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## Duodecimal

Base 12, a.k.a. duodecimal, has a lot of nice features: addition and subtraction mod 12 are familiar from clocks, many common fractions are simple (1/3 = 0.4, 1/4 = 0.3, etc.), the repeat length rule has no common exceptions, and advanced divisibility testing is nearly as easy as in base 10.

The one disadvantage I can see is that there's no analogue to the lovely fact that 210 is close to 103. In other words, log 2 isn't an especially round number. As a result, the concept of kilobyte simply doesn't translate! The best candidates are the facts that 27 is close to 122, 211 is close to 123, and 218 is close to 125 (see The Twelve-Note Scale), but the first two aren't especially close (off by 13% and 19% respectively) while the third is too large to be useful, and still isn't all that close (5%, vs. 2% for 210 and 103).

A second sad result is that the McIntosh-Doerfler algorithm doesn't work, both because log 2 isn't round and because log 3 isn't independent of log 2. It's a subtle matter, actually. When we add or subtract log 2 and log 3, we're implicitly using factors of 2 and 3 to construct a fraction with known logarithm. Because of the base, we get to include factors of 5 for free. So, to do the same thing in base 12, we should add or subtract log 2 and log 5 and get our factors of 3 for free. Unfortunately, log 5 isn't a nice round number either.

Speaking of 2 and 3, in base 12 all other prime numbers end with 1, 5, 7, or B; see Other Bases for more about that. Here I'm using A and B as digits with values 10 and 11, just as I do in hexadecimal and everywhere else.

Now let me return to the subject of clocks. In base-12 military time, the first digit of the hour is also an AM-PM flag; for example, 17:00 is 7:00 PM. However, although hours work nicely in base 12, minutes are a little strange. Instead of 60 there are only 50, so half an hour is 26 minutes and quarter hours fall on 13 and 39. I think maybe I'd take a “decimal” approach instead, with 17.30 as 7:15, 17.60 as 7:30, etc. In that system, 0.1 would be a nice five-minute unit (just as on a clock) and 0.01 would be 25 seconds, or roughly half a minute. (And 0.001 would be about 2 seconds, and clocks could have four hands instead of three!)

Finally, here are a few notes about base 12 in language and culture.

First, I'd like to take back what I said in In Mathematics about base 12 in the Anglo-Saxon past. Here's what Wikipedia has to say on the subject at the moment (from Duodecimal).

Germanic languages have special words for 11 and 12, such as eleven and twelve in English, which are often misinterpreted as vestiges of a duodecimal system.[citation needed] However, they are considered to come from Proto-Germanic *ainlif and *twalif (respectively one left and two left), both of which were decimal.

On the other hand, even if we trace the twelve-ness of months and hours back to the lunar cycle and the twelve-ness of inches and ounces back to the Roman uncia, it still seems pretty remarkable to me that we have special names for a dozen and a gross.

Second, in addition to the Arabic and Chinese digits that they normally use, the Chinese also have two ancient alternate sets of digits, the ten heavenly stems and the twelve earthly branches. The two can be used separately in certain contexts, but they can also be used together, one stem with one branch, and then they increment together to form a cycle of length 60. The digits aren't used with positional notation (as far as I know), so we're not actually talking about base 12 or base 60, but who cares, it's still fun.

Third, the people in Greg Egan's latest book The Clockwork Rocket use base 12. How strange that I was already thinking about that when I read the book!