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 "24_{10}-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 would go from 0 to 50_{12}. Only real
advantage is slight aesthetic improvement for those splitting minutes
and hours into chunks of five whose results would be 10_{12},
20_{12}, 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, 27_{12}, the twelves
column never sees a three. Again, perhaps curiosity value only.

Result: **marginal**.

360_{10} is 260_{12}. 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
260_{12} 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
A_{12} 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
0.6_{12} 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 100_{12} or 144_{10} 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 40_{12}%.

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 5,000_{10}!

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: 40_{12}% and
20_{12}%.

Result: Overall, **solid improvement**

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?