Wednesday, June 17, 2009

Sumerian base sixty, and how it haunts us.

I like that everyone in the comments yesterday gets the significance of base sixty. Or at least, has the same reaction to it that I do, which is, "WHY IN...??!!??" The reason I know what little I know about number theory and history is due entirely to the Sumerians. I first heard about the base sixty thing in College Part One, and have been digging around through history and tech books ever since, trying to figure out why exactly they did it. No one really knows. There are half a dozen theories, but I've got my own. (Big shock.)

But first, a word or ten about the Sumerians themselves. 'Cause, see, they were really weird.

The biggest deal, when it comes to the Sumerians, is that nobody knows who they were. That is to say, they just popped up in the Middle East one day seven-ish thousand years ago. Culturally, linguistically, they're different than the people who lived around them. It's thought they came from central Asia, somewhere near modern Kazakhstan, but nobody really knows for sure. This is useful to know, because it plays a part in some of these number theories. Plus I like historic mysteries, so I almost HAVE to mention it.

Sumerians, and most other middle-eastern civs of the day, ran on clay. They were in a double river valley (Tigris and Euphrates, cradle of civilization, blah blah, don't you remember this from sixth grade?) and so they had all the clay they could use, and they did use it. For everything. Building, storage, cooking, decorating, and most importantly for us, record keeping. They didn't write on paper. They wrote on lumps of clay. If those lumps of clay were then fired (either on purpose, or by accident in the equivalent of house fires), they literally lasted forever unless someone ground them to dust.

Politically, the Sumerians used a theocratic city-state system. Which means each city-state was run by a PRIEST, and the cities were centered around a temple, not a palace. Each city was dedicated to a different god, and it's thought the pantheon's doings in later years reflect the political jockeying of the city-states in the very early years. Because this civilization lasted three thousand years, in different forms.

And depending on how you look at it, it survives today, right here on your computer, in the lower right corner if you're running Windows.

So, why did the Sumerians use base sixty? Well. There are theories. Lots of theories. In short:

-It has been suggested (since the fourth century, no less) that base sixty was chosen because it has a huge number of divisors (numbers it can be divided by), and therefore was 'easiest to use'. I've had modern mathematicians offer this as a theory, also, when I've discussed it with them. The problem with this theory is, civilizations 'choose' which base number system to use when they're counting goats on their fingers; they have no concept of what a divisor IS at the time the base system is established.

-The system is based on the days of the year. 365.24something days, rounded down to 360, and from there sixty becomes a 'natural' unit. This has the same problem as the first theory: when they were establishing the base system for their civilization they couldn't have known how many days there were in the year. They needed the advanced math to figure out the days in the year, BEFORE they could choose base sixty. There are related theories of base sixty established by 'natural' reckoning, like the hours of the day, the visible diameter of the sun, or number of planets (that they could observe with the naked eye). You still need advanced math to figure this stuff out BEFORE you decide to use base sixty.

-It is based on their system of weights and measures. Which totally screws up cause and effect, like the earlier ideas.

-It arose when the Sumerians moved into the area, and combined with another civilization that was already in the area, and somehow base ten plus base six (and why in hell would they be using base six? It just triggers more questions) equals base sixty. Sure, you can get base sixty easily from that doing the MATH, but culturally it makes no real sense. Usually one civilization snuffs out the other - at least culturally, if not literally - and moves on from there.

-It's really base twelve (duodecimal), which isn't totally unheard of in other parts of the world, like Western Europe (a dozen donuts, anyone?)

-Mystical reasons. The mathematicians were also priests and used numerology all the time. Different gods were represented by different numbers, and the upper god of Heaven (An/Anu) was sixty. Again there's a big question of cause and effect going on.

What do I think? As usual, I think it's a combination of all these reasons. Or most of them.

Going from what I know of the Sumerian numbering system (and you're gonna have to take my word for it, or read the article, 'cause I don't wanna be here all day), I think they were running equally on base ten and base twelve, and the sixty came up just as an easy factor for both those systems. (5x12=60, 6x10=60, that's the lowest number where the two systems 'meet' neatly.) It would make sense then, that their most 'good' god, the highest, most celestial, would be assigned the number they would see as wholeness, the one that is the neatest factor of both numerical systems they were using.

Base ten was probably the original system, the holdover from goat-counting on their fingers, and was, for the most part, used for practical purposes.

Base twelve was probably created by the priest/ruler class. I bet it was used, originally, for 'holy' purposes, for all those reasons listed above. 360 days in the year, six visible planets, etc. It looked to the priests like the natural world ran on sixes and twelves, so that's what they used. Note that the stuff we still use base twelve for is the stuff they would have considered holy - time, planets, astronomy, geometry.

As for everything else, like their accounting (boy howdy, do we have piles of their accounting tablets)? It's a jumble of the two systems. There are symbols in their writing system for 1, 10, 60, 600 (10x6), 3,600 (60x60 or 36x10). And so on. All other numbers were written as multiples of these symbols. What's that look like to YOU? To me that looks like base six/twelve and base ten jumbled together.

Now. This probably sounds insane, like a really dumbass way to run a civilization. But you know what? We're still dealing with vestiges of this dumbass system every day. And not just when you buy a dozen donuts instead of ten. The Sumerian system evolved over three thousand years, and was picked up by other civilizations (notably the Phonecians), and from there the Greeks, then the Romans, and here we are today, seven thousand-odd years later, still looking at dozens (see that? A special word for groupings of twelve, in an allegedly base ten civilization) of holdovers from those crazyass Sumerian priests/math geeks.

Look at your clock. Twenty-four hours in a day? Sorta. It's even more base twelve than that. Twelve hours of sunlight, and twelve hours of darkness. That's how it was originally figured. The clock - such as it was, usually a sundial - was 'reset' every sunrise and sunset. In older texts, even Roman ones, there are references to 'summer hours' and 'winter hours'. Because the days, regardless of length, were divided into twelve equal parts, and since the days are longer in summer, so are the hours. It doesn't stop there. Sixty minutes in an hour, sixty seconds in a minute. When we hit sixty, we start over at one. No one ever stops to think "why don't we go to a hundred before we start over?" or at least not very often.

Not only are there 360 degrees in a circle, it has carried over in the modern world to latitude and longitude, which is measured in minutes and seconds, all on base sixty just like time minutes and seconds on a clock. Yes. Modern GPS systems ultimately run on base twelve, like the ancient Sumerians.

Twelve months in a year. An arbitrary division, but twelve was seen as the 'neat' number to divide it into. Thirty days in a month; half of sixty. (We've since tinkered with the calendar, adding days and moving them around here and there, but the basic organization of it remains the same.)

And for Americans, the Imperial measurement system is a jumble of base ten and twelve. Twelve inches in a foot. 36 inches in a yard (3x12).

I imagine for the ancient Sumerians it was just as simple to switch back and forth, and they did it just as easily as modern humans, with just as little thought to it.


If we're still having fun with this, and everyone's interested, tomorrow I can do zero, and how it's not really the big deal it's made out to be - sort of.

For anyone finding this SUPER fascinating, I suggest "The Universal History of Numbers", by Georges Ifrah. (Oh, look! He has a new book out!) Unfortunately it is a translation, and therefore a bit dry, but it's still a really amazing book.


historicstitcher said...

The 12 hours for night/day carried over through to the middle ages, and into the Renaissance, until they figured out how to make a clock that could keep track of time more accurately.

Prior to the invention of the clock, folks listened for the church bells, which would be rung every "hour" (again, priests controlling time!) and the markets would open and close, and other indicators of the day were marked by the church bells.

It wasn't until 1577 that someone invented the minute hand! (I built a reproduction of a 15th century, single-handed clock that hangs in my living room. I can attest to the... shall we say... imprecision... of its timekeeping)

When clocks become popular (around 1884!!!) then we had to make time zones and stuff. "Normal" people didn't have or need clocks.

In fact, if you think about it, "normal" people lived by the daylight until the Industrial Revolution and the need to be "on time" for work! Until then, we did what needed done, and "time" didn't matter! (Can we go back to that, please?)

Excellent post again, Julie. Loving it! I love seeing your geek-side interpretations of scientific things, and then I see the social side of it. Let's write a book!

Oh, and as for the Sumerians and their base 60 - is there any indication that the numbers system has anything to do with the fact that there are 12 major Sumerian cities (each with an associated god) and that one could count on one hand (five) for each city, using the god-name like we use "tens"? Then it would be sort-of base 12, expanded to base 60 with the "full" counting, right? (Like Eridu-one, Eridu-two, Eridu-three, Eridu-four, Eridu-five, Larsa-one, Larsa-two, etc??)

I think I think too much sometimes.

Louiz said...

Very interesting, and another book for the list....

Shoveling Ferret said...

Your discussion of number systems is wonderful and what I have to say below doesn't really affect it, but I thought you might be interested:

Actually, the notion of the Sumerians as a group that came into southern Mesopotamia roughly around the same time as the appearance of cities (ie 7000 years ago) is a bit dated. More recent publications on the "Sumerian Problem" call into question whether Sumerian language use is indicative of ethnic group based on anthropological studies of language and ethnicity as well as a reexamination of the available linguistic and archaeological evidence. The archaeological evidence, albeit sparse for the earliest periods in the south also calls into question the idea of a large-scale migration of Sumerians into the area. Even the appearance of Akkadian as possible evidence of a second migration of a different group can be explained instead as simply the effect of writing appearing later in northern Mesopotamia.
A lot of the Sumerian-migration hypothesis was based on a mistaken understanding of the geology of the lower Tigris-Euphrates basin wherein it was assumed that the extreme southern portion of the area (Sumeria) couldn't support a population until relatively recently - ie around the time of the appearance of "Sumerian" cities. Instead, it seems that Sumeria could have supported a population much further back into the past than originally thought, also suggesting it more likely that the Sumerians were (relatively) indigenous.
Study of both Akkadian and Sumerian seem to indicate that the people themselves used "Akkadian" and "Sumerian" to refer to written and spoken languages and to regions but NOT to separate ethnic groups.
Part of the issue too is how very popular migration/diffusion hypotheses were in archaeology and history, especially of the Old World for much of the 20th Century. Those ideas tend to get ingrained and perpetuated.

I can hook you up with a PDF article that provides a pretty decent summary and bibliography of the Sumerian problem, as well as an interesting discussion of the archaeological study of ethnicity if you'd like.

PinkPorcupine said...

My husband would like to thank you, sincerely. I've always been a "words" person. He's a math person, theory to be exact. He can't explain without getting super technical, which means I don't understand. But this? I get this. :) Thanks!

Anonymous said...

Oooh, please do the one on zero! I just got done doing L'Hopital's Rule (division by zero, and how you can solve that in calculus) at the end of school, so I'm curious as to where it came from and everything. With you as the author, I'm sure it'll be interesting. :)

Roxie said...

The Summerians were actually aliens who colonized the T/E valley in a "return to the simple life" utopian experiment. Their landing base was an island called Atlantis which eventually collapsed due to tectonic tremors produced by their advanced engineering. They had six fingers one each hand. Why look for complicated explanations when the simple answer is so elegant?

historicstitcher said...

I take it all back - Roxie is right. The simple answer is always the right one.

Besides, who can deny the aliens-with-six-fingers theory????

TinkingBell said...

Love this - but don't forget some of the naturally occurring 6 and 12 - snowflakes, nautilus, basalt, crystals etc - these are more often 3/6/12 than they are 5/10/20

Loving these posts - see - never come away without learning something!

Ann said...

Wow, you have crazy-smart readers. The fact that they can just come up with lots of info re. timekeeping and Sumerians off the cuff (whereas I have trouble remembering my authors' academic affiliations) makes me wonder if I'll be up to the task of getting that masters in cultural anthropology I've been thinking about.

Anonymous said...

well you didnt talk about what clocks they used to do math so if i could rate it on a scale from 1-10 i would rate it a 3.

Anonymous said...

12 months in a year was positively NOT an arbitrary division. Did you ever notice the Moon?

Julie said...

The lunar year has thirteen months.

Thong Hoi Wei said...

About the 60 minutes, 60 seconds... There's 100 split seconds if that's any consolation.

Anonymous said...

actually you can count pretty well in sixes on your fingers if you count no fingers as 1 and this system is in use in the middle east.

Also, dozens are a useful organic measurement, divisible into 2,3,4,6 as you say. do you often cut a cake into five pieces? or usually four or six? we don't really do 5s much so there's no real reason for base 10 to be necessarily more natural than base 12.

John said...

Julie, the length of the lunar cycle is 29.5 days, not 28. The ancient Babylonians overcame this problem by having alternating months of 29 and 30 days (as do the Jews who adopted the Babylonian Calendar as being more accurate than the earlier Canaanite one) This gave 534 days to a year, 11 days short. So every 3 year they inserted a duplicate month making a 13th month. But then there would be 2.75 days to many in the 3rd year. These would accumulate to require still another calendrical adjustment. Check out the Sumerian site at