Why measurement units are built in tens

0:00You almost certainly own the figure this whole video is about. It is an ordinary ruler, the kind that lives in a pencil case. Take one centimetre of it, and blow it up until you can see inside. There are ten small divisions inside it, evenly spaced from one end to the other. Ten millimetres, filling one centimetre exactly. Not nine, not eleven. That is a relation you have known for years, and it has been sitting there doing nothing.

0:33Because look at what it actually says. One centimetre is being cut into ten equal parts. You have seen that move before. Quite recently, in fact. Cutting a unit into ten equal parts is exactly what makes a tenth. So a millimetre is not merely a small unit that happens to be handy. A millimetre is a tenth of a centimetre. It is a place. The same tenths place you have been writing in all along.

1:05Which means writing one millimetre in centimetres needs no working at all. It is nought point one centimetres. One in the tenths place, because that is where tenths go. And five millimetres is nought point five centimetres. Five tenths. Nothing was multiplied. Nothing was divided. The count was written down, and the point was put where the unit says it goes. Try one that runs past the unit. Twelve millimetres.

1:38That is twelve tenths of a centimetre, which is more tenths than fit inside one. Ten of those tenths make one whole centimetre, and two are left over. One centimetre and two tenths. One point two centimetres. And notice what happened to the digits. One, two. They did not change. The point simply landed between them. Now run it backwards. Five point six centimetres, in millimetres. Take the same two digits, and move the point back out to the end. Fifty six millimetres.

2:15The conversion never touches the digits. It only ever decides where the point goes. Since millimetres are the smallest thing on your ruler, it is worth knowing what they are for. Here is a strip two millimetres wide, drawn at its true size. A single sheet of newsprint is about five hundredths of a millimetre thick. A human hair is around a tenth of a millimetre across, so roughly twice the newsprint.

2:46The shell of the smallest known land snail is about seven tenths of a millimetre. One of the smallest ants measures under one millimetre, nose to tail. And a mustard seed sits at one to two millimetres, which makes it a giant in this company. For scale, the line a fine pen draws is about half a millimetre wide; a medium one, a whole millimetre; a bold one, one and a half.

3:14So most of that list would fit inside a single pen stroke. Now go the other way, and take a bigger unit. The metre. A hundred centimetres fill one metre exactly, which is another relation you have known for years. So a centimetre is a hundredth of a metre, and by now you know what that buys you. One centimetre is nought point nought one metres. The one goes in the hundredths place.

3:44Ten centimetres is ten hundredths, which is nought point one nought metres. And fifteen centimetres is fifteen hundredths. You can read that two ways and they agree. Ten hundredths and five hundredths. Or one tenth and five hundredths. Nought point one five metres. Same number, arrived at down two different roads, because it was one number all along. And there is a third relation hiding in the two you already have.

4:16There are a hundred centimetres in a metre, and ten millimetres inside every one of those centimetres. So a thousand millimetres fill one metre. Which makes a millimetre a thousandth of a metre. Nought point nought nought one. Now look at the three of them side by side. The metre is the unit. The centimetre is the hundredth. The millimetre is the thousandth. They are not three separate facts you have to remember.

4:45They are three places of one chain, with names painted on them. And that is why none of this ever needs new arithmetic. The same thing is true of weight, so watch it happen once more. A thousand grams fill one kilogram, which makes a gram a thousandth of a kilogram. Now here is two hundred and fifty four grams. Write it in kilograms. You could reach for a rule about dividing. Do not. Read it instead.

5:17Two hundred and fifty four grams is two hundred and fifty four thousandths of a kilogram. And a count of thousandths is something you already know how to break up. Two hundred thousandths, and fifty thousandths, and four more thousandths on top of those. Which is two tenths, five hundredths, and four thousandths. Nought point two five four kilograms. And the digits, again, never moved. One thing here bothers people, and it is worth clearing up.

5:52Ten grams. Write it in kilograms. Ten grams is ten thousandths of a kilogram, so you write nought point nought one nought. That last zero looks like a mistake, and it is not. It is there because the counting started in thousandths, and a gram is the third place along. The zero is a record of where you began. Of course you may tidy it away. Nought point nought one is the same weight, exactly.

6:22Ten thousandths and one hundredth are one number, which we already know, because ten thousandths make a hundredth. Neither form is wrong. One of them is just remembering more about how it got there. Here is my favourite picture in all of this. Five heaps of rice on one shelf. The first weighs ten kilograms. The second, one kilogram. The third, a tenth of a kilogram, which is a hundred grams.

6:54The fourth, a hundredth. Ten grams. The fifth, a thousandth. One single gram. Each heap is ten times the one after it, all the way down the shelf. Now weigh the whole shelf at once, by adding all five of them together. The answer is eleven point one one one kilograms. And that is not a coincidence, it is the whole idea in one number. Every single digit of it is one of those heaps. Read the number, and you have read the shelf.

7:28Money does exactly the same thing, and almost every currency in the world does it the same way. A hundred cents make one. So a cent is a hundredth, which by now should feel completely unsurprising. So here is seventy five cents, and the job is to write it with a point. Seventy five hundredths. Or seventy hundredths and five hundredths. Or seven tenths and five hundredths. Nought point seven five.

7:59The very same working as fifteen centimetres, on a completely different kind of thing. And that is worth stopping on for a second, because it is the reason any of this was worth learning. Length, weight and money are not three systems. They are one system, measuring three things. Last thing. Here is an old handwritten bill from a cafe, from around nineteen seventy. It has two money columns ruled down it, one for whole units and one for cents.

8:34Three coffees. One in the left column, fifty in the right. Two teas. Nothing, and twelve. One cake. Nothing, and fifteen. Add the right column. Fifty and twelve and fifteen is seventy seven. Now add the left column. Just the one. So the whole bill comes to one, and seventy seven. Now write that the modern way. One point seven seven. Those two digits after the point are the right hand column. They always were.

9:06The ruled line and the point are two ways of drawing the same boundary, and the boundary is where the unit ends.