A hundredth part, and continuing the split

0:00Here is a strip of paper, and against the ruler it measures eight units and nine tenths. Fold it exactly in half, and cut along the fold. How long is each half? Eight units and nine tenths, shared between two pieces. The eight whole units split cleanly. Four each. That leaves nine tenths to share between two, and nine will not split cleanly at all. Four tenths each, and one tenth left over in the middle with nowhere to go.

0:32So each half is four units, four tenths, and then a bit more. And the ruler has nothing at all to call that bit. Put the cut edge against the scale and look at where it actually lands. It is past the mark for four units and four tenths. It has not reached the mark for four units and five tenths. It sits between them. And not just anywhere between them, but exactly halfway.

1:01The scale can tell you which two marks it lies between, and then the scale has run out of things to say. You have met this difficulty before, one level up. When the whole units were too coarse to report a length, cutting each unit into ten fixed it. Now the tenths are too coarse. And the fix, it turns out, is exactly the same fix. Take one tenth, one of those small parts, and cut it into ten equal pieces of its own.

1:33Then do that to every tenth along the whole ruler. Each new piece is a tenth of a tenth. And it has a name. One hundredth. You can see straight away where that name comes from. There were ten tenths inside one unit, and every one of them has just become ten pieces. Ten tens is a hundred. So a hundred of these new pieces fit inside one whole unit.

2:00That is the entire idea, and there is nothing else hiding inside it. The handle that turned once has simply been turned again. Two trades fall out of that, and they are the only rules you need. Ten hundredths make one tenth. And a hundred hundredths make one whole unit. Watch the first one. Here are ten hundredths, laid end to end. They reach exactly as far as one tenth does. Not nearly. Exactly.

2:32So ten of them may be handed in for a single tenth, and a tenth may be broken back into ten of them. That is the same trade as ten tenths for a unit, and the same trade as ten ones for a ten. It has never once changed. Only the size of the parts has. Every trade in this whole system happens at ten, all the way up and all the way down.

2:59Now go back to that cut edge, stuck halfway between two marks with no name. The gap it was sitting in has been cut into ten, so there are marks inside it now. And the edge lands on one of them, dead on. The fifth. So each half is four units, four tenths, and five hundredths. There are other ways to say that, and they are all the very same length.

3:24Four tenths and five hundredths is forty five hundredths, so it is four units and forty five hundredths. Or forget the units entirely and count nothing but hundredths, from zero. Four hundred and forty five of them. Check that against the strip we started with. Eight hundred and ninety hundredths, halved, is four hundred and forty five. The fold was exact the whole time. We simply had no parts small enough to write it down with.

3:57Here is a piece of wire, and it has three names as well. Measured the obvious way, it is one unit, one tenth, and four hundredths. Now roll that tenth into hundredths. One tenth is ten hundredths, and four more makes fourteen. So it is one unit and fourteen hundredths. And roll the unit in too, since a unit is a hundred hundredths. One hundred and fourteen hundredths. Three names. One wire. Nothing was added and nothing was thrown away.

4:31Moving between them is only ever that trade, run forwards or run backwards. Which name you use depends entirely on what you are about to do next. For telling two lengths apart, the plain count is usually easiest, and here is why that matters. Three lengths. Three tenths. Three hundredths. And thirty three hundredths. They are built from the same digit and they look like close neighbours. They are not.

5:03Three tenths is thirty hundredths. Three hundredths is three of them. One of those is ten times the other. Not close. Not nearly the same. Ten times. And thirty three hundredths, the one that looks biggest, is only three hundredths past the first. Try another set. Three and six tenths. Three and six hundredths. Three and six tenths and six hundredths. The same six in three different places, and moving it one place to the left makes it worth ten times as much.

5:41Now add two of these together. Fifteen units, three tenths and four hundredths, plus two units, six tenths and eight hundredths. Line up the parts that match. Units under units, tenths under tenths, hundredths under hundredths. Fifteen and two is seventeen units. Three and six is nine tenths. Four and eight is twelve hundredths. Twelve hundredths. Which is a perfectly real count, and also more than a tenth's worth. So trade. Ten of them cross over as one tenth, and two stay behind.

6:19Nine tenths, plus the one that just arrived, makes ten tenths. And ten tenths is one whole unit. That unit crosses over too. Seventeen becomes eighteen, and the tenths column empties completely. Eighteen units and two hundredths. Here is the thing worth noticing, and it is really the point of all of this. Put that sum beside a perfectly ordinary one. Four hundred and eighty three, plus two hundred and sixty eight.

6:52Three ones and eight ones is eleven. Too many for the column, so ten cross over as one ten, and one stays. Eight tens and six tens and the one that arrived is fifteen tens. Too many again, so ten cross over as one hundred. Four hundreds and two hundreds and the one that arrived is seven hundred. Seven hundred and fifty one. Now look at the two of them side by side. Both overflow in the right hand column. Both overflow in the middle one.

7:28Both hand one part upward, twice, at exactly the same two moments. It is not a similar method. It is the same method, working on parts of a different size. Subtraction runs those trades backwards, and there is one moment in it worth slowing right down for. A board twenty five units and nine tenths long. Cut off six units, four tenths and seven hundredths. Start at the small end, with the hundredths.

8:01Zero hundredths, take away seven. There are none sitting there to take. So go one place up and break a tenth open. Nine tenths becomes eight tenths and ten hundredths. The board has not changed length. It is the same board, written a different way. Ten take seven is three hundredths. Eight take four is four tenths. Twenty five take six is nineteen units. Nineteen units, four tenths and three hundredths.

8:34One more, where you have to do that twice over. Fifteen units, three tenths and four hundredths, take away two units, six tenths and eight hundredths. Four hundredths cannot give up eight. So break a tenth. Three tenths becomes two, and four hundredths becomes fourteen. Fourteen take eight is six hundredths. Now two tenths cannot give up six. So break a unit. Fifteen becomes fourteen, and two tenths becomes twelve.

9:07Twelve take six is six tenths. Fourteen take two is twelve units. And beside it, six hundred and fifty three take two hundred and sixty eight, breaking open twice in exactly the same two places. So here is where all of this leaves us. The unit was cut into ten, and then the tenth was cut into ten again. And nothing anywhere says you have to stop. Cut a hundredth into ten and you have a thousandth. There is no last cut waiting at the end.