The base twelve lens

In Déjà twelve we saw how twelve cropped up in a number of places. Then we examined how the importance of primes in the divisors of bases makes life easier in division.

Now we'll revisit the clock, months, degrees, percentages and the more advanced topic of prime numbers looking at them in base twelve.



Imagine a clock face with only single digits all the way around it, without the clumsy 10 and 11. The only double digit number would of course be 10 centred at the top of the dial. A designer's dream!

Result: solid aesthetic difference.

A picture here would be nice, even just with A and B

Similarly, digital clocks operating in "2410-hour" mode would go from 0 to 1B. Ah ha! The current confusion of 18:00 being 6pm would entirely disappear: 16:00 would be 6pm. Fantastic! As a tiny child I still to this day remember our entire family tearing through Heathrow airport in a state of panic because we, fairly fresh from S. Africa's then-only am/pm system, had misinterpreted the 24h flight time as pm and were two hours late for an extremely expensive flight. Four tickets on the line. Easy mistake to make, but ah, for base twelve then...

Result: huge improvement of usability.

Minutes and seconds

Minutes and seconds would go from 0 to 5012. Only real advantage is slight aesthetic improvement for those splitting minutes and hours into chunks of five whose results would be 1012, 2012, etc.

Result: marginal


Like hours, they could be numbered with a single digit perhaps making the lives of typographers fractionally easier. But since numbers are rarely used for months we probably wouldn't see this advantage often. Maybe a computer scientist might be interested.

With months having at most thirty one days, 2712, the twelves column never sees a three. Again, perhaps curiosity value only.

Result: marginal.


"duodecades", centuries, millenia, and other big celebrations. Reduced frequency of partying a consequence


36010 is 26012. Common activities are splitting circles and right-angles into halves, thirds, quarters, etc.

Description Ten Twelve
Circle 360° 260°
Right angle 90° 76°
Half right angle 45° 39°
Third right angle 30° 26°
Quarter right angle 22.5° 1A.6°
Fifth right angle 18° 16°
Sixth right angle 15° 13°
Eighth right angle 11.111...° B.3°

It's perhaps not immediately obvious this makes life easier, aside from the eighth of a right angle case where a recurring fraction appears. Both base ten and base twelve sit well with a system split into 26012 parts.

A couple of things worth noting: almost all the base twelve variants end in 0, 3, 6, or 9. To base ten'ers that isn't particularly attractive but to base twelve'rs they're round numbers, it's a series of quarters: recall the positions of those numbers on a clock face. It's similar to base ten'ers seeing a number ending in 0 or 5 and knowing it's a number that won't present any problems when we start dividing it by five or ten. The other minor point is that while A12 looks odd to the base ten crowd who aren't used to digits aside from 0 through 9, it is no different from the other twelve digits. In other words, it's not odd-looking in base twelve. And 0.612 is of course just a half.

Result: again, interesting, but marginal


"Percent" comes from the latin root centum meaning a hundred. It's a convenient way of measuring a division at a fairly granular level without using fractions. 1% is quite a small chunk of something. But why a hundred, and indeed what is a hundred? Looking at the choice of a hundred we need to examine the rationale for percents in the first place: we want to represent a fine grained split of something without using fractions. A tenth is too course so we try one power higher: ten times ten, a hundred.

(Incidently, there is a permille measurement which uses 0-1000 but it's rarely seen. The symbol has an extra tiny 'o' next to the one on the bottom-right of the % symbol.)

This same technique works as well for twelve, except we end up with higher granularity measuring 10012 or 14410 chunks. More importantly, there is the benefit of twelve having many divisors namely that our "pertwelvand" can be used to represent halves, thirds, etc without itself using fractions. A third in base ten percentages is 33.333...% as we saw in grads & percentages. No matter how high you take the power to get more granular measurement you'll never be able to express a third of something with whatever-ages, under base ten. With base twelve it's simply 4012%.

Here is a comparison of whatever-ages in base ten and twelve. Most of this should be starting to look quite familiar by now.

Description Ten Twelve
Half 50% 60%
Third 33.333...% 40%
Quarter 25% 30%
Fifth 20% 24.972497..% (eugh!)
Sixth 16.666...% 20%
Eighth 12.5% 18%
Ninth 11.111...% 14%
Tenth 10% 12.4972...%
Twelfth 8.25% 10%

For the same reasons we saw in division, fractions are a whole lot more easily expressible in base twelve percentages than base ten, with the same caveat about five. Note though that in base twelve, saying 25% is a very, very close approximation of a fifth to the point we're hitting most typical engineering manufacturing tolerances in terms of how far off we are from 24.972%. If saying 33% for a third is a 1% error, saying 25% for a fifth is less than a 0.02% error, or one in 5,00010!

A note on packaging with discounts and offers

Frequently you'll see such enticements as "33% extra!" or "17% off!". Why these odd numbers? They're really trying to say a third extra or a sixth off but having to compromise because of the numbering system we base ten'ers use. With base twelve it could be said accurately and with round numbers: 4012% and 2012%.

Result: Overall, solid improvement


Measuring hours in base twelve would give us a fantastic relation between the standard (am/pm) and military (2410h) systems reducing confusion and increasing readability while the easing the job of simple math counting hours. A clock face might even look more appealing.

Percentages get a solid boost in readability making for a clear representation of fractions while increasing the granularity of measurement to eke out a little more use from them.

Months, minutes, seconds and degrees of angle don't enjoy the same big benefits but hopefully were at least intriguing to examine with new eyes!

But what do prime numbers look like in base twelve?