Why any number system needs a fixed, ordered sequence of symbols

0:00Two herds of cows stand in two fields. One is yours. One belongs to your neighbour. Here are three questions, and you have to answer all of them without using a single number. Have all your cows come back? Does your neighbour have more than you? And if so, how many more? No counting out loud. No numbers anywhere. That sounds like a trick. It is not — all three can be answered exactly, and people answered them this way long before anybody had a word for eight.

0:35The method they used is still, underneath everything else, what counting is. Here is the move. As each cow walks into the pen, you plant one stick in the ground. One cow, one stick. Another cow, another stick. You are not counting. You have not said a number. You are making a pairing — every cow matched to exactly one stick, and no stick used twice. When the last cow is in, you have a bundle that is not a picture of the herd but is exactly the same size as it.

1:11Eight cows in the pen; eight sticks in the ground. And now that bundle can go where the herd cannot. It can be carried, kept overnight, and brought back tomorrow. So: have all your cows come back? Walk them in again, and pull one stick for each. If a stick is still standing at the end, a cow is missing. Nothing left standing, nothing lost. Now the harder question. Lay your bundle beside your neighbour's and pair them off, stick against stick.

1:45Yours runs out. Three of theirs are still standing. That answers both questions at once: they have more, and they have three more. Nobody said a number. Nobody needed one. Which brings us to what counting actually is. When you count a herd out loud, you are doing exactly the same thing. You are pairing the cows against a sequence you agreed on before you started. One, two, three, four — those are the sticks, said instead of planted.

2:19The first cow gets the first term, the second cow the second, and so on down the line. And the term you stop on is the answer. That is the entire procedure. A number system is nothing more than a sequence to run that pairing along. Which means it can be made of almost anything — so the real question is what it has to be like. First requirement, and it sounds trivial until you break it.

2:49The order of the sequence has to be fixed, and agreed, before anybody starts. Suppose you and I both count the same three cows. You use a, b, c. I use c, b, a. You stop on c. I stop on a. Same herd, same pairing, two different answers. Neither of us made a mistake — we simply never agreed on the order, and the order is the only thing that makes the last term mean anything.

3:19A sequence whose order can shift is not a number system at all. But here is what does not matter, and this is the surprising half. The order you visit the cows in. Take six things and count them left to right. Now right to left. Now in any jumbled order you like. Six things can be walked through in seven hundred and twenty different orders. Every single one of them stops on the same term.

3:50That is not obvious, and it is the reason counting is worth anything. The sequence has to be rigid. The collection can be approached however you please. So: a fixed order, and a one-to-one pairing. What else does a sequence need? Let us try building one out of something everybody already knows by heart. The alphabet. a, b, c, d, e — a perfectly good sequence, and its order is fixed and agreed the world over.

4:23Count five cows and you stop on e. Count twenty-six and you stop on z. Count thirty, and you stop. There is nothing after z. Four cows have no term to pair with, so there is no answer at all — which is not the same as a wrong one. The alphabet has a ceiling, and the ceiling is twenty-six. That is fixable, and the fix is worth watching closely.

4:52When you run out of letters, do not stop. Start again, and keep track of having been round once. After z comes a-a. Then a-b, a-c, on to a-z, then b-a. Two letters carry you as far as z-z, which is seven hundred and two. And then three letters begin. The ceiling is gone. This sequence never runs out. Notice what it cost: nothing. The same twenty-six signs, used in strings.

5:22That is the trick behind every large number system there has ever been. Now a system built out of written signs instead. One stroke is one. Two strokes, two. Three strokes, three. Then a new sign for five — and four is written as one-before-five. Ten gets a sign of its own. Twenty is two of those. Follow that rule and the first twenty come out on their own, with nothing memorised.

5:53I, II, III, IV, V, VI, VII, VIII, IX, X. Then XI, and on up to XX. This is a serious number system — fixed order, one term per number, and it ran a continent for centuries. So put it through the same test the alphabet just failed. Does it go on forever? Do not guess. Climb. Write one, then two, then three, and keep going until the system cannot do it any more.

6:28It manages a thousand with a single sign. It manages two thousand, and three thousand. It gets to three thousand nine hundred and ninety-nine — the largest sign three times over, and then the rest. And at four thousand it stops, because there is no sign bigger than the one for a thousand. So this system has a ceiling as well. A far higher one, but a ceiling. Sticks, meanwhile, never run out. You can always plant one more.

7:01Which makes it sound as though sticks win. They do not, because there is a second test. Is it short enough to actually use? Write thirty in sticks: thirty marks, and you must count them again to read it back. Write thirty in the signs: three marks. Write thirty the way we write it now: two. And the signs are sometimes wonderfully short. A thousand is one mark. But not reliably.

7:30The most expensive number under their ceiling is three thousand eight hundred and eighty-eight, and it costs fifteen signs. Over that same range, ten digits and place value never need more than four — eleven signs of difference at the worst point. So put the three methods against the two tests. Sticks: they go on forever, and they are hopeless to read. The alphabet: easy to say, and it stops dead at twenty-six.

8:02Written signs: easy to say, and they stop too. Now look at what that grid is actually telling you. One of them passes the first test and fails the second. Two pass the second and fail the first. So neither test implies the other. Unending does not mean usable, and usable does not mean unending — and a system worth having has to be both. Letters running on into strings is the first thing here that manages it.

8:35One last thing, and it is about how long all of this took. Names for large numbers came a very long time before signs for them. There were words for one, ten, a hundred, a thousand, ten thousand — each one ten times the last. That list ran all the way up to ten to the twelve: thirteen names, climbing one power at a time, before anybody wrote a digit.

9:03The writing came later. Ten signs, including a mark for nothing, first appear written down around the year two hundred and fifty — the zero a simple dot. An astronomer sets them out around the year five hundred. They reach the Arab world by about eight hundred, and a book there argues for them soon after. They reach Europe by about eleven hundred, and around twelve hundred somebody there writes a book arguing for them too.

9:35And then Europe takes another four hundred and fifty years to actually adopt them. That is the longest single wait anywhere on the route — and it comes after everyone involved had already been told. Fourteen hundred years, from the first written zero to ordinary everyday use. Which is worth remembering the next time something obvious takes a while to catch on.