# Talk:Division (mathematics)

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## Page title

Given Addition, Multiplication and Subtraction are all named without appending (mathematics) at the end... and the likely thing people are trying to find when they go to a page called Division will be the mathematical version... can't we just have Division go straight to this page with a thing at the top for disambiguation? --93.97.23.110 (talk) 02:22, 4 January 2011 (UTC)

Yes: your proposal would be more consistent with Wikipedia's style. --Hroðulf (or Hrothulf) (Talk) 10:30, 6 January 2011 (UTC)
While I'm generally in favour of making this move, one counterargument is that for addition, subtraction, and multiplication, all other uses of the word stem from or are to some extent indistinguishable from the mathematical sense of the word, while division as a mathematical concept stems from the more general concept of dividing something into smaller things. A set of 10 can be divided into 2 new sets, one with 7 and one with 3. This is very different from the mathematical sense, and some may consider this a compelling reason to keep Division as a disambiguation page, with the mathematical sense explicitly distinguished as such, while keeping the other operations where they are. That being said, the asymmetry does bother be, and in my opinion the move should be made. -Ramzuiv (talk) 02:34, 25 September 2019 (UTC)
Such a move has no chance to reach a WP:consensus, as the word has many meanings that have nothing to do with mathematics (14 meanings in the Wiktionary entry), and the mathematical meaning is certainly not the WP:primary topic. The primary topic, if any, is probably "The act or process of splitting anything". D.Lazard (talk) 08:06, 25 September 2019 (UTC)

## Remove the division graph

I suggest that we remove the division graph at the start of the page. It doesn't convey any useful information about division, and the way the points are connected is wrong.

184.66.4.239 (talk) 21:19, 10 October 2014 (UTC)

Done D.Lazard (talk) 22:01, 10 October 2014 (UTC)

## the limit of a quotient of functions

This reversion removed a proposed addition to the article about the limit of a quotient of functions. That is an easily sourced subject, so this is a request for a section on the limit of a quotient of functions. Here are two sources that can be used as a basis for the section:

NB: The limit theorem for a quotient of functions is mentioned in Limit of a function#Properties, but without sources.

--50.53.34.137 (talk) 14:15, 21 October 2014 (UTC)

## School-book division

As far as I can see, there is no presentation of the "tool" that children use in dividing:
____
)
Kdammers (talk) 14:24, 11 April 2016 (UTC)

This is in Long division. D.Lazard (talk) 16:35, 11 April 2016 (UTC)
True, but it belongs here as well. The tool is introduced in short division, and for amny people it is the common tool used with actually doing division.. Kdammers (talk) 22:24, 15 April 2016 (UTC)
It very much belongs here, including the (obvious) extension for polynomials, and factorization of polynomials, as this forms the basis for modern cryptography (for example, to identify each beacon in the GPS satellite system, or the CDMA coding algos for cell phones, etc.) I say "obvious" because long-division base-10 can be changed to base-x and one gets the polynomials. At least a cursory, passing mention of primality, prime ideals, reducibility, quotient spaces, quotient rings, etc. belongs here as well, right?84.15.187.18 (talk) 10:27, 17 June 2016 (UTC)
I have added {{main|Long division}} at the top of Section "Computing". D.Lazard (talk) 14:31, 17 June 2016 (UTC)

## History section is lacking

Although this is a vital article and the topic has a very long history, there is no history section here. I'll add a short history section (in fact a stub) in Euclidean division. This could be a starting point for this article, but an expanded and sourced version is needed here. I have not the needed historical knowledge for doing it myself. D.Lazard (talk) 13:14, 25 September 2016 (UTC)

## Divided by - what is a succinct synonym?

20 ÷ 5 isn't dialectical English. When you write something, if possible write the wordish for of it.

twenty divided by five or ? — Preceding unsigned comment added by 2A02:2149:8464:BA00:BCE5:B794:8895:1FC4 (talk) 10:06, 15 February 2018 (UTC)

Good point. Fixed. D.Lazard (talk) 10:37, 15 February 2018 (UTC)

## Division is anticommutative

Division is anti-commutative. Given that ${\displaystyle 1}$ is the right identity for division

${\displaystyle x\div 1=x}$

then

${\displaystyle x\div y=1\div (y\div x)}$

or in fraction form

${\displaystyle {\frac {x}{y}}={\frac {1}{\frac {y}{x}}}}$

Matt (talk) 11:41, 9 March 2018 (UTC)

To Matt: It is not useful to duplicate your posts (here and in my talk page)
The assertion that division is anti-commutative is wrong, as anticommutativity is a well defined concept in mathematics. Division would be anti-commutative if one would have
${\displaystyle {\frac {y}{x}}=-{\frac {x}{y}}}$
for all x and y.
This is a true property that one gets the multiplicative inverse when commuting the operands of a division. This could be added to the article, but this should be written in a less confusing way (less jargon, more text, less formulas and less complicated formulas). D.Lazard (talk) 12:23, 9 March 2018 (UTC)
To D.Lazard: Limiting anticommutativity to just the additive inverse makes no sense, it is the simplest and most common but not the only case. Both are forms of the more general case; with operator ${\displaystyle \cdot }$ and right identity under operator ${\displaystyle \cdot }$ (a.k.a. ${\displaystyle I_{(\cdot )}}$) anything that in the general case conforms to this equation
${\displaystyle x\cdot y=I_{(\cdot )}\cdot (y\cdot x)}$
can be considered to be anticommutative.
There is a general pattern here and if multiplication can be considered commutative (just as addition is commutative) it doesn't make sense to say that subtraction is anticommutative and division is not. Matt (talk) 18:33, 9 March 2018 (UTC)
While there is a pattern here, it is not the one you have settled on. The negative in the anti-commutative law that you are interpreting as additive inverse is not really that, as can be seen when you examine the more general n-ary definition. It is actually the sign of a permutation and can be positive or negative. It just happens to be negative in the common two variable case. Your attempt to redefine the meaning of anti-commutative will fall upon deaf ears and certainly has no place here on Wikipedia. --Bill Cherowitzo (talk) 19:07, 9 March 2018 (UTC)
I do not think that Matt refers to "additive inverse", but rather to "multiplicative inverse". So he arrives at "anticommutativity" of division, with respect to "multiplicative inverses", which is trivial in the context of division being the "inverse operation" of multiplication (${\displaystyle \cdot }$). Certainly, "multiplying" something with its "multiplicative inverse" yields the "multiplicative unity". The pertinent WP-article Anticommutativity is not very explicit about a canonical or generally accepted setting. Personally, I prefer the notion "antisymmetry" for the setting described there (cf. "totally antisymmetric symbol"). My question at the ref-desk also did not return ultimate answers. Purgy (talk) 16:26, 10 March 2018 (UTC)
Aside from what Purgy has said above (I almost entirely agree with his description of his understanding of what I was trying to say) I consider the general n-ary definition as a moot point for two reasons a) the four basic operators I have mentioned are all binary, and b) if you consider either lambda calculus or Turing / von Neumann machines (where almost all the world's research effort on computability has gone) n-ary computations can always be reduced / decomposed into a sequence of binary operations. In the case of basic arithmetic and algebraic operations they can be reduced even further to the Peano axioms and the Successor function (itself often used for Church encoding in pure lambda calculus systems). Matt (talk) 16:24, 12 March 2018 (UTC)
Most of above discussion is WP:OR. In particular operator arity and computability have nothing to do with anti-commutativity. The point is: are there reliable sources that define anti-commutativity in terms of a multiplicative inverse? I do not know any, and I am quite sure the if there are some, they do not belong to the main stream of mathematics. Thus, unless somebody provide reliable sources, the extension of anti-commutativity to multiplicative inverses is WP:OR and does not belong to Wikipedia. D.Lazard (talk) 17:18, 12 March 2018 (UTC)

I have added an introduction section to the article because as it is, the beginning space was too complicated for something to work correctly as a start. As it was, it did not adequately describe division in a way that was approachable; trying to adequately describe the entire process of division without becoming to long was not something that it could do. Because division is more complex than the other elementary operations, it is too difficult to shove into the beginning. I have also merged the properties section into the intro to show how division works (the "properties" section was so far only one property, and not sufficient anyway). I would greatly appreciate any extra help in making the article more approachable without losing information. Thank you! IntegralPython (talk) 21:02, 23 October 2018 (UTC)

This article is already on my mind for a long time, but I am not sufficiently versed in the appropriate literature to make any changes on my behalf. I would like to revert the recent edit "making the text less cluttered", because I think its content should be discussed beforehand: there is no "remainder" in other basic operations at all, "keeping" the remainder may be usual lingo, but is imho un-mathy, creating the rationals therefrom is strange to me, "division by zero is only defined..." leads astray (similar to "summing" divergent series to -1/12), ...
• report the didactic stories invented to engage the beginners (see also Quotition and partition),
• extend on the "operation" division (I am unsure to which extent there is mathematically RELEVANT literature on this) as a "basic" algebraic structure with e.g. "right-distributivity" (sic!), besides the important Euclidean "divides" with "remainder" in the integer environment,
• refer to a function implementing multiplicative inverses, or simply demand their existence, with mentioning the final target of getting rid of "division" in all formal treatments on operations on rational/real/complex/quaternion-numbers,
• delve into other settings (polynomials, algebra, ..., (wheels?))
I certainly would like to help. Purgy (talk) 08:25, 24 October 2018 (UTC)
The present version, has two major issues: the second paragraph is about teaching or learning division and does belongs to the lead (and, maybe, to this article, as consisting mainly of original research about the psychology of students). The third paragraph begin by the undefined concept of "mathematics of fields". If this means field theory, this is too technical for appearing here. If this means something else, the formulation is clearly not acceptable.
I'll try to improve this lead. D.Lazard (talk) 09:24, 24 October 2018 (UTC)

## Symbols

I ended up here because I was idly curious about the background or history of the ÷ symbol and whether there was any truth to my perception that it's been largely disappearing in recent decades. Of course, I didn't find much about that, but I did find a line (under "Of integers") reading "Names and symbols used for integer division include div, /, \, and %". If that % wasn't a typo of ÷, it's in need of explanation. --01:00, 29 December 2018 (UTC)71.234.116.22 (talk)

The % appears only in the section on Euclidean division of integer, and more specifically in the paragraphs on division on computers. It is true that some computer languages use % for Euclidean division. As far as I know, this notation is not used outside computer programming. Nevertheless, this paragraph could be made clearer.
The symbol ÷ is called obelus, and there is an article about it, and it's history. Your perception of its disappearition is true, and is clearly stated in the article: "This form is infrequent except in elementary arithmetic. ISO 80000-2-9.6 states it should not be used." In fact, mathematicians do not use ÷ since a long time. I do not know whether it becomes less used in elementary arithmetic. If it is the case, a probable reason is that / is simpler to type on a keyboard than ÷. D.Lazard (talk) 09:48, 29 December 2018 (UTC)

## Wikiprojects B class checklist

I'm going through the criteria for this page to be considered class B, and I'll just put some notes here for my thoughts as I go through the checklist. For Verifiability, I have started working my way through this page, placing markers on statements that may be worth verifying. We need much more verification before this article is in a good place.

As regards structure, I think the structure of this article is perfectly suitable, and have marked it as such, but it annoys me that the structure of this page and that of multiplication are quite different, so I would like to propose taking the effort to align this page's structure with that of multiplication, as well as subtraction and addition.

As far as accuracy, I find no reason to complain

I would like to see more content on this page. Off the top of my head, it seems worthwhile to mention the formal definition of arithmetic, division as it regards dimensional analysis, and probably also quotient sets. If you can think of anything else worth adding, please let me know.

Grammar and Style is good

This article seems quite lacking in support material. Some topics that stand to be illustrated include euclidean vs. rational division, illustrating both quotition and partition, and perhaps other interpretations of division, why commutativity and association don't hold, why it is left-distributative but not right-distributative, manual methods of division, and division of complex numbers and matrices. All of these topics have some way they can be visualized, and such illustrations will greatly aid understanding.