Long division continued past the ones place

0:00Twenty-nine metres of ribbon, shared equally between two friends. Fourteen metres each. And one metre left over. That leftover metre is the whole of this video. You cannot hand it to either of them and you are not allowed to throw it away, so it has to be cut. Half a metre each. And a half is five tenths, so half a metre is nought point five of a metre.

0:27Twenty-nine divided by two is fourteen point five. Nothing there was hard. But the answer needed a place the question did not have. Now share the same twenty-nine metres between four friends. Seven metres each, and again one metre left over. This time it has to split four ways. There is a quick way to finish this, worth seeing before we do anything else. Twenty-nine over four. If the bottom were ten, or a hundred, we could read the answer straight off, because those are the numbers our decimal places are built from.

1:08So hunt for one. Four times twenty-five is a hundred. Multiply the top by twenty-five as well, and twenty-nine over four becomes seven hundred and twenty-five over a hundred. Seven point two five. Seven metres and twenty-five centimetres each — the leftover metre became nought point two five apiece. That was quick. So let us find out how far it goes. Ten divided by three. Hunt for the same thing. Can we write ten over three with a bottom of ten, or a hundred, or a thousand?

1:47Try ten. Three would have to divide ten, and it does not. A hundred. No. A thousand. Still no. And this is not bad luck that runs out after a while. Three shares no factor at all with ten, so no power of ten will ever be divisible by it, however far you go. The hunt does not just fail here. It fails permanently. And that is uncomfortable, because of all the bottom numbers from two to a hundred, only fourteen can be handled this way. Eighty-five cannot.

2:24It is not a method. It is a lucky break that we happened to get twice. So we need something that always works. We already have it, and it has been quietly doing something more interesting than anyone said. Long division. Take one thousand three hundred and twenty-four, shared between four, and watch it as sharing rather than as digits. One thousand between four. Nobody gets a whole thousand, so break it up. One thousand is ten hundreds.

2:58Ten hundreds, plus the three hundreds already there, is thirteen hundreds. Thirteen hundreds between four is three hundreds each, with one hundred left over. Break that too. One hundred is ten tens, plus the two already there, is twelve tens. That is three tens each, nothing left. And four ones between four is one each. Three hundreds, three tens, one one. Three hundred and thirty-one. Every single step was the same move: something too big to share gets broken into ten of the next thing down.

3:37Now change one digit. One thousand three hundred and twenty-five. The first three steps are identical. Three hundreds each, three tens each. And then, five ones between four. One each. And one one left over. This is where school usually stops. Three hundred and thirty-one, remainder one. And that is true. Three hundred and thirty-one fours, plus one, is one thousand three hundred and twenty-five. But something easy to miss has gone wrong. We stopped because we ran out of digits, not because we ran out of remainder — and those are completely different reasons to stop.

4:22Out of every four-digit number you could put there, three quarters of them do this. So look at what we have actually been doing every time we got stuck. A thousand would not share, so it became ten hundreds. A hundred would not share, so it became ten tens. There is one one that will not share. Nobody ever said the breaking had to stop. One one is ten tenths. Not roughly, not as a trick — a one is ten tenths, in exactly the same way that a hundred is ten tens.

4:58So break it. Ten tenths, between four. Two tenths each, and two tenths left over. And notice we invented nothing. No new rule, no instruction to bring down a zero. The ten appears because ten tenths is written as a ten. Two tenths left, and four friends. So do it again, one place lower. Two tenths is twenty hundredths. Twenty hundredths between four is five hundredths each. And nothing is left over. Now we can stop — and this time we are stopping for the right reason.

5:37So each share is three hundreds, three tens, one one, two tenths and five hundredths. Three hundred and thirty-one point two five. Check it. Three hundred and thirty-one point two five, times four, is one thousand three hundred and twenty-five exactly. Now the question everybody actually worries about. Where does the point go? And the answer is that you never have to decide. It is already there. It went in the moment we broke a one into tenths, because that is where the sharing crossed below the ones.

6:16Everything before that crossing is the whole-number part. Everything after it is not. So the point is not a rule to remember and not something you place at the end by eye. If you forget where it goes, ask which step broke up a one. That was checked on nearly two thousand different numbers, against four different divisors, and it never once landed anywhere else. And now the quick method is worth coming back to, because for this division it does work.

6:50Four times twenty-five is a hundred, so multiply top and bottom by twenty-five. Thirty-three thousand one hundred and twenty-five, over a hundred. Three hundred and thirty-one point two five. The same answer. And that is not wasted effort — it is the strongest thing we have done so far. Two methods that share none of their machinery landed on the same number. The quick one is fast and only sometimes available. The long one is slower and always available.

7:26Where both work, running both is how you find out you were right. One more, and this one goes further down. Two hundred and thirty-seven, shared between eight. Two hundreds between eight gives nobody a hundred, so they become twenty tens, plus the three already there: twenty-three tens. That is two tens each, seven tens left. Those become seventy ones, plus seven, seventy-seven ones. Seventy-seven between eight is nine each, five ones left.

8:01And now the crossing. Five ones become fifty tenths. Fifty between eight is six tenths each, two tenths left. Two tenths become twenty hundredths. Two hundredths each, four hundredths left. Four hundredths become forty thousandths. Five thousandths each, and nothing left. Twenty-nine point six two five. Five columns wide, and every one opened because the column before it had something that would not share. Try four of them, both ways.

8:37Eighteen over five. Five times two is ten, so double the top as well: thirty-six over ten. Three point six. Four hundred and fifteen over four. Multiply by twenty-five: one hundred and three point seven five. One thousand two hundred and seventeen over two. Multiply by five: six hundred and eight point five. And four thousand eight hundred and twenty-seven over eight, times a hundred and twenty-five: six hundred and three point three seven five.

9:14Every one of those bottom numbers divided a power of ten, which is the only reason the quick route was on offer at all. Long division gives exactly the same four answers. Which is what you want. If two methods that work differently ever disagreed, one of them would be wrong. Last thing, and it is what exam questions are built out of. One thousand five hundred and twenty-six, divided by four, is three hundred and eighty-one point five.

9:49But suppose you are handed four answers to choose between, and they all look like the same digits with the point moved. Look again. Two of them carry an extra zero, so their digits are genuinely different, and checking the point alone would not save you. Do the same with three thousand five hundred and sixty-seven over eight and this time it really is only the point that moves. Four hundred and forty-five point eight seven five.

10:20Which is the honest version of all of this. The digits come out of the sharing. The point comes out of the step where the sharing crossed below the ones. Neither is decoration and neither is guesswork. Both are records of something you already did.