Divisions that never end

0:00Ten divided by three. Let us actually do it, one regrouping at a time. Ten ones, shared among three. Three each, and one left over. That one cannot be split three ways as it stands, so break it. One becomes ten tenths. Ten tenths among three. Three tenths each, and one tenth left over. Break that one too. Ten hundredths among three. Three each, one hundredth over. And again. And again.

0:31Something has already gone wrong here, and it is worth naming exactly what. Look at the leftovers rather than at the answer. One. Then one. Then one. Then one again. It is the same number every single time. And that is not bad luck. Ten tenths among three leaves one tenth for the same reason ten ones among three leaves one one. The pieces keep getting smaller, but the arithmetic does not change at all.

1:04So the leftover is never going to be nothing. And a division only finishes when the leftover reaches nothing. This one cannot finish. Not because we ran out of patience, but because there is no step that could ever stop it. That deserves saying carefully, because it is the whole idea. At any step of a long division, only two things matter. What you are dividing by, and what is left over.

1:35The divisor never changes. So the leftover on its own decides the next digit, and the next leftover. Same leftover, same everything that follows. Which means the moment a leftover turns up for a second time, the working from there is a copy of the working from the first time. Not similar. A copy. So it is committed, for ever, from that moment on. That was checked on every division of one by the numbers up to sixty, run four hundred steps deep. Eighteen thousand times a leftover came round again, and every time the digit that followed was the one it gave before.

2:21So what is the answer to ten divided by three? Three point three three three, and the threes never stop. We write it with dots on the end, and those dots are not laziness, and not an approximation either. They are a claim. They say we know exactly what comes next, all the way down. Which is worth separating from something people often confuse it with. Not ending is not the same as not being known.

2:51Every digit here is completely determined. There are simply infinitely many of them. Now a harder one. One divided by seven. One whole among seven. Nothing each, and one whole left over. Break it. Ten tenths among seven. One tenth each, three tenths over. Those three become thirty hundredths. Four each, two over. Twenty thousandths among seven. Two each, six over. Sixty among seven. Eight each, four over. Forty among seven. Five each, five over.

3:31Fifty among seven. Seven each — and one left over. One. Which is exactly where we started. So write the leftovers out in order. One, three, two, six, four, five, and back to one. Six of them, and then the chain closes on itself. Do not draw that as a row. Draw it as a ring. Because that is what it is. Six positions, and a marker that walks round them and arrives back where it set off.

4:02And look at which six numbers turned up. One, two, three, four, five, six. Every leftover that could possibly exist below seven, each one exactly once. The ring did not happen to close. It closed because it had run out of anywhere else to go. Now put the digits beside the leftovers. One, four, two, eight, five, seven. Every leftover on the ring produces exactly one digit, so the digits make a ring of their own, turning in step with the first.

4:38When the leftovers come back round, the digits have no choice but to come round with them. One divided by seven is nought point one four two eight five seven, repeating for ever. Six digits, because six leftovers. The length of the repeat was never a mystery about the digits. It is a fact about the leftovers. And now the general argument, which costs almost nothing. Divide by seven, and every leftover is smaller than seven.

5:12So a leftover can only be one, two, three, four, five or six. Six possibilities, and that is all there are. Now take seven steps. Seven steps and six possible leftovers, so one of them has to appear twice. That is not a fact about seven. Divide by anything, and there are always fewer possible leftovers than the divisor. So either you reach nothing and stop, or something repeats. There is no third thing that can happen.

5:46It even tells you how long the repeat can be. Never longer than the divisor minus one. And that limit is exactly right, not a safe over-estimate — seven, seventeen, nineteen and six others below a hundred all reach it. But it is a ceiling, not an answer. Ten divided by nine repeats after a single digit. A hundred divided by eleven takes two — nought nine, nought nine, and you have to count that leading zero or you will get it wrong.

6:21So knowing that a division repeats tells you nothing at all about how soon. Which brings us to something that looks like a conjuring trick. Take the block. One four two eight five seven. Multiply it by two. Two eight five seven one four. By three. Four two eight five seven one. By four, by five, by six — and every time you get those same six digits, in the same cyclic order, started somewhere else.

6:53Nothing new ever appears, and nothing is ever lost. It is not a trick, and we have already drawn the reason for it. Two sevenths, three sevenths and the rest all travel the same ring. They simply get on at a different stop. Then multiply it by seven. And the pattern breaks. One four two eight five seven times seven is nine nine nine nine nine nine. Which is not a rotation of anything. It is six nines.

7:26But it is the most sensible answer it could possibly have given. Seven sevenths is one whole. And nought point nine nine nine, repeating, is also one whole. So the block times seven filling every place with nines is just the arithmetic agreeing with itself. The trick and the thing that breaks the trick turn out to be the same fact. Seven does this. So does seventeen — one divided by seventeen repeats after sixteen digits, the longest it possibly could.

8:03Below a hundred there are nine numbers like that. Below ten thousand, four hundred and sixty-seven. They keep turning up. So are there infinitely many of them? Nobody knows. That question was asked in nineteen twenty-seven, and it has not been answered yet. Not hard. Open. A century of mathematicians have not settled it. Which is worth sitting with for a moment. You can walk to the edge of what anyone knows, starting from a long division you could do on paper.

8:39One question left, and it ties the whole thing off. Which divisions do end? Halves end. One half is nought point five. A quarter is nought point two five. An eighth, a sixteenth, a thirty-second — every one of them stops. Fifths end too. One fifth is nought point two, a twenty-fifth is nought point nought four, and so on down. And notice how many places each one needs. A quarter needs two, an eighth needs three, a sixteenth four — the count just follows how many twos are underneath.

9:18But thirds do not end. Sevenths do not. Elevenths do not. So what is special about two and five? Ten is two times five. A decimal is a fraction whose bottom is a power of ten, so the only bottoms that can ever fit are the ones built out of ten's own factors. A division ends exactly when its divisor, reduced, is made only of twos and fives. Every other division there has ever been repeats — and now you know what it is repeating, and why.