Talk:Division (mathematics)

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Perhaps we should have some explanation of restoring and non-restoring division algorithms?

(Edit: yes, please! I can't find an explanation anywhere. Please, Obi-Wan-Wiki, you're my only hope...)

Should there be any reference to a military unit sense?

See Division (military) the disambig page Division has links to both meanings. --Salix (talk): 20:48, 8 October 2008 (UTC)

Should we write a division algorithm?

Let's have the correct definition of division![edit]

Intermediate Algebra by Barnett and Kearns has the actual definition of division and the explanation as to reason for division by zero is not defined:

"We say that a divided by b equals c if and only if there exists a unique value c such that b times c equals a."

Division by zero (b=0) is not defined because if a is nonzero, c doesn't exist, and if a is zero, c is not unique.

B.Wind 21:38, 24 December 2005 (UTC)

The division by zero rationale is useful, I agree, and maybe we can work it into the article. The information in the first statement (the "definition"), though, is already adequately covered in the article intro. Also, as phrased above, it is not really a definition in the sense of "Division is…"; rather it makes a statement about something "we" say in relation to division. So I greatly prefer what is in the intro right now.
There is also no "we" in an encyclopedia, despite the fact that this article and many other math related articles often egregiously violate this principle. Although it is hard to break the habit of using the lecture style when discussing technical subjects, at least make an attempt to make important declarations without directing the reader's attention ("note that"/"notice how"), giving the reader advice ("one should"/"it should be noted that") or referring directly to a speaker or listener ("we say that" etc.).—mjb 18:01, 17 June 2006 (UTC)
Division is only defined in terms of multiplication. It is used in its own right for concepts such as sharing, measurement and grouping. Lets not try to say that the definition is anything more than the inverse of multiplication, unless someone proposes an axiomatic set theory definition that is more precise. --Hroðulf (or Hrothulf) (Talk) 15:16, 12 February 2010 (UTC)

Integer division?[edit]

In the section "Division of Integers", Patrick recently replaced

Give the quotient as the answer, so . This is sometimes called integer division.

so that (with some copyedits by Oleg Oleg Alexandrov) it read:

Two versions of what is sometimes called integer division:
(a) Apply the floor function to the quotient, so .
(b) Truncate the quotient, so .
These coincide if the integer to be divided are positive.

I've reverted for a couple reasons. First of all I can't infer the general rule from the example given in (a). For example what is 26/10 equal to? 3 or 2? Also why introduce -26/10, when the original example was 26/10? (Can someone supply a source for this definition of integer division?). Secondly (b) is a rewording of the original, but I think the original is better. And again why change the sign of the example? Finally I don't know what is meant by "These coincide if the integer to be divided are positive".

Paul August 20:45, 16 June 2006 (UTC)

By "These coincide if the integer to be divided are positive" I meant the two definitions (a) and (b). But I agree that the whole thing doesn't make any sense. Oleg Alexandrov (talk) 21:04, 16 June 2006 (UTC)
[1] and [2] say that we have to truncate a negative quotient, while [3] says we have to apply the floor function. See also [4]. --Patrick 07:31, 17 June 2006 (UTC)

Thanks Patrick for providing these sources, I will add some of these to the article, when I get a chance. Your new edit is much better so thanks for that also. Besides apparently Mathematica, and Python, are there any other examples where integer division is implemented by rounding toward negative infinity? Paul August 18:04, 17 June 2006 (UTC)

There's an interesting comment on this by Italo Tasso on this page, which suggests that Perl also may have this. The treatment of the quotient, which is naturally the main focus of this article, is intimately related to the remainder and/or modulo congruence value (see also Modular arithmetic#Remainders). I believe some programming languages in fact provide both types of remainder. The GNU multiple precision arithmetic library provides both kinds of quotient. -R. S. Shaw 05:17, 18 June 2006 (UTC)
I disagree that Mathematica and Pythons are examples of rounding toward negative infinity. First of all, I don't think we have access to exactly how both these things implement integer division. Second, the effect here is not "rounding" but basically choosing the quotient so that you have a positive remainder. If you choose to always have a remainder below one million, you can find an appropriate quotient to satisfy the condition. 10:30, 8 September 2007 (UTC)

The general rule of the (a) part is the quotient results in a positive remainder. So, -26/10 = -3 with a remainder of 4. -26 = -3 * 10 + 4. This comes from one of definitions of remainder. (The one that guarantees a positive remainder less than the divisor.) The other definition provides for (b). -26/10 = -2 with a remainder of -6. -26 = -2 * 10 - 6

According to Division algorithm the quotient of -26/10 is -3 (remainder 4) while the quotient of 26/-10 is -2 (remainder 6). (And the quotient of -26/-10 is 3 and NOT 2, remainder 4.) It does not seem this type of understanding could be gained from the current section. It seems necessary for the section to cover negatives because the remainder and quotient each change based on the input signs. Someone should create this information.

Integer division in programming seems to be a special topic and not worthy of too much attention in this section of Division(mathematics).

I would have corrected the errors in the edit, not reverted. The error being that the quotient of -26/10 is (probably) -3 so "truncate" in (b) is wrong, and "flooring the quotient" is wrong since it is already an integer.

I'll give it a try if no one does a thing. 10:30, 8 September 2007 (UTC)


why is there no divide key on keyboards? Wtatour 23:57, 28 June 2006 (UTC)

Historically, both "/" and ":" have been used as division signs. Personally I only use the latter when setting up informal division calculations. So, it depends on what you mean by "divide key". --Frodet 16:09, 30 September 2006 (UTC)
I think that there should be a '÷' key on keyboards. It would make sums much more easier to type. --lEN2323 18:22, 6 January 2007 (UTC)

Article (or section) request[edit]

I think "Dividing by 0" is a valid discussion topic that worth at least a section on this article. Or even expland into a full article that consist of varies methods of proving that "Dividing by 0" is undefined. Lightblade 17:24, 16 November 2006 (UTC)

There's a link to Division by zero at the end of the lead section. I agree that it should have a section in this article as well; why don't you try writing it? Melchoir 19:02, 16 November 2006 (UTC)
Thanks for the link! I didn't see it down there. Lightblade 21:38, 16 November 2006 (UTC)

'No general method for integrating quotients of functions'[edit]

Article: "There is no general method to integrate the quotient of two functions." I disagree strongly, but I'm not sure how to better word it (I tried). My main quibble is that the article means to say that there is no general method particular to quotients. Of course, by treating a quotient of functions as a product with an inverted term (when doing so is possible), general product methods like integration by parts work fine. I just don't want a student to encounter a fractional integrand (with divisor zero on a set of measure zero or whatever) and panic because 'there's no general method' when integration by parts would work fine. Suggestions? 03:20, 18 October 2007 (UTC)

Integration by parts does not "work fine" on every quotient. Of course it works in some cases. Eric119 22:21, 18 October 2007 (UTC)

Case of negative quotient in lede[edit]

The second sentence of the lede says "Division can be described as repeated subtraction." This is false when the quotient is negative, such as when dividing -8 by 3 or 8 by -3. For example to divide -8 by 3 one must add 3 to -8 three times in order to get the remainder r into the range 0 ≤ r < 3. Likewise to divide 8 by -3 one must add -3 to 8 however many times it takes (presumably either two or three) in order to get the remainder into the desired range, whatever that is. Is this a case where it is preferable to have a false statement than to complicate matters with the truth, or does accuracy take precedence in Wikipedia? --Vaughan Pratt (talk) 05:01, 1 December 2009 (UTC)

"Co-divide" of matrices[edit]

Recent additions by an unregistered user appear to be sourced from

which are abstracts of conference submission pre-prints.

  1. I don't think this is yet peer-reviewed, so it doesn't meet the community definition of a reliable source
  2. For this vital core article, I think we should wait until such concepts become fully established in the literature, rather than post information from every single article. We should bear in mind that there are surely millions of words written about division, and one encyclopedia article can only summarize a few of those words, and that we are editing for an audience of general readers, not math professionals.

All of the new content was reverted by various editors. I removed the latest addition, which also had a gallery of examples that showed only placeholder images.

--Hroðulf (or Hrothulf) (Talk) 08:55, 31 August 2010 (UTC)

Exact mathematical definition?[edit]

given here? —Preceding unsigned comment added by (talk) 17:49, 16 October 2010 (UTC)

Page title[edit]

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? -- (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)

Remove the division graph[edit]

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

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

the limit of a quotient of functions[edit]

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.

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

School-book division[edit]

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? (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[edit]

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)