The Mayan system, and a placeholder symbol for zero

0:00Everything so far has come from one part of the world, and the ideas travelled between the places that had them. This one did not travel. On another continent, with no contact of any kind, the same idea arrived: let a mark's position say which landmark it counts. That is the strongest evidence there is that place value is a discovery, not a habit of one culture. And they went one step further than the people who thought of it first. They gave the empty place a mark of its own.

0:37A shell, standing for nothing. So this should be the happy ending. It is not, and the reason is one number on a list. Here is the list of landmarks they grouped by. One. Twenty. Three hundred and sixty. Seven thousand two hundred. A hundred and forty-four thousand. Read the steps between them, one at a time. One to twenty is twenty. Twenty to three hundred and sixty is eighteen. Then twenty, then twenty.

1:10Twenty, eighteen, twenty, twenty. One of those is not like the others, and it is the second. For this to be a base, every step has to be the same number, and it is nearly the same number. Twenty twenties is four hundred. Four hundred is not on the list, and where it should have been there is a three hundred and sixty instead. That is the whole of what goes wrong, and it goes wrong quietly.

1:40Start with what they had to draw. Three marks and no more. A shell, worth nothing. A dot, worth one. A bar, worth five. One, two, three, four are that many dots. Five is a bar. Six is a bar and a dot. Ten is two bars. Nineteen is three bars and four dots: seven marks, and that is the most any single place ever needs. Every value a place can hold is drawn this way and read back, and no two look alike.

2:14Three marks, and everything a place can hold. Now the places. They wrote them one above another instead of side by side. The bottom row counts ones. The row above counts twenties. The row above that counts three hundred and sixties. Same idea, turned through a right angle: position still says which landmark, only now it is height. One rule governs the rows, and it is the rule you would expect: no place may hold the step to the next rung.

2:49The ones place stops at nineteen. The twenties place stops at seventeen, because eighteen twenties is a landmark. Every number below a hundred and forty-four thousand, written this way and read back: all of them come back right, and not one place comes out overfull. So the writing works. Watch a real one. Three rows. From the top: four dots, then one dot above two bars, then a shell. Take them one row at a time, and start from the bottom.

3:25The shell says the ones place is empty. Nothing there. The middle row is two bars and a dot, which is eleven, and it counts twenties: eleven twenties is two hundred and twenty. The top row is four dots, and it counts three hundred and sixties: four of them is one thousand four hundred and forty. Add the three: one thousand six hundred and sixty. And that is the number written beside it as the total. The rows and the total agree.

3:59Underneath the numeral there is a line showing the working, and one term of it is wrong. The first two terms are right: four times three hundred and sixty, and eleven times twenty. The third term should be nothing times one, because the bottom row is a shell. What is written instead is three times ten. Both halves of that term are wrong: the count, and the landmark too, because ten is not on this ladder at all.

4:29Add the line up as it stands and you get one thousand six hundred and ninety. Which is thirty more than the total sitting right beside it. Stop and notice how you just caught that, because the method is worth more than the slip. You did not need to know what the right answer was. You needed two things that were supposed to say the same number, and you checked whether they did.

4:56The rows say one thousand six hundred and sixty. The total says one thousand six hundred and sixty. The working says something else. Two out of three agree, so the odd one out is the working, and you can go and find which term did it. A number written down twice by different routes is a number you can check. That is not a trick for spotting slips. It is most of what checking anything ever is.

5:27Back to the one number that is wrong on the list. In a system that really bundled by twenty, the third landmark would be twenty twenties, which is four hundred. It is three hundred and sixty. So where does four hundred go? It goes past the third rung, not onto it. Four hundred is one three hundred and sixty, and two twenties left over. Three rows to write a number that a genuine base twenty would write in two.

5:57Nobody made a mistake here. The choice is deliberate, and it is usually connected to their calendars — a suggestion, not a settled fact. It is a design decision with a price, and the next two minutes are the price. Here is the thing a base gives you that nothing else does. Take a number, move every place up one rung, and put an empty place underneath. In a real base twenty, that is multiplying by twenty, always.

6:31No multiplication is carried out. You just move the marks. Try it here. Move a stack up one rung and each place gets multiplied by its own step — and the steps are not all the same. For some numbers the shift multiplies by twenty. For others it multiplies by eighteen. Below four thousand, the shift is a multiplication by twenty for two hundred and thirty-nine numbers, and by eighteen for seventeen.

7:02For the other three thousand seven hundred and forty-three, shifting the marks up a rung is not multiplying by anything whole at all. One rung out of place, and moving the marks stops meaning one thing. The same failure shows up wherever the landmarks have to be multiplied together. There are five landmarks, so there are twenty-five ways to multiply two of them. In a genuine base twenty, fifteen of those products are landmarks again, and the ten that are not have run off the top.

7:38On this ladder, thirteen land back on it and twelve do not, and some of the twelve are nowhere near the top. Twenty times twenty is the very first one you would try, and it misses. So multiplying two numbers stops being bookkeeping with landmarks and starts being ordinary work. You can still compute. It just stops being easy, and easy was the entire point of having a base. None of that touches the shell, and the shell is worth measuring on its own.

8:13Take every number below seven thousand two hundred and write each one three ways. With a shell in every empty place, no two numbers share a picture. Not one collision. Leave the empty places out instead, and one thousand and fifty-eight of those numbers share their marks with some other number. Run all the marks together with nothing separating the places, and four thousand five hundred and forty of them do.

8:43Seven hundred and twenty of those numbers have an empty place somewhere in them, and the shell is the only thing standing between you and losing them. So the mark for nothing is not decoration. It is what makes the positions readable. Which leaves two separate things on the board, and they have been getting confused with each other this whole time. Place value is one idea: position says which landmark a mark counts, and an empty position gets a mark of its own.

9:18A base is a different idea: every landmark is the same step above the one below it. You can have the first without the second, and here is a civilisation that did, and wrote every number it needed. What they gave up was not the writing. It was everything that comes from the steps being equal. The only way to tell two ideas apart is to find something that has one and not the other.

9:47This is that something. And it leaves an obvious question: what happens when a system finally has both?