Division, a comparison

There's been a large amount of talk so far and we're at the point where we can see and understand the real honest-to-goodness benefit of using base twelve. Below is a list of the decimal and duodecimal results of one divided by another number (the number's reciprocal).

How to read this table

The table shows the floating point represention of one over "n" and compares how many digits in the mantissa (the part of the number after the point). The mantissa will be composed of a fixed number of digits, and sometimes additionally a series of digits that repeat (recur) endlessly. For us as humans when dealing with these numbers we want as few digits as possible, and none recurring. A number with recurring digits in the mantissa is not expressible in floating point, so a number with recurring digits is very bad.

Ten...type Twelve...type Twelve better?
11integer 11integer same
20.51, fixed 20.61, fixed same
30.333...1, recurring 30.41, fixed yes!
40.25...2, fixed 40.31, fixed yes!
50.21, fixed 50.2497...4, recurring no
60.1666...1, fixed + 1, recurring 60.21, fixed yes!
70.142857...6, recurring 70.186A35...6, recurring same
80.1253, fixed 80.162, fixed yes!
90.111...1, recurring 90.142, fixed yes!
100.11, fixed A0.124972497...1, fixed + 4, recurring no
110.0909...2, recurring B0.111...1, recurring yes!
120.333...1, recurring 100.11 fixed yes!


It's quite clear from the table that divisions are easier in base twelve than base ten.

Base twelve runs into trouble in base ten's sweet spots, five and ten. In most other places though, it's better with either fewer fixed places, or replacing recurring answers with fixed. Seven in particular is a problem for both. All the prime numbers (two, three, five, seven, eleven, ...) are problems in any numbering system that doesn't have that prime as one of its divisors. Twelve has primes two (twice) and three as its divisors and thus can express fractions as floating point whose denominator is a product of these numbers. And indeed most of the small numbers are covered: two, three, four, six, eight and nine are all products of just two and three.

Percentages, hundreds, and grads revisited

So armed with this knowledge we can look back on percentages & the grads angle unit and see why they have trouble with thirds: base ten doesn't have three as one of its divisors and three is prime hence it will end up with a nasty recurring result.

Similarly, to get rid of the "problem" of five we would have to use a base that has five as one of its divisors. Ten is as we know one such example but it suffers from lack of other divisors. So to combine the advantages of base twelve (two * two * three) we would at a minimum require two * three * five = thirty. Thirty however only has two as a divisor once so the extremely common measurement, a quarter, (one divided by [two times two]) isn't compact. Thus we'd have to use sixty: two * two * three * five. Sixty can indeed express quite a huge range of fractions neatly but it requires sixty different digits! Mind you, in English we have twenty-six characters in our alphabet, the Chinese two hundred and forty one and the Japanese full Kanji thousands so it's far from inconceivable this could be workable. In fact, the Babylonians used base sixty and we indirectly use base sixty when dealing with degrees as angles.

Now that we're starting to get comfortable wearing the base twelve lens let's take some common and not so common collections of numbers and see what they look like. On to the base twelve lens.