# Why did Sumerians use base 60 mathematics?October 21, 2007 9:55 AM   Subscribe

An hour has 60 minutes and a minute has 60 seconds because the Sumerians used a base 60 counting system. Why 60? A plausible explanation is that they could count to 12 with one hand, and to 60 with both hands. Alternate explanations from the MacTutor History of Mathematics archive.
posted by russilwvong (47 comments total) 11 users marked this as a favorite

if it were true that they counted to 12 with their hands, it seems pretty likely that they would have a special symbol for 12, not for 10.
posted by empath at 10:16 AM on October 21, 2007 [1 favorite]

Plus, why not include the base knuckle? using that method, I can count to 16 with one hand.
posted by delmoi at 10:25 AM on October 21, 2007

Great information, but the chummy asides in the first link are almost too much to bear.
posted by migurski at 10:35 AM on October 21, 2007

Not if this hurt their brains like it does mine. I go with the QWERTY effect. Some Sumerian dude named Illb Stega had a pile of 60 rocks, and it took him a minute to throw them.
posted by weapons-grade pandemonium at 10:42 AM on October 21, 2007 [1 favorite]

Fun and interesting! Thanks russil.
You get the sexagenary (? sp) cycle in traditional Chinese time-counts too, and a twelve-hour day.
Made me wonder if the Heavenly Stems and Earthly Branches were based on a similar count plus the ten of all digits but can't seem to find an opinion.
posted by Abiezer at 10:49 AM on October 21, 2007

That first link doesn't look very reliable to me.

My recollection is that the 12/24 hour day is due to Egyptian calendrics -- it's related to stars that rose at the beginning of every 10-day period in the year. (There were 36 "decan" stars/star groups.) 18 of these stars would rise during a 12-hour night, but 3 were assigned to dusk and 3 to dawn, leaving 12 of these decans rising during an evening. When the calendar was regularized, this became our 12 hour system.
posted by cgs06 at 10:51 AM on October 21, 2007

"Do you seriously think that if humans had 8 fingers instead of 10 a “dime” would have 10 pennies? No, it would have 8, and there’d be 8 dimes to the dollar."

Er... what?

"Under the old [British] currency of pounds, shillings, and pence, the pound was made up of 240 "old pence" (denoted by the symbol d), with twelve old pence in a shilling and 20 shillings (denoted by the symbol s) in a pound."[1]

And 12 is a very divisible number - much more so than 10; That may contribute to it's use.
posted by Auz at 10:51 AM on October 21, 2007

Wow, that's absurd. I mean, the "they counted to 12 on their fingers and 60 using both hands". Way to retrofit an argument to a fact.

Twelve was used because it's easily divisible into halves, thirds, quarters, and sixths.

Sixty because it's additionally easily divisible into fifths, tenths, and twelfths.

Being able to easily divide things into like-sized parts is very useful in a wide variety of circumstances.
posted by Flunkie at 10:57 AM on October 21, 2007 [1 favorite]

It probably had more to do with weights and measures rather than something abstact. A person could probably carry about 60 units of something, close to a pound. These would have been everyday calculations counted over and over, bundled accordingly.
posted by Brian B. at 11:08 AM on October 21, 2007

Flunkie has it. 60 has factors 30 20 15 12 10 6 5 4 3 2.
posted by anthill at 11:11 AM on October 21, 2007

when i clicked on "count to twelve with one hand" i thought maybe sumerians had six fingers, including thumb. the piece's reference to a canadian dollar being worth 64 cents dates it somewhat.
posted by bruce at 11:50 AM on October 21, 2007

However, the argument "12 was used because it is easily dividable" sort of begs the question. Why 12 instead of 24? Why 60, and not 120, or 240, or 480?

On the other hand, empath does have a point. The evidence from the second link (where they have a special symbol for 10s, not for 12s) seems to contradict the premise that the Sumerians counted in base 12.
posted by moonbiter at 11:50 AM on October 21, 2007

On the third hand, I just noticed that the second link is talking about the Babylonians, and not the Sumerians, who seem to have done things a bit differently.
posted by moonbiter at 12:02 PM on October 21, 2007

Here's what I want to know. Why are there 24 hours a day in China?

It's easy to find references to the date that China and other east Asian countries adopted a Western calendar, but it's hard to find references to their adopting Western time. For example, China officially went to the Gregorian calendar for official business on January 1, 1912, and it became commonly used in 1929 -- but I find no mention of any changing of clocks. One would think that would be a big deal. Imagine trying to change your worldview to include seconds of different length, perhaps as part of some lunatic rush to metricize the world. There would be outright rebellion. So either the clocks did not have to change, or they did not exist at all -- and is it thinkable that advanced civilizations such as China and Japan could have no concept of measured time before 1912?
posted by darksasami at 12:02 PM on October 21, 2007

I can count to 21, but only if I'm naked. (And it's not really cold out)
posted by Eekacat at 12:06 PM on October 21, 2007

Somebody told me people in India still use the knuckle counting method today, but I've never been so not sure if that is true.

However, the argument "12 was used because it is easily dividable" sort of begs the question. Why 12 instead of 24? Why 60, and not 120, or 240, or 480?

Because 60 is the first multiple of 12 also divisible by 10, so you can use it in both systems. Sumerians were smart. Still, any base 12 system destroys base 10 systems in usability. Base 10 really blows because you can only divide it into halves and fifths without going into decimals and fifths are useless, base 8 or even a base 6 system would be more useful.
posted by afu at 12:08 PM on October 21, 2007

Wikipedia Knows All
The hour was originally defined in ancient civilizations (including those of Egypt, Sumer, India, and China) as either one twelfth of the time between sunrise and sunset or one twenty-fourth of a full day. In either case the division reflected the widespread use of a duodecimal numbering system. The importance of 12 has been attributed to the number of lunar cycles in a year, and also to the fact that humans have 12 finger bones (phalanges) on one hand (3 on each of 4 fingers). (It is possible to count to 12 with your thumb touching each finger bone in turn.) There is also a widespread tendency to make analogies among sets of data (12 months, 12 zodiacal signs, 12 hours, a dozen).

The Ancient Egyptian civilization is usually credited with establishing the division of the night into 12 parts, although there were many variations over the centuries. Astronomers in the Middle Kingdom (9th and 10th Dynasties) observed a set of 36 decan stars throughout the year. These star tables have been found on the lids of coffins of the period. The heliacal rising of the next decan star marked the start of a new civil week, which was then 10 days. The period from sunset to sunrise was marked by 18 decan stars. Three of these were assigned to each of the two twilight periods, so the period of total darkness was marked by the remaining 12 decan stars, resulting in the 12 divisions of the night. The time between the appearance of each of these decan stars over the horizon during the night would have been about 40 modern minutes. During the New Kingdom, the system was simplified, using a set of 24 stars, 12 of which marked the passage of the night.

posted by public at 12:14 PM on October 21, 2007 [2 favorites]

Plus, why not include the base knuckle? using that method, I can count to 16 with one hand.

I gather the theory is that they did use the base knuckle, but not the thumb, as the thumb was used for pointing at the knuckles. I wrote about decimalisation and the alternatives on my own site earlier this year. Why we use one counting system over another in various contexts is very much tied to arbitrary historical events. For example, the US began using decimal currency mostly because Thomas Jefferson pushed for it at a time when only Russia's currency was already decimal, since Peter the Great declared it so.
posted by scottreynen at 12:22 PM on October 21, 2007

Because 60 is the first multiple of 12 also divisible by 10, so you can use it in both systems.

Precisely, worth-repeatingly, so. 12 is the smallest number that offers the utility of thirds, halves, and quarters. Isn't that also the basis for the 3:2 ratios at the heart of the Pythagorean tuning? Was 3:2 important in Sumerian culture as well?
posted by mwhybark at 12:44 PM on October 21, 2007

I'm now wondering if soft clay record keeping had anything to do with base 6. There are six sides to a square, which could be used to stamp their numbers. They could have used multiple dies, but one die seems to work too. The shape of their singular would accommodate a layered stamping process, with a stamp for 1, 3, 4, 5, and these overlapping to be stamped twice or three times to produce the other digits. The 10 stamp could replicate 20 and 30, with a 40 stamp changing the shape to allow 50 and 60 in combination with the 10. At no time would more than three stamps be required to produce either side, nor more than five stamps total for any number 1-60, except 39.
posted by Brian B. at 12:48 PM on October 21, 2007

darkasami - I suspect it wasn't too big a wrench fro China to move from a twelve hour system to a 24-hour one. They already split the hours further into ? which were the length of our quarter-hour (varied a bit over history) and I seem to recall you could say first half of the two-hour period or second half too. Bit of the top of my head that, though.
posted by Abiezer at 1:05 PM on October 21, 2007

I can count to one hundred using both hands. Learned it back when I was in preschool, actually.

As for divisors, that's true, but honesly I can't help but wonder if its more rooted in superstition than math. People back then didn't believe in their religions, they *BELIEVED* in their religions. At the very least I'd bet that there were religious reasons along with mathematical reasions.

And, for all that people can, correctly, argue that base 10 has limits, we're rather stuck with it. I'll bet we see spelling reform in English before we see any significant population adopting a different base. Given that it'd make sense to switch over to metric time, but I suspect we won't see that anytime soon either.
posted by sotonohito at 1:07 PM on October 21, 2007 [1 favorite]

eek - Unicode lost again - it said ? kè where the question mark is.
posted by Abiezer at 1:08 PM on October 21, 2007

"Now actually, that is not the answer that I had in mind because the book that I got this problem out of wants you to do it in base eight. But don't panic. Base eight is just like base ten really, if you're missing two fingers. Shall we have a go at it? Hang on..."

I love Tom Lehrer.
posted by ZachsMind at 1:42 PM on October 21, 2007

You know who else could count to twelve on one hand?
posted by champthom at 2:20 PM on October 21, 2007

The divisibility argument is very persuasive. Food must have been the most commonly divided commodity. Everybody knows that numbers are an abstraction, but even fingers are an abstraction if they're simply a way of keeping track of twelve onions, sixty olives, or four chickens.
posted by weapons-grade pandemonium at 3:03 PM on October 21, 2007

When you hold your hand at arms length and align the bottom of your hand to the horizon, the width of each finger represents 15 minutes of travel as the sun sets. Yes, this explains nothing.
posted by StickyCarpet at 3:07 PM on October 21, 2007

champthom: "You know who else could count to twelve on one hand?"

Anne Boleyn?
posted by ZachsMind at 3:10 PM on October 21, 2007

I like this kind of stuff, but agree this is an overly simplistic explanation. For one thing, he misses the very plausible influence of the sundial in determining time-increments.

I would bet that a four-directional sundial created by two straight lines (4 right angles on a circle) would have made it quite likely that the day-time would be measured in a number divisible by four. Why 12 won out over 8 and 16 could be attributed to the digits of the fingers, but I think there are plenty of other ways to go - if they had a right angle as 90 degrees, for instance, then dividing it by 3 makes more sense than dividing it by four... I don't know enough about sumerians, but this seems just as intuitive.
posted by mdn at 3:13 PM on October 21, 2007

But the sun isn't even there, StickyCarpet. It's a half finger below where you see it.
Does that help?
posted by weapons-grade pandemonium at 3:14 PM on October 21, 2007

OMG, in 2001 the Canadian Dollar was only worth US\$.64!!!
posted by wayside at 3:14 PM on October 21, 2007

I've always thought that the Sumerians got base 12 from the fact that there are on average 12 lunar cycles a year. Once you start farming it makes sense to keep track of what time of year it is and the most obvious way of doing this is watching the moon. Once they had a 12 month year they would have worked down from there in the rest of their time keeping systems. It's just speculation but it makes sense.
posted by afu at 3:40 PM on October 21, 2007 [1 favorite]

I can count to one hundred using both hands.

If I were more dextrous, I could count to 1023 on both hands just by using binary.
posted by ROU_Xenophobe at 3:43 PM on October 21, 2007

They used base sixty because time is cubic, and 4*4*4=60.
posted by Citizen Premier at 4:42 PM on October 21, 2007 [1 favorite]

If you can count to 12 on one hand, wouldn't it be 144 you could count to on both?
posted by nebulawindphone at 4:58 PM on October 21, 2007

ah hell....what's 4 between friends? 4*4*4=60
posted by MNDZ at 5:03 PM on October 21, 2007

They used base sixty because time is cubic, and 4*4*4=60.

ah hell....what's 4 between friends? 4*4*4=60

is this a joke / reference to something, or are you both just weirdly bad at math in the same way?

posted by mdn at 5:12 PM on October 21, 2007 [1 favorite]

ack, forgot to close that italic, sorry.
posted by mdn at 5:13 PM on October 21, 2007

The Universal History of Numbers is an excellent read on the world's various historical origins for numeracy and number forms, and detailed examinations of the advantages and disadvantages the systems gave their users. It's an excellent bit of readable research.
posted by five fresh fish at 6:34 PM on October 21, 2007 [1 favorite]

Oh, and all your questions in this thread are answered in that book. If someone happens to have easy access to it, it'd be easy enough to summarize what the author feels is historically accurate and what's reasonable conjecture.
posted by five fresh fish at 6:36 PM on October 21, 2007

I don't think the divisibility argument works for me. People use that because they don't want to switch to the metric system. 2.5 is as easy as 1/4, and it makes calculations easier.
posted by Mr. Gunn at 6:51 PM on October 21, 2007

I don't think the divisibility argument works for me. People use that because they don't want to switch to the metric system.

Or, perhaps people don't want to switch to the metric system because it makes divisibility more complicated? The claim is that metric is simpler, but it is only simpler because it's what you're used to. There is no inherent benefit to base 10, whereas something that works out from base 2 naturally splits in two, which is useful when we divide things in half. This may not be useful for what you do, but talk to people who actually deal with measurements, and find out if keeping track of 1/64 of a unit or .0156125 of that unit is more appealing...

Carpentry works in 8, 16, though also 12, 60... computer numbers are all 128, 256 etc because they're binary. 10 stands out as a pyramid number (1+2+3+4), and we could make some kind of claim about 5 being a nice foundational number for humans to think in, so 5 by 2 having an advantage for that, but really you could say that about 4 just as easily, & in terms of pure mathematic usability, 8 seems superior. What it comes down to is the metric system is just based on our having ten fingers, and if it is just as easy to deal with 2.5 as 2, or 1.25 as 1 (if we compare a quarter or an eighth of 8) then how can you argue it's harder to multiply 8s than 10s?You just get used to it, like memorizing your times tables, or knowing the numbers that computer memory comes in.
posted by mdn at 7:35 PM on October 21, 2007

Somebody told me people in India still use the knuckle counting method today,

Yeah, I still count like this when i need to track something greater that 10.
posted by dhruva at 8:16 PM on October 21, 2007

The Sumerians probably used an abacus, said to derive from sand and pepples. Maybe it was round, with 12 spokes or grooves.
posted by Brian B. at 8:41 PM on October 21, 2007

Oh, and all your questions in this thread are answered in that book. If someone happens to have easy access to it, it'd be easy enough to summarize what the author feels is historically accurate and what's reasonable conjecture.

According to Georges Ifrah, author of The Universal History of Numbers, the Sumerians were the only ancient culture to ever use a base 60 system. Why and how did they developed this oddity? It's a mystery. He discusses a few theories, even a mystical one, but only to dismiss them. Then he gives his own theory, which seems to be the very same theory loosely treated in the first link.

He speculates that base 60 was a compromise between two cultures of counting, one with base 5 counting and one with a base 12 system derived from counting on one hand as described in the link. He posits that word for the number ten being the same word for fingers does not denote a decimal system but a simple fact that humans have ten fingers. He explains that the Sumerian words for one, two, and three were legacies from a primal counting system of one, two, and many. The words for six through nine were elisions and contractions of the word for five and the words for one through four. So this hinted at an earlier culture of base 5 counting. 5 x 12 = 60, so base 60 was the compromise between base 5 and base 12. Ten was chosen as the basis for writing out larger numbers because it would produce smaller number strings and would be familiar to base 5 counters. 10 was also used as an aid in dealing with base 12 orders of magnitude in their oral counting traditions.

After all this, he dismisses his own theory as a fiction. There is no evidence to the origins of the base 60 system.
posted by effwerd at 9:05 PM on October 21, 2007

Similarly to sotonohito counting to 100 on both hands trick, I came up with a way to count to twenty-five using just my hands when I was in grade school (I was all proud because I thought of it on my own, but I later discovered that I had 'invented' something that had been thought of many, many times before me.)

Right hand ticks of 1, 2, 3, 4, 5, for every completed right hand, the left hand ticks off a finger, so left hand 2, right hand 3 equals 13.

Combining 'my' idea with the knuckle counting trick gets me up to 65 on my hands.
posted by quin at 9:39 PM on October 21, 2007

If I were more dextrous, I could count to 1023 on both hands just by using binary.

I actually do this, and I'm not particularly dextrous. Just finger touching table/desk/leg/etc.=1, finger not touching=0.
posted by DevilsAdvocate at 10:51 AM on October 22, 2007

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