Dividing when the dividend has a decimal

0:00A ribbon three point nine metres long, to be cut into ten equal pieces. How long is each piece? You can feel the answer coming. Three point nine, shared among ten, is nought point three nine. Each piece is thirty-nine centimetres. And almost everybody gets that right by moving the point one place left, without ever being able to say why that works. So let us find out why — because the reason turns out to be the reason everything else here works too.

0:35Take the point off first. Three point nine is thirty-nine tenths. Thirty-nine over ten. And dividing by ten is the same as multiplying by one tenth. So we have thirty-nine over ten, times one over ten. Tops together, bottoms together. Thirty-nine over a hundred. Watch what actually happened there. The top never changed. Thirty-nine went in and thirty-nine came out. The only thing that moved was the bottom, which picked up a zero.

1:10Thirty-nine over a hundred is nought point three nine. Now cut the same ribbon into a hundred pieces instead. Same start. Thirty-nine over ten, times one over a hundred. Thirty-nine over a thousand. The top still has not changed. The bottom has picked up two zeros this time instead of one. Nought point nought three nine. Three point nine centimetres per piece. And here is a trap worth stopping on. A hundred has two zeros, so it is tempting to say the answer has two decimal places.

1:50Count them. Nought point nought three nine has three. The two zeros tell you how far the point travels. They do not tell you where it ends up, because the one place three point nine already had is still there underneath. So look at what the shortcut is actually doing. Three point nine. Nought point three nine. Nought point nought three nine. Three, nine. Three, nine. Three, nine. The digits never move. They never even change.

2:23What changes is what each one is worth. The three was three ones, then three tenths, then three hundredths. Every digit dropped one rank, and then another, because that is what sharing among ten does to a place-value unit. A whole column of digits dropping one rank at once — with the digits standing still, that is exactly a point moving one step left. It is not a separate trick. It is the same regrouping, applied to every column at once.

2:56That was checked across eight thousand cases, and the digits came out identical every time. So we can fill in a whole table without doing any dividing at all. Start with eighteen point seven. Divided by ten, one point eight seven. By a hundred, nought point one eight seven. By a thousand, nought point nought one eight seven. By ten thousand, nought point nought nought one eight seven. One, eight, seven. All the way down. Only the point is walking left.

3:33Twenty-one point one does the same thing. Two point one one, then nought point two one one, and on. And so does nought point one three, which starts below one already and just keeps going. Nought point nought one three. Nought point nought nought one three. The zeros in front are holding the empty places open. But two of the rows are not given a starting number at all. One of them says only this: divided by a hundred, the answer is two point one four six.

4:08So what did it start as? Run the arrow the other way. If dividing by a hundred moved the point two places left, then getting back means moving it two places right. Two point one four six becomes two hundred and fourteen point six. Check it. Two hundred and fourteen point six, divided by a hundred, is two point one four six. It works. The last row gives even less. Divided by ten thousand, the answer is nought point nought nought five eight.

4:44Four places right. Five, eight — and it lands as a whole number. Fifty-eight, with no point at all. Which raises a fair question. If fifty-eight has no point, where does the rule start? At the right-hand end. A whole number is a decimal whose point has nothing after it, so we do not write it. One hundred and twenty-three divided by ten. Put the point at the end, step it once left. Twelve point three.

5:17Twelve divided by a thousand is more interesting. Three steps left, but only two digits. So zeros get written in front to hold the empty places open. Nought point nought one two. They are not padding. They are place holders, and leaving one out makes the answer wrong by a factor of ten. Same digits, same rule. It never mattered whether the number started with a point or not. Now the harder-looking case. Nine and a half kilograms of sugar, packed equally into four bags.

5:55Nine point five, divided by four. And four is not ten, so no shortcut is available. This has to be done properly, by sharing out place by place. And notice something before we start. Nine point five has an ones column and a tenths column, and no tens column at all. Nine ones between four is two each, with one one left over. That one becomes ten tenths, and five tenths are already sitting there, so fifteen tenths.

6:28Fifteen between four is three each, three tenths left. Those become thirty hundredths. Seven each, two hundredths left, which become twenty thousandths. Five each, and nothing left. Two point three seven five kilograms in every bag. Now, was any of that new? Look at where the point came from. It went in when one one became ten tenths — the same test as always. The only difference is when that happened.

7:02Dividing a four-digit whole number, that crossing comes at the fifth step out of six. You work through thousands, hundreds, tens and ones first. Dividing nine point five, it comes at the second step out of four. Same event. Same method. It simply arrives sooner, because there are fewer columns above the point to get through. A point in the number you are dividing does not change the procedure at all. It changes where the procedure starts.

7:37Push that as far as it goes. Nought point nought six, shared among five. The whole number is already under one, so the sharing starts below the ones before anything at all has happened. Read it out in places. Zero ones. Zero tenths. Six hundredths. Zero ones between five is zero each. Nothing to share and nothing left over. Zero tenths between five is zero each as well. Six hundredths between five is one each, with one hundredth left.

8:12That becomes ten thousandths, which is two each, and nothing remains. Nought point nought one two. Look hard at that answer, because it contains something easy to throw away. There is a zero sitting in the tenths place. It is very tempting to read that as nothing happening, and to leave it out. But it is not nothing. It is a record. It says that when the tenths came round, there were none to hand out.

8:42Take it away and you get nought point one two. Which is ten times too big. Nought point one two, times five, is nought point six — not nought point nought six. A zero inside a number is not a spacer. It is the statement that this place is empty, and every place left of it has been dealt with. It is the one digit in an answer that people delete, and it is load-bearing.

9:12One last thing, and it is the payoff. Divide one hundred and thirty-two by four. Thirty-three. Now thirteen point two by four. Three point three. One point three two by four, nought point three three. Nought point one three two by four, nought point nought three three. Every answer is a tenth of the one above, and the digits never changed. One division, done once, and four answers out of it.

9:43But do not let that look tidier than it is. Try it with a divisor of eight. One hundred and twenty-six over eight is fifteen point seven five — already two places, and each step down adds another. By nought point nought one two six over eight, the answer needs six decimal places. So shifting the number you divide does not keep the answer the same length. It keeps the answer's digits.

10:12And that is the whole of it. The digits come from the division, the point comes from where the sharing crossed. Move one and you have not touched the other.