Talk:Long division/Archive 1

Problem
There's a problem with this Bold texte xample. In step 3, you say "place a zero to the right of [it]". It's only zero here because of the particular example, the general method is to "pull down" thBold textd next digit in the dividend above. That doesn't make this example incorrect - but it doesn't make the method clear for any number. If you take the step to mean "add a zero" you won't get the right answer. edita cerna i love u p 2003 (UTC)


 * I just clarified it. Dysprosia 06:31, 7 Sep 2003 (UTC)

Multiple digit divisor
This also doesn't clarify a method when using a multiple digit divisor. —The preceding unsigned commenilovet was added by you24.11.140.191 (talk • contribs) 10:39, July 28, 2004 (UTC)

Request for improvement
An anonymous editor added this comment to the top of the article:


 * --This article needs to be improved. It explains a mathmatical method that is usually used by those in mandatory education in a style aimed at the intelligent, this is near useless to all guests.

Michael Hardy 22:06, 4 December 2005 (UTC)


 * Yes, the example is poor, and can be better done with simple division --scruss 12:34, 19 February 2006 (UTC)


 * I've changed the example, to be a bit more instructive I think. Paul August &#9742; 18:46, 7 March 2006 (UTC)

Another Request
Maybe somebody could state a different method of doing long division (but keeping everything that's currently there), as this may be very confusing to somebody learning it for the first time with only a general concept of math. Thanks.. anything would be much appreciated.

Maybe an easy remembering methid like DMSB (Dirty Mice Smell Bad)? as in D(irty)=Divide M(ice)=Multiply  S(mell)=Subtract  B(ad)=Bring down? 130.13.98.191 19:33, 21 March 2007 (UTC)

Symbol Name
Can't seem to find any reference outside this article to the term perquin, or its supposed coiners. --scruss 12:34, 19 February 2006 (UTC)
 * I've removed the "perquin" term as dubious, MathWorld in their long division article says "The symbol separating the dividend from the divisor seems to have no established name". I suspect this was added as a sneaky vandalism. Paul August &#9742; 18:46, 7 March 2006 (UTC)
 * It was. The MathWorld article notes two common names, however, and we should mention them as well. — Llywelyn II   00:11, 12 February 2016 (UTC)

Remove "python implementation"
The python implementation is useless. The entire division is done in one simple line, using python's built in division: digit=numerator/denominator It doesn't explain in any way how to do division, just how to add decimals if you already have a working integer division function. --Ropez 16:53, 19 April 2006 (UTC)

Short division
Just wondering whether a comparison to short division is in order? Actaully there is no short division page on Wikipedia, perhaps this is in order too. :)


 * That's actually why I decided to check out this page, wondering whence the "long" derives --Belg4mit 16:38, 13 April 2007 (UTC)


 * Me too :-) It'd be interesting to know. Maybe because it takes longer than simply using a computer? (*smirk*) -- Coffee2theorems 20:38, 5 August 2007 (UTC)


 * The short/long distinction is apparently only one of style, not substance: in short division, you don't write down the intermediate steps .  However, Knuth (TAOCP, v2, exercise 16 on page 282) and others (  et al) have used the phrase to name an instance of division where the divisor has a single digit (or 'limb', in the parlance of some multiple-precision arithmetic packages).  So the current example in this article could be said to be "short", not "long", division.  mdf 16:21, 13 September 2007 (UTC)


 * I created an article on short division back in December 2007. It has since been updated by other editors a number of times. I'll add a link to it in this article. SimpsonDG (talk) 15:12, 16 July 2019 (UTC)
 * There is already a link in the lead of the article. D.Lazard (talk) 17:53, 16 July 2019 (UTC)

Standards based mathematics reform
This section is rather weak. Who uses "R" for remainder"? The remainder is the remainder, not a variable. Simple. And what, praytell, are teachers teaching if not this? --Belg4mit 16:38, 13 April 2007 (UTC)


 * What I find curious there is the claim that mathematicians don't use long division. Well, sure, they tend to use CASs, but what if you're away from your computer and just need to divide a few polynomials (for which simple calculators don't work)? I've never seen anyone use anything else but long division on paper (or blackboard) for that. As for "R", I guess it refers to the "r" in a=qd+r, where it is indeed a variable.. But who cares what you call the variable, and isn't "r" the usual convention anyway? Maybe it means that you shouldn't use a capital "R" but a small "r"? -- Coffee2theorems 20:38, 5 August 2007 (UTC)

My hands are up
I can't do long division and this example did not help in the slightest. I think the explanation should be changed/modified. Perhaps some colour coding.

I'm a relitavely intelligent guy. I have taught myself programming but I could not learn from this example.

Dopple 15:12, 10 September 2007 (UTC)


 * I'll second that. I've never heard the term long division before but I think there was a similar method with a much different name that I didn't get when my parents tried to teach. Since not understanding that I've done calculus and such (I'm not sure about the English terms for most of what I at least used to know) and I feel no smarter for having read the two examples. Some seriously verbose explanations for each step in the examples are needed. As it the article is now I'd have to spend quite a few minutes doodling around on paper to even begin to understand and I doubt I'd even get it in the end.81.228.145.179 (talk) 22:26, 7 July 2008 (UTC)

the fifth grade teachers manual
fifth grade manual? from what country/state? Could we get a citation for that? Pete Iriarte 19:20, 7 November 2007 (UTC)

History of Long Division
I came to this article looking for insight into the history of the method used for long division. Would someone please add a history section? —Preceding unsigned comment added by 76.97.204.233 (talk) 20:16, 16 December 2007 (UTC)

Long division is understood by most adults, including teachers and parents, who have been taught elementary arithmetic.

I don't think this is true and none of the sources mention it. Most of the adults I know used to understand long division, but now have almost completely forgotten how to do it. I gave it a citation needed, but perhaps this should just be removed.

Rootwhisk (talk) 18:57, 4 August 2008 (UTC)

Seriously. Who actually uses long division? There are faster ways to get the sorts of answers it gives, even relying only on mental math. The polynomial long division algorithm is useful, but arithmetic long division is not. The only reason this has relevance is because schools teach it for the sake of tradition, mainly with the result of teaching students just learning arithmetic to be afraid of math. —Preceding unsigned comment added by 67.170.34.181 (talk) 02:49, 18 May 2010 (UTC)


 * According to Florian Cajori, it goes back to Luca Pacioli at least. .  Dicklyon (talk) 00:21, 18 October 2009 (UTC)

Exercises
Long Division Practice Worksheets: Printable long division worksheets —Preceding unsigned comment added by DanielAjoy (talk • contribs) 06:09, 17 October 2008 (UTC)

Examples with bigger divisors needed; animation slowdown needed
Of the three numerical examples given in the various notational systems, only one of them, in European notation, has a divisor of more than one digit (and that one is accompanied by no explanation). This is unfortunate because people who know long division never use it unless the divisor has at least two digits.

Could somebody put in an example with at least two digits in the divisor? Maybe it could be based on the animation at the beginning of the lede. I'd do it myself except that my imaging skills are zilch.

Also, could someone figure out how to slow down the animation in the lede by a factor of about ten? Right now it goes so fast that I can't begin to follow it (even though I know and use long division).Duoduoduo (talk) 00:18, 8 July 2011 (UTC)

"Latin America" (except MEX/COL/BRA) method almost like Germany/Poland/et.al.?
Well, from my (German) standpoint, I can only say the method used in Latin America (except for Mex./Col./Bra.) is identical to that in Germany, Poland and some other countries, except for the division sign, which is written as a ÷ there, and we'd write it as a colon. Let me inform you that the minuses you put in so kindly are usually omitted, so that our "Hühnerleiter" (German colloquial word, literally chicken's ladder) looks EXACTLY like that from non-Mexican, non-Colombian and non-Brazilian Latin America. And this means: we could actually summarize those and would not need to use a separate section for Germany (et. al.) as we have now. -andy 77.7.108.7 (talk) 08:24, 12 July 2011 (UTC)

country-specific?
The technical way to do lang division changes from year to year, teacher to teacher, book to book, state to county to primary to school secondary school. The country-specific stuff in this article is arbitrary nonsense, and should be deleted. --129.13.72.198 (talk) 17:17, 27 August 2011 (UTC)

Long division of complex numbers
At the start of the article it states "In arithmetic, long division is a standard procedure suitable for dividing simple or complex multidigit numbers". Shouldn't the use of the term "complex" in this situation be avoided to differentiate from complex numbers (numbers with an imaginary and real part)? — Preceding unsigned comment added by Paulsomething1 (talk • contribs) 18:06, 16 September 2011 (UTC)
 * Yes, I agree. --seberle (talk) 20:15, 16 September 2011 (UTC)

Proof?
Shouldn't this page have a proof of the algorithm? like the one seen here:

http://www.mathpath.org/Algor/algor.long.div.htm
 * In the long division procedure, the dividend must equal the sum of the remainder and all the numbers that have been subtracted.
 * But the numbers subracted are d×qi with place value 10i. So
 * a = (d × qn)10n +(d × qn-1)10n-1 + ... + (d × q1)101 + (d × q0) + r, where 0 ≤ r < d
 * = d × (qn10n + qn-110n-1 + ... + q1101 + q0) + r
 * = d × qnqn-1...q1q0 + r
 * = d × q + r,   0 ≤ r < d.
 * Enter the Division Algorithm!
 * According to it, q and r must be unique. That is, the q we have found in the long division is indeed the one and only value possible, namely the quotient of a when divided by d.
 * Now that we know why long division works, it is easy to extend to dividends that are not integers. Suppose a = 758.9 and d =5. Then a/5 = (1/10)(7589/5) so that we carry out the long division involving two integers and then divide the answer by 10 which is accomplished by moving the decimal point left. Finally, noting that the Division Algorithm is valid in any base, we can extend these arguments to any base just as well for base 10.
 * Now that we know why long division works, it is easy to extend to dividends that are not integers. Suppose a = 758.9 and d =5. Then a/5 = (1/10)(7589/5) so that we carry out the long division involving two integers and then divide the answer by 10 which is accomplished by moving the decimal point left. Finally, noting that the Division Algorithm is valid in any base, we can extend these arguments to any base just as well for base 10.
 * Now that we know why long division works, it is easy to extend to dividends that are not integers. Suppose a = 758.9 and d =5. Then a/5 = (1/10)(7589/5) so that we carry out the long division involving two integers and then divide the answer by 10 which is accomplished by moving the decimal point left. Finally, noting that the Division Algorithm is valid in any base, we can extend these arguments to any base just as well for base 10.

Or http://planetmath.org/encyclopedia/ProofOfLongDivision.html — Preceding unsigned comment added by 24.246.34.77 (talk) 16:58, 15 December 2011 (UTC)

I'm not really sure how to format it correctly in the wiki. — Preceding unsigned comment added by 24.246.34.77 (talk) 16:56, 15 December 2011 (UTC)

To be honest, the first proof for long division kind of lock you in the decimal system where you crank out the quotient digit one by one. I'm sure that similar approaches can be extended to the decimal or other numeral systems, but that seems more like a roundabout way of doing it.

A more satisfying way (to me at least) of establishing the validity of long division and its related methods is by mimicking the division procedure itself (as in this guide for example). That way it escapes the confine of traditional long division, and can be used to explained any extension or variant of long division if needed to.

45.72.231.39 (talk) 19:26, 21 June 2019 (UTC)

LaTeX?
It would look nicer without those dotted line boxes... If no objections I'll do it eventually. F = q(E+v×B) ⇄ ∑ici 08:59, 9 April 2012 (UTC)

Mixed mode
I've added a section on mixed mode arithmetic. I well remember slogging through dividing a quantity by a number at age 10, but I cannot for the life of me remember how to formally set out the division of two quantities to produce an number. Put simply, how do you calculate 12 tons 6 cwt / 3 tons 2 cwt 5 lbs? Martin of Sheffield (talk) 23:18, 6 November 2012 (UTC)
 * One way would be to multiply the dividend by a power of ten when it's less than the divisor. So to get a/b you say the answer is (1/10).(10a/b). You could also divide the divisor by a power of ten, but that's obviously a lot more complicated and might not even work if it's not divisible by ten. Or the boring way of do it is to calculate the equivalents in lbs and divide those two figures. Count Truthstein (talk) 19:49, 30 December 2012 (UTC)
 * So working through your example, we have 3/2/5 into 12/6/0. You figure out that you can subtract it at least three times. Write 3 in the answer line. Multiply by 3 to get 9/6/15, and subtract from 12/6/0 to get 2/19/97. Now I multiplied by 100. This means the next step will give 2 more digits. However, as you multiply by greater power of ten, it becomes harder to check whether the minor columns (cwt and lb) will overflow into the ton column, so it might not be worth it. So this gives 200/1900/9700, which we convert into 299/6/68. Now we do 3/2/5 into this figure. Multiplying 3/2/5 by 96 gives 288/192/480 which we turn into 297/16/32. Write 96 into the answer, with the 6 in the 0.01 column (so the answer is 3.96) Subtracting, we get 1/10/36, which < 3/2/5 so the choice of 96 was correct. (I checked that 3.96 was the right answer by doing 27552 lb/6949 lb on a calculator.) Count Truthstein (talk) 20:44, 30 December 2012 (UTC)
 * Thanks for that. I did it long hand and the method seemed familiar, can anyone help since this is way too close to WP:OR.  The way I set it out was:

t - cwt - lb                    3.965 3-2-5)12    6    0          9     6   15      -           20   190  970     x10                 8  896     adjust                198           9   180               29    18   74           27    18   45     -            20    00  290    x10                  2  224    adjust            20     2   66            18    12   30     -             10   100  360    x10                   3  336    adjust                  103             5   100                   15     3   24             12     8   20    -              2    15    4    which is more than half so round up.             =============
 * Start by estimating 3 as the first digit, write 3 * 3-2-5 under the dividend and subtract. Multiply by 10, and then adjust (8 * 112 = 896 and 9*20 = 180).  I have left out carries and borrows from this version, apply them as normal when doing it long hand for real.  The next digit is 9, and so 27-18-45 is subtracted from 29-18-84 to get 2-0-39.  Multiply by 10 and adjust as before.  Now subtract 6*3-2-5=18-12-30, multiply by 10 and adjust.  The final 5 leaves a trivial remainder.
 * The method is clear, checkable and extensible to any mixed mode, though clearly "scrap" calculations might be necessary if the divisor included difficult values (17 cwt or 47 ton for instance). Martin of Sheffield (talk) 23:59, 30 December 2012 (UTC)
 * It is original research, although I pretty sure that this is the way to do it. I don't know if anybody ever wanted to do this kind of thing, though. BTW I think there is a mistake in what you've written above. Your first subtraction should give the result 2-19-97. I've then figured the second subtraction should give 2-0-29, and the third should be subtracting off a multiple of 4, not 5, to give 2-15-4.
 * The way you've multiplied by 10 is better than the way I did it, as it nicely shifts the columns one to the right. You could write the addition to the answer above the new zero as well, which would help you to match up which parts of the calculation led to the digits in the answer. PS I'm going to have check my calculations again as they don't seem to match up with what I did above... Count Truthstein (talk) 21:09, 31 December 2012 (UTC)
 * Well spotted, in a sense that proves the method, it is easily checked. I have reworked it (I know that normally you shouldn't revise comments, but there is little point in leaving an error), the error you spotted was a copying error, but then I found another! :-( As to whether anyone ever did this, yes they did.  I can remember working out how many people you could pay at one rate with a given sum and how many lorries or railway wagons were needed to carry a certain quantity of coal.  It was the formal method I was having problems remembering, and I'm now about 75% sure that this is what we did.  Ah well, there's only about 40 minutes to midnight, so I'd better get back to the family.  Happy New Year all! Martin of Sheffield (talk) 23:21, 31 December 2012 (UTC)

127 ÷ 4
IP user 81.243.130.87 has been doing some useful tidying up, and since this is his first edit I've welcomed him. However I'm a little concerned that he is systematically combining the first two steps from 31.         4)127.00      12        (12 ÷ 4 = 3)       07       (0 remainder, bring down next figure)        4       (7 ÷ 4 = 1 r 3 )                        3         to      31.  <     4)127.00 124                                    3 Whilst I accept that dividing 4 into 127 is trivial, such simplifications obscure the method. I would suggest that 123,456 ÷ 789 would could not be done using the shortcut as a first stage by most people! I've no desire to WP:BITE, but I do feel that unless this is clearly explained as a shortcut it will mislead rather than inform. What do others think? Martin of Sheffield (talk) 08:33, 21 May 2013 (UTC)

What I learned in Austria
In an Austrian middle school (Hauptschule) I learned a version that is basically the Mexican method with German arrangement:

127 : 4 = 31,75     07       30        20         0

Can anyone confirm or debunk this? Judith Sunrise (talk) 15:37, 10 July 2015 (UTC)
 * Dividing by 4 isn't really long division, it's short division. How would you have laid out 123,456 ÷ 789? Martin of Sheffield (talk) 23:04, 10 July 2015 (UTC)
 * Basically the same way:

123456 : 789 = 156,     4455                 (1x9=9 and 5 is 14 remember 1, 1x8+1=9 and 4 is 13 remember 1 ... copy down the 5) 5106               (5x9=45 and 0 is 45 remember 4, 5x8+4=44 and 1 is 45 remember 4 ... copy down the 6) 372 rest          (or keep going for decimals)

Judith Sunrise (talk) 20:46, 11 July 2015 (UTC)
 * The question of "long" or "short" should not be about the divisor but about the notation/algorithm. We got clearly demonstrated a "short" version here, and in Austria as well as in Germany there is also teached a "long" version, mainly meant to be easier for beginners. (My User Page: https://de.wikipedia.org/wiki/Benutzer:Kurzsprung)
 * Kurzsprung (talk) 15:10, 30 September 2016 (UTC)

Agreed, but dividing by 4 is so trivial I couldn't see if full working would be shown. Using the UK layout:

4) 127     31 r3

but 156 r372 789)123456      789        4455       3945         5106        4734          372        ==== The long/short difference is clearly shown.  I assume a similar shorthand exists for short division under the Austrian system.  I'm slightly surprised though that not all the working is shown - no problem if calculators are used but essential for longhand. Martin of Sheffield (talk) 17:06, 30 September 2016 (UTC)

Doubts concerning "Latin America (except ...)"
What should "Latin America (except Argentina, Bolivia, Mexico, Colombia, Paraguay, Venezuela, Uruguay and Brazil)" be? The countries listed as exceptions surely contribute a vast a majority to the area of Latin America and to its population, too.

The above quoted description was probably meant as "Latin America (in particular ...)". But what about Chile and the many clearly not negligible countries in Middle America then? I am not well informed enough to apply corrections myself to the article but I would suggest to rework the quoted part.

My User Page is assigned to the German wikipedia: https://de.wikipedia.org/wiki/Benutzer:Kurzsprung. Kurzsprung (talk) 15:04, 30 September 2016 (UTC)

An "exception" to the algorithm
Hi, I, like many people, could have come to this article in search for the explanation to some "exceptional" cases like this?

201/4 where

201 |4 -20  5,25  0010    -8     20    -20     00

Wich doesen't return the correct resut 50.25

If I'm correct, this way I did the division is "incomplete".

The correct way would be

201.00 |4 -20     50.25  001   -0    10    -8     20     -20      00

Maybe the article should adress this kind of case. — Preceding unsigned comment added by Oani000 (talk • contribs) 17:44, 7 November 2016 (UTC)


 * Your first example contains an error. It should have been:

201 |4 -20  50 r. 1 001  -0    1

or else supply the additional zeros to calculate the decimal. Look closely and you'll also notice 0010 - 10 = 2 which is news to most arithmeticians.

Keeping track of the columns is essential, that is why squared paper is used. Engineering offices used to have calculation books with squared paper on one leaf for exactly this reason. Martin of Sheffield (talk) 19:11, 7 November 2016 (UTC)


 * Corrected the 0010 - 10 = 2, my mistake. I wrote this example because, at least in my education, I learnt that every time you "give" a zero to the rest to continue the algorith a comma must be added to the answer. But you are right. My intention was just to adress this possible little mistake.

Oani000 (talk) 20:26, 7 November 2016 (UTC)


 * I may be wrong, but I think you are still missing the point about the spacing. In your first example look at the second line.  The "20" is actually 20 tens, not 20 units, so the answer is 5 tens, not 5 units.  You must pay strict attention to the columns and their meanings.  For that matter the second example would have got you in trouble at school.  To preserve the meanings of the columns you should be writing:

201.00 |4 -20     50.25  001   -0    1.0    -.8     .20     -.20       0
 * at the risk of being very picky, you are also switching between the continental practice of using a decimal comma and the English practice of using a decimal point. You really must stick to one of the other if confusion is to be avoided. Martin of Sheffield (talk) 23:27, 7 November 2016 (UTC)

No no, your are exactly right. See, I've got my graduate degree, in education in Physics incredibly, and I've never see someone being tought the algorithm with this degree of "precision". That was my point, I'm teaching a kid right now and he did the same mistakes I've done my whole life dividing numbers. Back at high school, I've would correct my mistake using another division method and think the algorithm didn't work for every case, wich is incorrect. Maybe then this kind of little mistake I made should be adressed in the article. By the way, I've interviewed my friends at Facebook and almost all of them didn't use the algorithm. The majority did (200/4) + (1/4), maybe because they did the same mistake as me. Even my dad, 20+ years engineer got 5.25 dividing without paying atention. Perhaps there's a flaw in how division is taught in my state here in Brasil. Or It was just my bad.

Oani000 (talk) 16:28, 10 November 2016 (UTC)


 * There are some advantages to being ancient! Back in the '60s we had to learn manual methods at age 10.  No calculators, I bought my first one at university, so school work was manual, slide rule or log tables.  I suspect that manual methods are not properly taught anywhere now - most people have a better calculator in their mobile phone than existed on the planet back then.


 * It wasn't just simple long division though, we also had to master £sd and avoirdupois weights and measures. Personally I prefer the tableau method, IMHO it is clearer and less prone to just this sort of column mistake - but that could just be familiarity.  The basic algorithm works for all cases, it just gets more complex with increasing number of digits!  Homework might be a 6 or 8 digit number divided by a 3 or 4 digit number and it took a complete page of an exercise book.  Regards, Martin of Sheffield (talk) 17:29, 10 November 2016 (UTC)

In this example, 37 is to be divided into 1260257.
Please don't use the phrase "x divided into y". It is confusing and ambiguous and should be deprecated. Some people use it to mean y/x while others read it as "x divided into y parts" and so interpret it as x/y. John G Hasler (talk) —Preceding undated comment added 14:40, 17 February 2018 (UTC)
 * I agree. "Divided by" is more common, not ambiguous and less confusing, as corresponding to the order of terms in y / x or y ÷ x. I have edited the page accordingly. D.Lazard (talk) 16:06, 17 February 2018 (UTC)