Talk:Divisor

Divisor defined as denominator

I have added a second definition to the lead, covering the sense of "divisor" to as at the start of the Terminology section. I don't actually agree with my own addition: an article should be about a concept, and should not cover two independent concepts that happen to share the same name. I think that this article should cover only the sense in abstract algebra (possibly restricted to integers), with a hatnote with a redirect to the other sense. To get rid of the alternative definition will involve removing a few lines only. Comment? — Quondum 22:53, 1 August 2013 (UTC)

Don't worry, I've removed my additional definition, and made some changes to clear up some of the confusion between the concepts in this article. — Quondum 00:04, 2 August 2013 (UTC)

0 again

The question of whether or not zero is a divisor of zero has reappeared at the Math Project Talk page. The question interested me enough to go scrambling through a number of texts (algebra, number theory, intro to proofs). I found both sides of the issue equally represented. About half define a | b only for nonzero a. About half of those that permit a to be zero make no statement about 0 | 0, while the rest are quite explicit about this being true. The authors ran the gamut and included some very respectable mathematicians. Some quick examples:

Herstein is of the nonzero a camp and states that if integer m ≠ 0 then m | 0.

Paley & Weichsel permit a = 0 and say "As a consequence of this definition, we see that any integer a divides 0. Indeed even 0 | 0."

Niven is of the nonzero a camp and says "It is understood that we never use 0 as the left member ... in a | b."

Dean of the nonzero camp says "However, by definition, zero divides no integer."

McCoy of the nonzero camp says (paraphrased) "We could just as well allow a to be zero in the above definition, but this case is unimportant and it is convenient to exclude it."

The one thing that I did not find was any statement about uniqueness in the definition of divisor. Personally I kind of like that wrinkle in the definition and it may appear in some education literature, but I don't have a citation for it.

The current definition on this page only reflects one point of view on the issue and probably should be expanded to give a broader perspective. Bill Cherowitzo (talk) 04:01, 2 August 2013 (UTC)

I'm a little confused by your statement [a]bout half of those that permit a to be zero make no statement about 0 | 0. Surely 0 doesn't divide any integer other than 0. Did you mean "permit b to be 0" rather than "permit a to be 0"? --Trovatore (talk) 06:37, 2 August 2013 (UTC)

Sorry, I should have been clearer. What I meant was that these authors did not make any explicit statement about 0 | 0, permitting it to be inferred from the definition without further comment. The others brought this point to the fore. Bill Cherowitzo (talk) 19:16, 2 August 2013 (UTC)

I see nothing strange about so making and others not making a statement about a being zero: that an author bothers to make a specific statement about this special case when it is clear from the definition is not significant. To distil Bill's finding: the field is split between allowing and disallowing zero as a divisor (which is different from what 78.92.92.3 found at the Math Project page (though relative notability is always an issue).

How about this hypothesis: authors coming from the educational perspective and dealing only with integers might exclude zero as a divisor, because of potential confusion between the dual uses of the term "divisor" (one in abstract algebra, one in arithmetic), and authors in a more formal mathematical context, especially abstract algebraists, uniformly allow zero as a divisor.^{Only a hypothesis: not checked} Does the exclusion simplify the statement of any theorems? What convenience is gained by excluding it? If my hypothesis is valid, there may be an argument to relegate mention of zero being excluded as a divisor to a discussion or footnote, but not to treat it as having weight. — Quondum 14:31, 2 August 2013 (UTC)

It didn't really split that way and my impression (non-scientific) was that the algebraists were more likely to not allow zero. The data is pretty fuzzy though and I wouldn't make any hard and fast calls from it. Bill Cherowitzo (talk) 19:16, 2 August 2013 (UTC)

I was wondering about McCoy's "convenient" as well. (McCoy who, BTW?) I don't really see anything convenient about excluding it, except possibly that it would let you identify "a | b" with "b/a is a well-defined integer", but really I don't see how that enables any further convenience beyond its own bare statement. --Trovatore (talk) 18:03, 2 August 2013 (UTC)

Neal H. McCoy, Fundamentals of Abstract Algebra (1972). One convenience is that if you exclude it from being a divisor, then you don't have to do it when you define zero divisors. I've seen a couple of other statements in these comments which support the convenience. The only advantage that I can see for having 0 be a divisor of 0 is that then you can say that every integer divides 0 instead of every nonzero integer divides 0. A rather small gain in my mind. Bill Cherowitzo (talk) 19:16, 2 August 2013 (UTC)

Zero divisor is not to be confused with a divisor of 0.

Definition of zero divisor in a ring: $(a,b)$ is a pair of zero divisors ($a$ is a left-sided zero-divisor, $b$ is a right-sided zero-divisor) $\ \Leftrightarrow \ a\neq 0\land b\neq 0\land ab=0$.

An integral domain is a commutative ring with no (pairs of) zero divisors. Divisibility is usually defined in an integral domain (sometimes with unity). In this case, it is nonsensical to define zero divisor in terms of divisibility. So McCoy must have defined divisibility differently. How?

Also, what are the "other statements in these comments which support the convenience"? 78.92.91.246 (talk) 08:44, 3 August 2013 (UTC)

Well, no, there are various others. It makes the description of Euclid's algorithm cleaner, for one thing. It turns the natural numbers into a very nice lattice structure with 1 at the bottom and 0 at the top, and then gcd and lcm are just meet and join respectively, and you don't have to make special exceptions to explain that gcd(n,0)=n for every n including 0 (the correct choices for Euclid's algorithm). --Trovatore (talk) 19:26, 2 August 2013 (UTC)

Does the concept of "divides" not extend to more general rings (that include zero divisors) using the same definition? If one forms a ring as the direct product of several rings, then under homomorphisms, the exclusion must become pretty inconvenient. In particular, homomorphisms should preserve the "divides" property, whereas if we exclude zero, it wouldn't, because some homomorphisms will map some zero divisors onto zero. This reasoning is out of the air, so don't crucify me if I've got it wrong. But if I'm not off-beam, divisor does not behave nicely under generalization to more general rings unless we include zero. — Quondum 23:50, 2 August 2013 (UTC)

Gentlemen please. I am not arguing (pro or con) whether zero is a divisor of zero. In fact, I have not expressed my opinion on the matter. I made my initial comment because the discussion was getting to be lop-sided and I felt that some balance was needed. I shouldn't have to remind you that our job is not to improve, correct or beautify the mathematics we are discussing, but rather to report on what we find in the secondary sources without bias. The reality here is that the sources are split and whether zero is a divisor of zero seems to be author-dependent. This can not be ignored because one's favorite example has a pretty property only under one definition (and by the way, it is not universally held that 0 is a natural number, a key point in Trovatore's example), nor can it be dismissed as a fringe concept which messes up more advanced theory (see Divisor (ring theory)). We should strive to present (in this instance) both sides and not give undue importance to this rather trivial question. When a statement requires one or the other definition to be valid, it should be explicitly mentioned. This is not an onerous burden for an editor of a short encyclopedia article. Bill Cherowitzo (talk) 15:33, 3 August 2013 (UTC)

I generally agree that, given the sources mentioned, we probably do need to give a nod to both conventions. And this is beside the point, but no, the convention that 0 is a natural number is not actually essential to my example — you would just have to rephrase it as being about the "nonnegative integers". --Trovatore (talk) 21:00, 3 August 2013 (UTC)

I too agree with Bill that its seems clear that both conventions should be mentioned; the only potential discussion would now be around whether one was dominant. In the above side-track, it was clear to me that it was a side-track exploring the prettiness of the concepts, not the notability thereof (and I guess I was simply extending Trovatore's lattice concept in a way where one definition seems to be clearly preferable, but as noted, this is not relevant).

With such encouragement I have edited the page in line with my previous comments. As to my choice of order in the definition, this version required the fewest number of changes/comments in the rest of the article (3) and avoided a nasty one in the definition of trivial divisors. Bill Cherowitzo (talk) 20:01, 5 August 2013 (UTC)

A more subtle point: there is a difference between excluding 0 as a divisor and defining it as not dividing anything, and it would be helpful to distinguish these, not clear at present. It would need careful reading of the sources to see how each author regarded each case. The answer to the following question under the definition excluding zero should be clear: is 0∤0 true? (It is either true or undefined; I would hope that there will not be authors who consider it to be true.) — Quondum 10:19, 6 August 2013 (UTC)

A recent edit cleanly removed the exclusion of 0 as a divisor, but in so doing going against the consensus to represent both of the prevalent definitions. I do not wish to revert this, because I think it would make sense to reintroduce the exclusion as a modification of the less restrictive case (i.e make it the second definition). This will improve readability. It will also make dealing with my "subtle point" easier as a later caveat, rather than something extra to be "forgotten" when removing the zero exclusion. — Quondum 17:52, 27 October 2013 (UTC)

Definition of divisor at opening of article is not the most common

I am teaching some remedial math, and point students to wikipedia sometimes. Sending them here to learn the meaning of divisor would be a mistake; this page apparently deals with integer divisors, but the term is more general. I think a sentence briefly explaining this, with a link to the page on division would improve this page (see also the first comment by "mirwin" and the "wow" comment above). — Preceding unsigned comment added by 76.103.108.10 (talk) 21:27, 3 April 2014 (UTC)

The link is already there in the hatnote at the top. Please keep in mind that Wikipedia is not a dictionary, but rather a reference. You should be pointing your student to wikt:divisor for what you want. —Quondum 00:16, 4 April 2014 (UTC)

The unqualified page "Divisor" should explain the word in its most common sense. If the most common sense does not have it's own article like in our case here, the unqualified Divisor should redirect to the disambiguation page (Divisor (disambiguation)), where the most common sense defined the top, and all other meanings are listed as well. Shcha (talk) 15:25, 27 April 2023 (UTC)

Common definitions of "divisor"

The current article focuses only on the definition of divisor restricted to the field (er, ring) of integers. The section "Definition" states "Two versions of the definition of a divisor are commonplace".

This is patently silly. The word "divisor" is commonly used to refer to the "number on the bottom of a division", as it is in Wikipedia's own Long division article. "As in all division problems, one number, called the dividend, is divided by another, called the *divisor*, producing a result called the quotient."

One can appreciate that there are more restricted uses of a word in certain technical or academic realms. That does not mean that these realms dictate the meaning of the word over all other uses. This should be pointed out in the introduction. If a discussion of the more-specialized meaning is valuable, then it should be accompanied by an indication that it's a more specialized meaning.

This is, after all, the article titled "Divisor", not titled "Divisor (within mathematics, narrowly construed)" Gwideman (talk) 23:18, 11 November 2015 (UTC)

Both meanings of "divisor" are of common use. I have expanded the hatnote for clarifying the point. D.Lazard (talk) 15:40, 31 January 2017 (UTC)

Negation?

"a divides b" is abbreviated a|b

Is there an abbreviation for "a does not divide b"? 2.24.119.101 (talk) 17:23, 1 December 2015 (UTC)

Sure, $a\nmid b$. —David Eppstein (talk) 17:27, 1 December 2015 (UTC)

Complex!

This article shows why the civilian world has such a problem with math. If I ever go to hell I expect that I will be forced to read this article every day, and I *am* a mathematician. Something as basic as divisor shouldn't require a PhD to understand, or two minutes to figure it out from the noise.

I really hoped I could simply find the following sentence way up top in the article. Maybe even as a paragraph right above the table of contents. I submit this for consideration.

Simply speaking, for the equation c equals a divided by b (c=a/b), c is the quotient, a is the dividend, and b is the divisor.

Perhaps "Simply" could be "Generally".

If you just put that in the intro paragraph, not only will the suicide rate decline among math students, but the time it takes to simply see "which one is called the divisor" will be reduced to - a mere fraction.24.27.72.99 (talk) 04:27, 10 August 2017 (UTC)

That is a different meaning of the word "divisor" than the one this article discusses. Perhaps you stopped reading before you got to the second line? The one that says "This article is about a relation between two integers. For the second operand of a division, see Division (mathematics)." ? —David Eppstein (talk) 04:37, 10 August 2017 (UTC)

EDIT: I apologize for not responding earlier, David Eppstein. You are technically correct, in point of fact...yet...

I hope you'll agree, though, that the poorly assigned article title simply begs for this interpretation. Why in the WORLD would the article title state a general term ("divisor") - yet then discuss it only in a rare and limited usage? That's like titling an article "Red (color)" then only discussing the color "carmine" in the article, saying, "for mainstream use of the color 'red' see some other article." I believe that 99% of third grade arithmetic teachers would assert that the word divisor does not mean "the integer math subset meaning of the word."

This article is horribly mistitled :( 76.185.10.9 (talk) 15:57, 27 March 2018 (UTC)

So you are arguing that the other meaning of divisor (the second argument to a division operation) should be the WP:PRIMARYTOPIC? Maybe, but my feeling is that "divisor" by itself is more likely to have this meaning, and "divisor" with the other meaning is not likely to be the first word that introduces the topic of division, nor the first word that someone looks up when they look up "division". So even though the other meaning may be more common, this may be the meaning that makes more sense as the first result for searching for this specific word. —David Eppstein (talk) 18:57, 27 March 2018 (UTC)

Oh come on. If I want to find out what "divisor" means, I'm not going to search for division. Seriously? If someone want to find out what "divisor" means, "divisor" is the first word that someone looks up.2600:6C56:6600:1EA7:694D:FA90:265A:6C39 (talk) 13:53, 17 January 2019 (UTC)

Does this page have a bug or is it just me?? (Screenshot)

I am on firefox 56.0.1 (64-bit) on Win 7 and I get this red text on the page: Screenshot I have no idea what it is and what it means, so can someone fix it? Thanks, 79.101.241.42 (talk) 19:10, 25 October 2017 (UTC)

I'm no expert on MathML, but the WP markup I think is not the problem. Does the problem persist? Do you see it in the same location each time? The error message seems to be saying that the engine for processing <math> tags can't be reached for some reason. I'm using similar software and don't see the red error text. -- Elphion (talk) 04:04, 26 October 2017 (UTC)

Apparently, this is a problem of connexion between servers (html, latex and svg rendering). If the problem persists try purging the page (on my browser, button "purge" in menu "more" at the top of the page). If this does not work, verify if your version of Firefox is up do date. Otherwise you may ask your question at the help desk. D.Lazard (talk) 15:07, 17 January 2019 (UTC)

0 yet again

It seems to me that both definitions in the article as stated allow 0 | 0, and that where they differ is whether or not nonzero integers also divide zero. So I made some edits reflecting this. However, I notice that there are several footnotes in the article pointing out facts that require 0 | 0 to be true. Are there other common definitions that do not define divisibility when 0 is involved at all? Double sharp (talk) 03:46, 4 February 2019 (UTC)

This edit is not correct. If you restrict k != 0 in m*k = n, then 0|0 but no other m|0. Thus 0 is comparable with no other element in the lattice, and in particular is not the largest element. -- Elphion (talk) 05:44, 4 February 2019 (UTC)

@Elphion: You're absolutely right, of course. I thought of that after making the edit, but my bus reached my stop before I could fix it and it then slipped my mind. I've changed it to read "Given the definition for which m | 0 holds even for nonzero integer m", which should be correct now. Double sharp (talk) 06:46, 4 February 2019 (UTC)

In the meantime, I have taken the liberty of removing the two notes that call out statements that depend on 0 | 0, as both definitions agree that 0 | 0 is true. On the flip side I have added a parenthetical comment that the example 5 | 0 only holds in the first definition. Double sharp (talk) 06:49, 4 February 2019 (UTC)

I have edited the article before reading this thread. In fact, in number theory, zero is always excluded, when considering divisibility. So, there is no reason to give so much emphasis on divisibility of and by zero, and to consider the two definition as of equal importance. D.Lazard (talk) 10:59, 4 February 2019 (UTC)

@D.Lazard: OK, thanks for clearing this up. I have accordingly changed the qualifying first phrase in the "In abstract algebra" section to read "In definitions that include 0". Double sharp (talk) 16:14, 4 February 2019 (UTC)

It looks much better now. -- Elphion (talk) 19:40, 4 February 2019 (UTC)

Syntactically a divisor is...

Hi all, came here to check what a divisor and dividend is, wasn't entirely certain which was which. I expected to find a really simple description of "in a/b (or equivalently a÷b) then b is the divisor and a is the dividend". This is a superficial definition because it's purely a notational artefact, but still IMO useful. Thoughts? 79.75.100.60 (talk) 11:45, 8 September 2020 (UTC)

Added: just noticed someone's made a similar point <https://en.wikipedia.org/wiki/Talk:Divisor#Complex!> For a simple operation, a simple (even if facile) description should be available. And BTW I didn't know that in this example, c was called the quotient. — Preceding unsigned comment added by 79.75.100.60 (talk) 11:48, 8 September 2020 (UTC)

You are confusing two meanings of "divisor". I have modified the beginning of the article for trying to avoid such a confusion. In particular, the first line of the article (in italics) links now to Division (mathematics) where you will find the definition that you want. D.Lazard (talk) 13:18, 8 September 2020 (UTC)

I see. Thank you for sorting that out. 85.211.202.129 (talk) 21:46, 9 September 2020 (UTC)

Misleading Title

PLEASE...put this: "INTEGER ARITHMETIC ONLY" in the title. I read half the article thinking someone had lost their mind before looking up and seeing the italicized portion at the top. Please. Lots of people will walk away confused if they miss the easily overlooked italics. N0w8st8s (talk) 17:49, 3 November 2020 (UTC)

(a) It's not "misleading" -- this is the most common use of the word in mathematics. (b) It's already in italics and bold, twice at the head of the article. -- Elphion (talk) 00:54, 4 November 2020 (UTC)