Mesopotamian base-60: place value in a sexagesimal system

0:00Every system so far has bundled by ten. This one bundles by sixty. Nobody knows why, and the guesses are worth hearing. One: sixty sits close to the periods people were already counting — a month of about thirty days, a year of about three hundred and sixty. Two: sixty divides easily, which makes fractions comfortable. Three: an older list of landmark numbers — one, ten, sixty, six hundred, three thousand six hundred, thirty-six thousand — settled into the powers of sixty.

0:39Look at that older list for a second. It climbs by ten, then six, then ten, then six, then ten: two different steps, so it was never a base system at all. But take every second rung of it and you get one, sixty, three thousand six hundred. The base was already hiding inside it. And before this feels remote, look at a clock. One hour is sixty minutes. One minute is sixty seconds.

1:10Which means an hour is three thousand six hundred seconds — sixty sixties, the second landmark of this system, still being counted today. A day is eighty-six thousand four hundred seconds, and the only reason those numbers look odd is that we write them in ten. So the system did not die. Part of it is on your wrist. What it did with those landmarks, though, is the interesting part.

1:39It stopped drawing them. Start with what it had to draw. Two marks. An upright mark, worth one. A corner mark, worth ten. One, two, three are one, two and three uprights. Ten is a single corner mark. Eleven is a corner mark and an upright beside it. Fifty-nine — the largest it ever needs inside one place — is five corner marks and nine uprights. Fourteen marks, which is the most any single place will ever ask for.

2:14Two marks, and every number from one to fifty-nine, each one built additively: tens first, then units. Now go past fifty-nine. Six hundred and forty is ten sixties and a forty left over. Seven thousand five hundred and thirty is two of three thousand six hundred, five sixties, and a thirty. Written that way, six hundred and forty needs a sign for sixty, ten copies of it, and four corner marks. Fourteen marks in all.

2:49But look at what the grouping produced. A ten, and a forty. Two numbers, each of them below sixty, each of them writable with the two marks you already have. The seven thousand five hundred and thirty came out as three: a two, a five, and a thirty. One rule governs those groups, and it is the same rule as before. No place may hold sixty or more. Suppose you had written one of three thousand six hundred, seventy sixties, and a two.

3:23That is a perfectly good number — seven thousand eight hundred and two — but it is not a numeral, because seventy sixties includes sixty sixties. And sixty sixties is one of three thousand six hundred. Trade it: two of three thousand six hundred, ten sixties, and a two. Same number, and now every place is under sixty. That is exactly what the writer does anyway — group largest first, and no place ever comes out holding sixty.

3:56Every number below a hundred thousand, checked one at a time: not one place over. Here is the move everything so far has been walking towards, and it is a subtraction. Stop writing the landmark signs. Six hundred and forty was a ten, a sign for sixty, and a forty. Take the sign away and write just the ten and the forty, in that order. Nothing has been lost, because the ORDER now says what the sign used to say: the group on the right counts ones, the group to its left counts sixties.

4:36Fourteen marks became five. And the saving is not the point — the sign set is. The landmark signs were the thing that had to keep being invented. Positions do not have to be invented. There are always more of them. So reading one is a procedure, and it starts from the right. The rightmost group counts ones. The next one left counts sixties. The next counts three thousand six hundreds.

5:06Take two, five, thirty. From the right: thirty ones, five sixties, two of three thousand six hundred. Seven thousand five hundred and thirty, read straight off the board. You already do this. In ten, the rightmost digit counts ones and the next counts tens. The structure is identical. Only the base and the marks are different. A system that works this way has a name — two of them, in fact.

5:37Positional. Or place value. Both names say the same thing: a mark's position fixes which landmark it counts. And that is what makes the sign set finite. Two marks here; ten digits in the system you grew up with. The number can be as large as you like. The set of things you have to draw does not grow with it. This is the idea, and several places arrived at it independently — Mesopotamia, the Maya, China, India.

6:10It is also probably not how the Mesopotamians got there. Their signs for one and for sixty looked alike, and an accident of writing may have led them into it. Either way, they had it. What they did not have is the other half. Because if position carries meaning, then an empty position has to be visible. Theirs was a blank space. Write one: a single upright. Write sixty: a single upright, in the sixties place, with nothing in the ones.

6:45Write three thousand six hundred: a single upright, two places along, with nothing after it. Three different numbers, and on the clay all three are one upright mark and some empty air. Do it again with a corner mark and two uprights: twelve, six hundred and two, thirty-six thousand and two. Six numbers. Two shapes. That is one example. The question is how bad it is. So take every number below four thousand, write each one the way the blank leaves it — marks run together, nothing recording an empty place — and count.

7:26One thousand six hundred and thirteen of those numbers share their marks with some other number. Three thousand nine hundred and ninety-nine numbers produce two thousand eight hundred and sixty-three distinct written forms. The numbers are all different. The writing is not. And notice where the fault is. It is not in the base, and it is not in the two marks. It is that a gap has no width you can count. One gap and two gaps look the same, so a reader cannot tell them apart.

8:02Later on, they did something about it. A mark for the blank. Not a number — a placeholder, put where a landmark is missing, so the empty place becomes something you can see. And it works, exactly as far as it is used. Below four thousand there are fifty-nine numbers with an empty place INSIDE them, with something after it. With a blank, every single one of those fifty-nine is ambiguous. With the placeholder, none of them is.

8:35But it was used inside a numeral, and not at the end. So the sixty-six numbers whose empty places are all at the end are exactly where they were, and three thousand six hundred still reads as one. Put the mark everywhere, then. Every empty place, end or middle. Now one, sixty and three thousand six hundred are three different numerals, and that particular failure is gone. But run the count again and it is still not clean: one thousand four hundred and forty-eight of them still collide.

9:13Because there is one more thing a reader needs, and it is easy to miss. Fifty-nine takes fourteen marks. A place is a heap, and nothing says where one heap ends and the next begins. Two uprights might be two, or it might be one sixty and one. Give every place exactly one symbol — a single sign for each of the counts a place can hold — and the collisions go to zero.

9:41That is the last thing this story needs, and it is the thing the next system did.