# base-12 numeral system

What could drive a human civilization to use base-12 system? As far as I know there have been such civilizations and they probably used their phalanges to count (instead of their fingers). I have found several possible reasons:

• the need to divide by 3 (probably the reason romans used base-12 fractions); other than the obvious, this it is also useful for angles in geometry
• natural events (like 12 months in a year) may make the system more natural
• counting can be done with one hand (while the other is used for something else)
• making clocks with 12 hours is easier (AFAIK that's why the decimal time never took off)

As the system is almost never used today, I guess there must be some reason. My only guess is trade necessitated using a single system and base-10 somehow won.

What led to the universal usage of the base-10 system? Can two numeral systems survive when trading or will one dominate?

• "natural events (like 12 months in a year) may make the system more natural" 12 months in a year is an arbitrary concept. You could just as easily use any number of months that reasonably evenly divide the number of days in the solar year into a not-too-large set. – a CVn Feb 5 '15 at 21:44
• @martinkunev ...which don't line up with the solar year. – a CVn Feb 5 '15 at 22:19
• @MichaelKjörling twelve months is the closest you can get; in most cultures, the months are twelve: en.wikipedia.org/wiki/Month#Months_in_various_calendars | my point was if you are making a calendar 12 emerges as a good approximation of the moon cycles in a year and that's what had driven humans to divide the year in 12 months – martinkunev Feb 5 '15 at 22:42
• @Tonny I wasn't sure that counted, since as far as I can tell, the entire Imperial system is a mess. – Rowanas Feb 6 '15 at 8:32
• @Rowanas I grew up on metric, but had to live in the UK for a year some 25 years ago. I think "a mess" is quite an understatement... It is getting better these days, but they still got a long way to go... – Tonny Feb 6 '15 at 12:52

Why do we use base 10?

We use base-10 because it's the system of counting used by the dominant groups in the early history of Europe and the middle east.

That being said, we do use other bases from time to time, such as for counting time. Our early systems for counting time at night were based on the passage of stars. Interestingly, 24 stars were used to divide the night, though it was divided into 12 periods. This was carried through to divide the day by 12 periods as well. The Babylonians brought in base 60 to count minutes and hours, which we use to count angles as well, since our early angle-based calculations were often astronomical in nature. (source)

Base 12 can be counted on one hand, but it's less convenient to count on the tarsals since someone farther away can't see as clearly what number you're indicating. If people had six fingers per hand, of course, we'd probably all be using base 12 right now.

Problems with base 10

Base 10 isn't a terribly efficient base to use, mostly because of its prime factorization. Dividing a number by anything besides 2 and 5 leads to a repeating decimal. Since people tend to divide by small numbers more often than bigger numbers, a counting system with 2 and 3 as prime factors would lead to fewer repeating decimals.

Other bases

So what base should we use instead? Often times base 8 and base 16 are suggested, but these aren't really that useful except for their easy conversion to binary, since 2 is their only prime factor. Let's consider three other candidates: 6, 12, and 30.

Base 30 seems like a good choice because it has 2, 3, and 5 as prime factors, but it would require 30 different characters to write in. There also isn't a good biological way to count to 30 on your fingers. Increasing the size of a given digit also increases the difficulty of computing digit-wise operations. In base 10, a multiplication table of pairwise operations contains 100 values. In base 30, there would be 900 such terms to multiply.

Base 12 is much simpler. We lose the ability to divide cleanly by fives, but twos and threes are more common. Counting on tarsals gives us a convenient way of counting biologically. Overall, not a bad choice.

However, my first choice for a counting base would be base 6. It's divisible by two and three, and adding divisibility by four doesn't add that much to a counting system since its only prime factor is two. Large numbers would be a bit longer, around 44% longer than base 12 numbers, to be exact ($1/(ln(2))$), but we'd only have 36 elementary operations instead of 144. Base 6 also makes zero-based counting incredibly convenient to do on your hands. Since the digits in base 6 are 0-5, all of which can be displayed on one hand, the right hand can be used for the first digit and the left hand for the second. Holding up five fingers on each hand would be read as 'fifty-five', instead of as 10, though this is 55 in base six, which is equal to 35 in base 10. Twelve would be two sixes, written '20', which would give us time that's an even multiple of our base.

Since you can count on both hands, it's possible to divide any base-6 number by two or three on your fingers pretty easily. Take your number (say, 5), and represent it with your right hand, with nothing on your left: ----- ||||| (Those are up and down fingers.)

Now, we subtract fingers off our right hand one at a time until it's divisible by what you're dividing it by. If we're dividing by 2, add 3 fingers to the left for each finger you put down. If we're dividing by 3, add two. Once your right hand is divisible by the number you want, divide it and you're done.

So 5/2 goes like this: ----- ||||| -> |||-- ||||- -> |||-- ||---, which equals 2.3, or 2.5 written in base 10. For 5/3, we get ----- ||||| -> ||||- ||| -> ||||- |, which equals 1.4, and is a repeating decimal in base 10.

You can repeat this process to divide by 4, and add fingers to the left hand to do two digit numbers, though you'll have to include some toes if you want the remainder.

I always thought the reason we use base-10 is that's how many fingers we have, although I have no reference for that.

There are some significant advantages with doing either Base-8 or Base-16 in a computerized world. Base-10 doesn't divide well into binary systems, it leads to inaccuracies which can compound and lead to significant errors. Base-8 or Base-16 would avoid that... but Base-12 doesn't really help. You can get around those issues in Base-10 or Base-12 computing, but you lose storage and computing speed.

The only advantage of Base-12 is it divides evenly by four, which Base-10 doesn't, and we tend to like to divide time into fours (seasons, hours). But I'd rather go Base-8 or Base-16, you get a lot more with either of those than you'd get from Base-12.

• Unfortunately, ancient people did not have computers in mind when they started counting. If they did, surely my parents would appreciate two's complement far more when I try to explain it to them :D – DaaaahWhoosh Feb 5 '15 at 21:20
• Base8 is odd, because if you have a power of 2 number of bits, it doesn't divide evenly into base8 digits. Base16 works better (4bits = 1 digit) – Cort Ammon - Reinstate Monica Feb 5 '15 at 21:28
• 16 is ideal, but 8 is still far better than 10, and would be a more natural transition - no extra numbers, and you can view it as just not using thumbs to count with in terms of fingers. – Dan Smolinske Feb 5 '15 at 21:32
• Representing 1/5 or 1/(5*2) is base 2 will be problematic whether or not you represent those numbers as .2 and .1. Granted, without base 10 we wouldn't divide things by 5 as often, so it would be less of a problem. – ckersch Feb 5 '15 at 22:16
• The problem with power-of-two bases is that they don't divide 3s cleanly. With 3 being the second prime number, this is kind of a big deal. Base 12 fixes that problem and allows you to divide by 2,3,4, and 6 cleanly. If it weren't for human limits, base 30 would be even better. – Beefster Jan 8 '18 at 21:03

It's based in Mythology...Sumerian mythology to be exact. They had 6 pairs of gods ((1 female + 1 male) * 6 = 12). The use of 60 minutes is this 12 * the number of fingers on a hand. 360 degrees and the sectioning of the sky apparently originates from the same source. It's notable that they lack the number 0 in this numbering system. 12 months...there's actually quite a few reflections of this numbering system to us today...even eleven and twelve having specific words and not oneteen / twoteen likely comes from this.

Of course this is all from studying their mythology that comes from books that predate pretty much all written language, so our interpretation might be a long ways off. In any case, it was Sumerian passed through Babylonian that started us down the base 12 numbering system.

• Any idea if the Sumerian gods were based on stars rather than planets? I'm curious because the Eqyptians had 12 divisions of the night based on the risings of prominent stars a couple thousand years later... – ckersch Feb 5 '15 at 22:41
• The degrees are 360 because one degree change in the positions of the stars corresponds to (about) one day. They chose 360 (instead of 365 or 366) because 60 divides it. – martinkunev Feb 5 '15 at 22:48
• @ckersch Potentially...I know it's also where the 12 Zodiaks descend from based on the earths 26'000 year wobble (one zodiak sign being 2160 years) and I think that's what the Egyptian division stems from, no? – Twelfth Feb 5 '15 at 22:49
• @martinkunev - I'm not sure who 'they' is in your statement, but this 360 method was established in Sumeria, likely long before your 'they' ever existed. Early humans were much more in awe of the starry sky than today's humans are. With all the light pollution and our TV's and other entertainment, most of us are barely aware the stars exist...Imagine what you would know about the night sky if it's all you had in your lifetime. – Twelfth Feb 5 '15 at 22:53

It is well assumed that the only advantage of base-10 is the number of fingers on our hands.

There are actually cases where base-12 was used in languages, though they are few and far between.

It is believed that the advantages of base-12 would be immense, because of easy division and multiplication by 1, 2, 3, 4, 6, and 12. However, in most cases, the convenience of base 10 for a H. sapiens civilization has won out, for whatever reasons you like to believe.