTC)
Thank you for your edits. However, you are not mathematically correct.
The word "remainder" means "leftover", "residue". As such, the remainder of division of 5 by 3 is not 5, is not 8, it is 2, and only 2. As such, your addition of the section "Other definitions" in the remainder article is not mathematically correct.
The concept of remainder is well-defined in Mathematics. The only possible ambiguity is when you talk about negative numbers, there you have a choice in sign. But that is all. Please take a look at Division algorithm and Euclidean algorithm.
I think you confuse remainder with modular arithmetic.
You are right that the remainder can be defined for real numbers, I will add a section for this. But, still the same basic truth holds, the remainder is always smaller in absolute value than the quotient.
Tomorrow I will go back to remainder and correct things. Please do not take it personally. Oleg Alexandrov | talk 16:02, 30 Jan 2005 (UTC)
When you say remainder for reals does not make sense since reals can be divided without remainder, I think you are assuming too much. It is true that reals can be divided without remainder, but this is only if you require that the quotient also be real. The modulo operation in computing (at least according to my observations) finds the remainder, given that q is limited to the set of integers — even though a, n, and r can be real. Thus there is often a remainder, which of course must be real. I do not see this as violating any principles. qn+r=a is still true and, I assume, provable, for the given sets of values allowed for each variable. Yet for some reason it is offensive to you. I don't understand why. The modulo operation merely "solves for r", given real or integer a and n and certain bounds, one of which is q being {…,−1,0,1…}. These equations, taken with their bounds, are statements of fact; one is no more valuable than the other. Mathematicians use one set of bounds, computer scientists another. This may not always be the case in the future. - mjb 00:24, 31 Jan 2005 (UTC)
To say "it can be proved" is correct English. See for example this Britannica article. For some reason, it sounds more natural to my ear than "it can be proven". That's why I will change this in remainder. Oleg Alexandrov 05:07, 10 Mar 2005 (UTC)
Can we discuss this before getting carried away with the edits today? — mjb 22:51, 11 May 2005 (UTC)
Actually, I probably would have removed them all if I hadn't been in a bit of a hurry. But I'm not going to be adamant about this point. Michael Hardy 02:49, 12 May 2005 (UTC)
−
instead of a hyphen to represent minus or negative.Are there other guides we can refer to?
Regarding
or  
)… the pros are that it
The cons are that:
Is the formatting bug something you can live with? Is there a workaround?
I think it is reasonable to assume that standard algebraic notation is best for formulas, so two variables q and d being multiplied are probably best written as qd when presented in a formula. That is, when citing a formula, don't use ×
.
However, when explaining, for a general audience (not just mathematicians), the formula or giving examples of its application, I feel it is prudent to use the multiplication sign — "×", coded as ×
. So, I think these examples should remain as-is (although I added no-break spaces here):
Do others agree with this convention?
Wikipedia seems to be pretty good at dealing with unescaped "<" characters, but since the MediaWiki markup also uses HTML tags, I think it would be ideal to use <
when we want to represent the less-than sign, just like in HTML and XML.
Does anyone know of a style guide that addresses this issue and makes recommendations?
Grammar question:
Is either one acceptable? Does one sound better than the other? — mjb 08:56, 9 August 2005 (UTC)
I just added a paragraph to the article. There seem to be no consensus among programming languages implementations regarding the choice for the remainder when negative integers are involved.
In the article today −42 = 9×(−5) + 3
seems to be saying -42 divided by -5 has a quotient of 9 with a remainder 3.
This may be true, based on the article definition, but I'm not sure we should accept that. I think most people would accept that ±42 divided by ±5 would give some kind of 8 as a quotient, not a 9. Additionally, the article definition of remainder does not allow the ambiguity introduced in the section with the word ambiguity. Isn't THAT ambiguous? Something needs to be changed! I vote we find an actual source for the definition of remainder. 75.17.12.230 07:58, 8 September 2007 (UTC)
I've seen the phrase "divides into" in print before, as in "a does not divide into b", referring to the fact the remainder of a/b is nonzero. Alternatively, "divides evenly by" is used sometimes to the same effect.
Could there be a place for this information either in this article or in Division (mathematics)#Notation or Multiple (mathematics)? —Preceding unsigned comment added by Kostmo ( talk • contribs) 05:12, 15 February 2009 (UTC)
There are some other definitions of remainder that should be considered in this article:
I have been adding article links and references for these subjects. RockMagnetist ( talk) 18:58, 29 January 2013 (UTC)
I've written and edited middle school maths textbooks for students on two continents and taught post-high school maths on a third, and I've yet to encounter residue in a mathematical sense. Where is this used? Thanks. -- Unicorn Tapestry {say} 14:48, 18 August 2013 (UTC)
above is incorrect,misleading and needs to be edited by an expert. naming the "quotient" quotient or q is incorrect,or at least the link is incorrect, because in this modulo theory the quotient should be an integer and in the link provided it does not necessarily has to be an integer.for example. 7/2=3,5 . Here 7=a , 2=d . So to make modulo work quotient wich is defined in the link is 3,5 and by modulo theory it has to be an integer, number 3 and reminder has to be number 1 — Preceding unsigned comment added by 91.124.154.226 ( talk • contribs) 18:19, 25 September 2013
The starting paragraph
does not seem quite right since the remainder could be zero: 8/2 is an integer, 8 divided by 2 has remainder 0.
This early definition contradicts what is later explained in the page:
and
🥺 2409:4042:269F:45F7:8CC8:F7B7:B350:614E ( talk) 09:11, 19 April 2022 (UTC)