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I don't understand why long division would be a mystery. It's just repeated subtraction - you are finding how many times you can subtract the divisor from the d
by geomark 7y ago
I don't understand why long division would be a mystery. It's just repeated subtraction - you are finding how many times you can subtract the divisor from the dividend, in a systematic way.
- pjscott 7y agoIt's almost never explained with such lucidity. If you find a hundred people who can perform long division, I would be surprised if twenty of them have any idea how it works.
- dboreham 7y agoYou're never going to find 100 people who can perform long division.
- pjscott 7y agoSure you can! Just look at a typical fifth- or sixth-grade classroom. Most of them can perform long division, briefly, before the test. They may even remember it for several days after the test.
- mavhc 7y agoWhat none of them can tell you is why they've been taught long division, when short division is faster. And it's because you need long division to do division in algebra
- abakker 7y agoWhat is short division? I only remember being taught long division and having a calculator. Is there another method?
- Tomte 7y agoFascinating. I didn't know, either. After looking at the English Wikipedia I found that a) what I learned in school is long division b) the notation with the dividend and the divisor separated by a parenthesis and the result on top of a horizontal bar is totally alien to me c) I've never seen short division before d) the German Wikipedia doesn't even have an article about short division
- jacobush 7y agoThanks! What you said, except my own point d) that Swedish Wikipedia tells that apparently short division is also used in Swedish elementary schools now.
- jeltz 7y agoI went to elementary school in the 90s and we were taught both long and short.
- shantly 7y agoHoly shit, I didn’t realize the shorter way of writing long division that I invented in early elementary school because I got sick of taking up so much space and introducing copying error potential had a name!
- beatgammit 7y agoInteresting, I'll start using that short version now. I always hated the long version because it took up a ton of vertical space and I didn't need the extra verbosity. In fact, I'll probably teach me kid that method when he gets to division in school.
- marakv2 7y agoFrom my quick google it's arriving at the answer, without the step by step process of calculating each division. Edit: although if someone else can explain it better I'd really like to know. From what I have searched,(quickly mind you), it seems to be a mind set.
- 7y ago
- marakv2 7y agoBecause knowing both helps realise how it works. Yes it's faster to use short division, but long helps you se the step by step process, giving a better understanding.
- jeltz 7y agoYeah, when we were taught short and long division I had the same question and it was not until quite late in high school we got the explanation. I went to school in Sweden where we do teach all this things, but in a strange order.
- hashmush 7y agoWent to school in Sweden too in the 90-00s. I wasn't taught long division until I had to divide polynomials in my last year of high school, and that course was an elective.
- mavhc 7y agoApart from the long division method pointing out that division is just repeated subtraction, that's the only reason to use it.
- darkerside 7y agoI've looked at both Wikipedia pages and still can't really understand the difference. Is it anything more than notational? Either way, you're calculating divisors for each component of the problem, and it only seems to matter whether you calculate it on paper or in your head.
- leetcrew 7y agothe way I learned it, it's just a notational difference. either way, you are dividing the divisor into each place of the dividend (starting with the most significant digit) and carrying over the divisor to the next place. short division is just a more implicit way of doing this that saves space. the shorthand version doesn't really work for divisors with more than one digit though.
- fractaled 7y agoWhat doesn't work with short division and multi-digit divisors?
- leetcrew 7y agoit's not that the algorithm "breaks"; it's just that it stops being a convenient notation for humans. the "short" notation relies on the ability to squeeze the remainder in between the digits of the dividend. if you have a divisor greater than ten, it's possible to have a remainder with more than one digit. two or more remainder digits is a lot harder to write legibly between the digits of the dividend. you also end up subtracting larger numbers in your head, which is error prone. finally, most people only memorize the multiplication table up to 12x12 or so. once they can't simply do a lookup into their memorized table, they switch to more of a "guess and check" approach, where they will inevitably have to cross out or erase their previous work. in the end, it introduces a lot of unnecessary opportunities to make mistakes and hides stuff that people can't do reliably in their heads.
- shantly 7y agoIt’s notation only. I “invented” it as a kid to save space on division problems and reduce the chance of copying errors. It’s the same method as long division (and TIL it already existed!)
- Mirioron 7y agoI think I've only briefly seen short division once. It was long division until fractions here.
- jpindar 7y agoOK millennial
- ikeyany 7y agoMultiplication tables are the most lucid way to explain long division. In order to get from 5 to 40, you need to keep adding 5s until you can't. So to divide 40 by 5, keep subtracting 5s until you can't any longer.
- stjohnswarts 7y agoExactly, I find it far easier to explain LD as just fast subtraction combined with multiplication to my kid than common core methods which I had to stare at for a while. The most critical method here is how to break up components of a number into addition "parts" as well as decimal positions.
- _bxg1 7y agoIt wasn't until I learned about for-loops that I started thinking of multiplication as repeated addition and division as repeated subtraction. Before then I thought purely in terms of geometry.
- novok 7y agoI think it depends on how you were taught the concepts in elementary school. I think a lot of it is based on things like bar charts[0] and visual representations at first, so it's possible you were never taught how it was repeated subtraction and addition, how powers are repeated multiplication and so on. https://en.wikipedia.org/wiki/Singapore_math#Bar_modeling https://en.wikipedia.org/wiki/Singapore_math#Bar_modeling
- _bxg1 7y agoYep. I remember in college when I made that connection and realized you could create infinite "levels" of operation, with each repeating the previous one N times.
- pas 7y agohttps://www.youtube.com/playlist?list=PL3A50BB9C34AB36B3 https://www.youtube.com/playlist?list=PL3A50BB9C34AB36B3 For anyone interested in going down the rabbit hole of those infinite levels.
- rocqua 7y agoIf you do this, 2 and 2 will always make 4, no matter what level you go to.
- pbhjpbhj 7y agoUp-arrow notation ftw.
- all2 7y agoI'm missing the context here. The only arrows I'm familiar with are vector notation and ion spin notation.
- chrisseaton 7y agoBecause the algorithm is so astronomically complicated that people can’t see any pattern or reason in it anymore and it becomes an opaque mystery. I have a PhD in computer science but I cannot understand long division.
- dboreham 7y ago>I have a PhD in computer science but I cannot understand long division I think you're trolling us a little: you could sit down with pencil and paper and think about how it must work and 20 minutes later you'd understand it, surely.
- chrisseaton 7y agoI have no intuitive understanding or mental model of the algorithm. I could follow the instructions if they’re written in front of me but I have never managed to understand why they work or to memorise them. I have a similar mental block for standard deviation - I always seem to have another ‘but why?’ question that eventually people answer with ‘because!’ and then we’ve both given up.
- dboreham 7y agoTry this? https://web.stanford.edu/class/ee486/doc/chap5.pdf https://web.stanford.edu/class/ee486/doc/chap5.pdf Edit: standard deviation is just a fancy name for root mean squared over the sample offsets. The square is there to deal with the fact that there are positive and negative offsets. The mean is the whole point of the exercise, and the root is to negate the fact that we squared earlier.
- chrisseaton 7y agoBut why not use the abs function or operator rather than squaring and rooting? This is how I always find maths and particularly statistics - I always just want to answer ‘but why’ and then people lose interest in explaining after a while.
- CraneWorm 7y agoOh, right! Easy! Try it with 1/7?
- fyp 7y agoSchool usually start with remainders instead of trying to teach repeating decimals at the same time. So here the answer is just 0 remain 1. The same process still works if you want to calculate the decimals though. Just pretend you're doing (1000000/7) * (1/1000000) or however many digits you want.
- theli0nheart 7y agoShift the decimal point of the operand right by N places (where N is how many decimal places of precision you want), do the long division, then shift the decimal point of the result left by N places, and you have your answer.
- marakv2 7y agoI looked up long division thinking that I had missed something, I just want to chime in for support of what you said. Afaikr it's just division, how man X are in y.
- pbhjpbhj 7y agoLong division calculates the multiple at each step; short division just finds the remainder. That doesn't seem that different but when you come to do it algebraically you need to do long division. I did both at primary school (age 10), but it wasn't until 6th form (age 16) that we used long division again, those who hadn't done it at primary struggled a little. 083.28 2.26... ___ 7 |583.00 That's short division (sometimes called the "bus stop method" in UK), top line is answer [quotient], next line is "remainders". You say "7 in to 5 won't go; 7 in to 58 is 56 [just from knowledge of times tables, it's 56 tens your dividing], with remainder 2 [write remainder down, usually as a superscript to the dividend]; 7 in to 23 goes 3, remainder 2 [write remainder down]". Now you have the answer 583/7 is 83 remainder 2; but you can continue and divide the 20 tenths by 7, and so on. Long division: 083.28 2.26... ___ 7 |583.00 [<-dividend] 56 [=8x7] -- 23 [2 from subtracting answer to 8x7 {8 is put in answer line}, 3 from the dividend] 21 [=3x7] -- 2.0 [2 is remainder, 0 from dividend] 1.4 [=2x7 is remainder, 0 from dividend] --- .60 .56 [=8x7] -- 4 [is the remainder in 100ths] What we're doing is taking five-hundred and saying can we divide that by seven-hundred, we can't. So then we say well how about fifty-eight tens, can we divide that by seven-tens. The tens cancel each other out ( 10/10==1 ), so 58/7 = 8r2 but this is really saying 580/70 is "80 lots of 7" and 20/7 left over. So now we add that 20 units to the 3 units we have already from the dividend we started with, so now we need to do 23/7. And so on ... I'd do an algebra example but it's a pain in the arse just using ASCII.
- billfruit 7y agoWell if long division is unintuitive, the procedure to compute square-roots by hand is even more mysterious, as if it has transcended dealing with numbers and instead become a symbolic manipulation. Thankfully it is something I never had to apply, even once in adult life till now.
- hervature 7y agoHow do you compute square roots by hand? Isn't it just a rudimentary root finding algorithm?
- all2 7y agoThe only by-hand method for finding roots I know is "Newton's Method". I think I learned this in pure calculus? https://en.m.wikipedia.org/wiki/Newton's_method https://en.m.wikipedia.org/wiki/Newton's_method
- chongli 7y agoThe older method is attributed to Hero of Alexandria [1]. [1] https://en.m.wikipedia.org/wiki/Hero%27s_method https://en.m.wikipedia.org/wiki/Hero%27s_method
- billfruit 7y agoThere is a method known as the the method of 'long-division' for finding the square root, that was what I was alluding to.
- jonsen 7y agoSquare Root By Long Division: https://www.solipsys.co.uk/new/SquareRootByLongDivision.html https://www.solipsys.co.uk/new/SquareRootByLongDivision.html
- gshubert17 7y agoThe calculation of the square root, relies on this result from squaring a number N = 10d + e (that is, single digits d and e, but d is in the 10s place): NN = (10d + e)(10d + e) = 100dd + 20de + ee Group the digits of the square by two, and add extra values to the "divisor" so that the extraction of the square root in effect undoes the squaring above. See, for example, https://sciencing.com/calculate-square-root-hand-5081134.html https://sciencing.com/calculate-square-root-hand-5081134.htm...
- krilly 7y agoI was never told this, and I'm sure the reason why is because my teacher didn't have a clue either