Talk:Division (mathematics)

Division is NOT what this rubbish entry on Wikipedia tells you!

To say "15 divided by 4" means "the measure of 15 using 4 as unit and/or equal parts of 4 to complete the measure".

15-4-4-4-3 or 4-(-4)-(-4)-(-3) are the meaning of division. This and NOTHING ELSE.

Mainstream math academia haven't understood much mathematics, but especially do not understand the foundations of mathematics.

If division meant the nonsense of "equal sharing", then you cannot explain how pi = pi divided by 1.

Mainstream math academics are ignoramuses, especially those with Phds who blaspheme against me by calling themselves "mathematicians".

https://www.academia.edu/125757733/Realisation_and_Development_of_Number_in_1_Page

https://www.academia.edu/124828666/Gabriel_arithmetic_without_numbers_a_method_based_on_Thales_proportionality_theorem

The Importance of Learning the Right Way: What does division mean in mathematics and how does it affect all other arithmetic concepts? https://www.amazon.com/dp/B0DDT2K1PS

There is significant value in learning the correct way. Unfortunately, many educators overlook this principle, often leading to regret later on. True learning goes beyond the mere memorization of facts; it involves developing a deep understanding that facilitates easier and more effective future learning. It is important to recognize that knowledge acquisition often occurs through inference, especially when prior related knowledge is lacking.

Therefore, it is preferable to build knowledge incrementally to avoid incorrect inferences and to ensure that any conclusions drawn are more likely to integrate seamlessly into a solid foundational understanding, minimizing the need for future revisions. 2A02:587:AF29:EA00:D56D:70A0:564B:E204 (talk) 17:04, 28 December 2024 (UTC)

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)

Division is anticommutative

Division is anti-commutative. Given that $1$ is the right identity for division

$x\div 1=x$

then

$x\div y=1\div (y\div x)$

or in fraction form

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

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

To editor 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

${\frac {y}{x}}=-{\frac {x}{y}}$

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 editor 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 $\cdot$ and right identity under operator $\cdot$ (a.k.a. $I_{(\cdot )}$) anything that in the general case conforms to this equation

$x\cdot y=I_{(\cdot )}\cdot (y\cdot x)$

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 ($\cdot$). 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)

No, it's not "anti-commutative". The relevant properties are those of the inverse operation to multiplication in a group or - with application to arithmetic in mind - a commutative (a.k.a. Abelian) group. For general non-Abelian groups, the right quotient may be completely characterized by the properties: a/e = a, a/a = e and (a/c)/(b/c) = a/b. This can be further refined to a set of properties that eliminate explicit reference to the group identity, e, while yet implying the uniqueness of a group identity (and its existence, if the group is non-empty, as is normally always assumed). For Abelian groups, the relevant set of properties are a/(a/b) = b and a/(b/c) = c/(b/a), with the identity being (uniquely) definable as e = a/a. To ensure the group be non-empty, one can also add in the axiom e = e/e. In both cases, the defining properties of a group can be recovered from these by defining inverse by a⁻¹ = (a/a)/a and product by a b = a/(b⁻¹). The property you're referring to as "anti-commutativity" is actually the one I just cited: a/(b/c) = c/(b/a).

Adding an introduction

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), ...

I am not certain about, to which extent this article should, e.g.,

• 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.

The standard I'm using for accessibility for this article is "can a 5 year old understand this". Maybe 5 years is a little extreme, but at the very least a 7-year old should be able to follow a good chunk of this article, which covers what many people would consider an elementary subject. Of course, some things (such as polynomial rings in one indeterminate) will be mentioned that I wouldn't expect a kid to understand - for more technical things such as these, I ask "would my father (that is, a knowledgeable person with a tangential interest in math) understand this?". I've gone through the introduction, channeling my inner 5-year-old to ask as many stupid questions as I could, taking notes, and I edited the article based on most of the notes I made. Considering how much I did edit, I don't feel that this article is at the level of accessibility that it should be at.

While I did pause at the lack of definition of what arithmetic is, I didn't really define it in my recent edit. Perhaps it should be defined, but I felt comfortable leaving it as it was. I feel the explanations for quotition and partition can be improved. I suspect this part: "In a ring the elements by which division is always possible are called the units (for example, 1 and –1 in the ring of integers)" can be improved.

- Ramzuiv (talk) 03:43, 6 September 2019 (UTC)

Can someone please rewrite the intro

A little pressed for time here but can someone please rewrite the intro?  Current intro as of the time of this writing exemplifies the worst of Wikipedia intros:  at some point I assume someone once wrote a couple basic starter sentences, and then some later editor began adding all this extra info (perhaps to show off their knowledge) about ISO 80000-2 standards and usage of symbols in anglophone countries and info about deprecating the division sign in favor of the solidus character blah blah blah.  None of that stuff belongs in an intro lead paragraph about a general mathematical topic.  The intro paragraph should be like one, at most two, concise, succinct sentences about division, and then all the ISO 80000-2 stuff and discussion about anglophone countries and other ridiculousness can be moved way further down in the article.  Can’t express how much I hate intro paragraphs like this.  —PowerPCG5 (talk) 03:23, 29 July 2021 (UTC)

a = c / b means a × b = c

No. a x b comes from (a -:- 1/b) or (b -:- 1/a).

Multiplication is derived from division and not the other way around.

2A02:587:AF29:EA00:D56D:70A0:564B:E204 (talk) 17:08, 28 December 2024 (UTC)

Illustration is a pie divided by 4 to make 4 slices.
slice = pie/4
letting a = slice, c = pie, b = 4 gives a = c / b
$4\times slice=slice+slice+slice+slice=pie$
b x a = c
a = c / b means b × a = c
194.207.86.26 (talk) 11:16, 19 December 2021 (UTC)

This is in the 5th paragraph of the article. D.Lazard (talk) 11:43, 19 December 2021 (UTC)

The article says `a = c / b means a × b = c` but it should say `a = c / b means b × a = c` 194.207.86.26 (talk) 12:11, 20 December 2021 (UTC)

This does not matter, since $a\times b=b\times a.$ D.Lazard (talk) 15:20, 20 December 2021 (UTC)

It may as well be corrected though. Since division and multiplication are not always commutative. 194.207.86.26 (talk) 08:20, 2 May 2022 (UTC)

In the non-commutative case, the equivalence is obtained by multiplying by b on the right. So, in this case, this is "a = c / b means a × b = c" that is correct. In any case, the symbol × is used only for multiplication of numbers, which is commutative, and, in the non-commutative case, notation $cb^{-1}$ is generally preferred to $c/b$ for avoiding any ambiguity. For example, if B and C are matrices $CB^{-1}$ is often encountered, and $C/B$ is never defined. D.Lazard (talk) 08:54, 2 May 2022 (UTC)

a = c / b, which is the right-quotient, always means a = c b⁻¹ which is equivalent to a×b = c. To get the equivalent of b×a = c would require a notation for left-quotient: a = b \ c = b⁻¹ c. For commutative products c / b = b \ c, but the two should not be confused for one another, since such confusion does not scale up to other cases of the product and quotient where commutativity does not apply.

"Delen" listed at Redirects for discussion

An editor has identified a potential problem with the redirect Delen and has thus listed it for discussion. This discussion will occur at Wikipedia:Redirects for discussion/Log/2022 February 14#Delen until a consensus is reached, and readers of this page are welcome to contribute to the discussion. ~~~~
User:1234qwer1234qwer4 (talk) 06:07, 14 February 2022 (UTC)

missing basics

There are no illustrations of how division is done at the introductory (grammar school), with __ in the U.S. or other 'tools" in other countries.

    )

Kdammers (talk) 03:09, 20 March 2022 (UTC)

Look at the draft to see what I'm trying to show -- a figure a little like

the square root figure. Kdammers (talk) 03:12, 20 March 2022 (UTC)