Chinese rod numerals: base-10 place value, one symbol short

0:00China had two number systems, and they were for two different jobs. One was written with a brush, for setting a quantity down. The other was built from rods — small sticks, laid out on a ruled surface — and it was the one you calculated with. It is the rods we are following: in use from at least the third century, and still in use in the seventeenth. And here is the thing about it. It is base ten. It is positional. The digits go left to right, biggest first.

0:35It is the system you already use, with exactly one part missing. What they did instead of that missing part is the whole of this. Start with a single place, and one kind of object: a rod. One, two, three, four, five are that many rods, standing upright side by side. Six could be six rods. It is not. Six is one rod laid across the top, standing for five, with a single upright hanging under it.

1:08Seven, eight, nine are that same crossed rod with two, three and four underneath. So nine costs five rods — exactly what five costs, and no digit ever costs more than that. Without the crossed rod, nine would have taken nine. That is what the turn is worth, before it is worth anything else. Now the part that has no equivalent in the system you grew up with. Every one of those nine drawings is also drawn a second way: the whole picture turned through a right angle.

1:45The uprights lie flat. The crossed rod stands up. One turned is a single flat rod. Nine turned is a standing rod with four flat ones under it. Nine digits, eighteen drawings, and not one drawing appears in both rows. That last point is the one to hold on to. Look at any drawing and you can say which of the two rows it came from. Which is useless, until you know what it is for.

2:16It is for the places, and the rule is as simple as it could be. The upright drawings go in the ones, the hundreds, the ten-thousands. The turned drawings go in the tens, the thousands, the hundred-thousands. Upright at every second place, starting from the ones. Turned at the others. So the form is decided by the position, not by the value. A four in the tens and a four in the ones are the same digit drawn two different ways.

2:48And two places next to each other never, ever use the same drawing. Hold that. It is about to do a job nothing else is doing. Read one. Four groups of rods, left to right. Two flat rods. Then a crossed rod with one under it. Then three flat. Then four upright. The rightmost group counts ones: four upright is four. Next left counts tens: three flat rods is three tens, thirty.

3:19Next counts hundreds: the crossed rod with one under it is six, so six hundred. And the leftmost counts thousands: two, so two thousand. Two thousand six hundred and thirty-four, in eleven rods, and no place needed more than four of them. Now take the turning away and see what breaks. Suppose every place used the upright drawings. Write forty-one. Four is four uprights. One is one upright. Lay them down side by side.

3:54Five uprights in a row. Which is exactly, and identically, the drawing of five. One arrangement of rods, two numbers, and nothing on the board to say where one place stops and the next begins. Put the turning back and they come apart at once: four flat rods and then one upright. Same five rods. One of them lying the other way, and the ambiguity is gone. That is one number. The question worth asking is how much of the problem the turning actually removes.

4:31So take every number below ten thousand, lay it out both ways, and count how many cannot be told apart from something else. With one drawing throughout, seven thousand one hundred and sixty-six of them share their rods with another number. Nine thousand nine hundred and ninety-nine numbers, and only three thousand seven hundred and forty different arrangements between them. With the two drawings alternating, the number that collide drops to seven hundred and fifty-nine.

5:05Nine thousand four hundred and nineteen arrangements, nearly one each. That is what the turning buys, and it is almost everything. Almost. Because there is one thing the turning cannot do, and it is the thing they never solved. If a place has no digit, nothing goes there. A gap. The same gap Mesopotamia left, on the other side of the world. It is easier to see here — the nine drawings are close to one another in size, so a hole in a row of them shows.

5:41Easier to see is not the same as marked. A gap has no fixed width. On a board, rods get nudged. And a reader who cannot count the gaps cannot count the places. So which numbers are actually lost? Look at what the seven hundred and fifty-nine have in common. Every single one of them has an empty place somewhere inside it. Every one. No two numbers with all their places filled are ever confused. Not one pair, anywhere below ten thousand.

6:16Take one hundred and one. Upright one, nothing, upright one — and with nothing in the middle, the two uprights sit next to each other. Two uprights in a row is the drawing of two. So one hundred and one, two hundred, and two are one arrangement of rods wearing three numbers. The turning did its job perfectly. It marked the boundary between two places. It cannot mark a place that is there and empty.

6:48And here is a detail that shows exactly what the boundary marker is doing. Two thousand six hundred and nineteen numbers below ten thousand have an empty place in them. Only seven hundred and fifty-nine are lost. So an empty place is what it takes, and it is not all it takes. Compare one hundred and one with one hundred and six. One hundred and one collides. One hundred and six does not.

7:19Because six carries that crossed rod on top, and the crossed rod belongs visibly to its own place. It anchors where the digit starts. Redundancy in the drawing is standing in for a symbol that is not there. It works, and it works partially, which is what a workaround is. So run the count a third time, with one thing added. The same rods, the same alternation, and a mark — any mark — wherever a place is empty.

7:52Below ten thousand: nothing collides. Not one number. Nine thousand nine hundred and ninety-nine numbers, nine thousand nine hundred and ninety-nine arrangements. One mark, and every ambiguity in the system disappears at once. Not a smaller number of them. All of them. That is the distance between this system and the one you write in. Which is a strange and rather moving place to end up. Base ten. Positional. Left to right, biggest first. Nine digits built from one repeated object.

8:32Everything is there except a way of writing nothing. And to cover for it they invented something genuinely clever — carrying the place boundary inside the digits themselves, by drawing every digit twice. Clever work, done to route around a gap that one symbol would have closed. A mark for an empty place is not a decoration on a number system. It is a part of it. The system that finally treated nothing as a number, rather than as a hole, is the one every one of us was taught.