# six isn't even divisible by four though

if you've ever seen someone suggest that a base is better than decimal, they were probably suggesting dozenal (base twelve). the main thing dozenal has going for it is that twelve is divisible by two, three, and four. now, a base's factors are important for lots of things. the easiest one to explain is divisibility tests, so let's start there.

ten is two times five, so in decimal, you can tell if a number is divisible by two and/or five by looking at the last digit. DEC18 (30) ends with 8, so it's divisible by two. DEC475 (2111) ends with 5, so it's divisible by five. DEC40 (104) ends with 0, so it's divisible by two AND five. in general, a decimal number that ends with 0, 2, 4, 6, or 8 is divisible by two, and a decimal number that ends with 0 or 5 is divisible by five.

six is two times three, so in seximal, you can tell if a number is divisible by two and/or three by looking at the last digit. 14 ends with 4, so it's divisible by two. 453 ends with 3, so it's divisible by three. 40 ends with 0, so it's divisible by two AND three. in general, a seximal number that ends with 0, 2, or 4 is divisible by two, and a seximal number that ends with 0 or 3 is divisible by three.

ten isn't divisible by four, and neither is six. this means you can't figure out if a number written in decimal or seximal is divisible by four just by looking at the last digit. instead, you need to look at the last TWO digits.

some of you might remember a trick you learned for how to check if a number written in decimal is divisible by three. all you need to do is take its digits, add them together, and then check if that number is divisible by three. this trick works because one less than ten is nine, which is three squared.

this trick also works in seximal, not for threes, but for fives. if you add the digits of a seximal number together and the result is divisible by five, the original number was also divisible by five. this trick works because one less than six is five, which is five.

in decimal, small multiples of eleven are easy to spot, because they're two digit numbers where both digits are the same: DEC11, DEC22, DEC33, DEC44, DEC55, DEC66, DEC77, DEC88, and DEC99. there's a trick for checking divisibility by eleven that's less well known than the trick for threes but it works similarly: take the number and break it into two digit "blocks". add those blocks together, and if they equal a multiple of eleven, the original number was divisible by eleven. for example, DEC1045 (4051) -> DEC10 +DEC45 -> DEC55. this trick works because eleven is one more than ten.

in seximal, small multiples of seven are easy to spot, because they're two digit numbers where both digits are the same: 11, 22, 33, 44, and 55. just like decimal's elevens trick, there's a trick for checking divisibility by seven: take the number and break it into two digit "blocks". add those blocks together, and if they equal a multiple of seven, the original number was divisible by seven. for example, 1045 -> 10 + 45 -> 55. this trick works because seven is one more than six.

at this point, it stops being easier to look at divisibility tests and starts being easier to look at the related property of the representations of fractions.

in decimal, 1/2 and 1/5 both have simple one digit representations: <.5> and <.2>. 1/4 is a bit less simple: <.25>. 1/3 is even less simple, but it's still easy enough to remember: <.333...>. 1/15 is right around the level of complexity where it starts getting tricky: <.090909...>.

in seximal, 1/2 and 1/3 both have simple one digit representations: <.3> and <.2>. 1/4 is a bit less simple: <.13>. 1/5 is even less simple, but it's still easy enough to remember: <.111...>. 1/11 is right around the level of complexity where it starts getting tricky: <.050505...>.

why does all of this matter and what does this have to do with dozenal? well, by looking at how any given base represents fractions, we can see which numbers it sees as more important than others. decimal is all about twos and fives, but it acknowledges that threes, nines, and elevens are also important. seximal is all about twos and threes, but it acknowledges that fives and sevens are also important. dozenal, which indeed has more factors, is all about twos, threes, fours, and sixes, but it acknowledges that elevens and dozen ones are also important.

seximal | decimal | dozenal | |
---|---|---|---|

S tier factors (factors of the base itself) | 2×3 | 2×5 | 2²×3 |

A tier factors (factors of one less than the base) | 5 | 3² | 15 |

B tier factors (factors of one more than the base) | 11 | 15 | 21 |

C tier factors (factors of one more than the base squared) | 101 | 245 | 5×45 |

as you can see on this chart, all three of these bases consider twos to be extremely important, but dozenal gets bonus points for putting two squared on the same level. seximal and dozenal are tied for caring about threes, and as was already mentioned, dozenal wins at fours, though seximal and decimal aren't far behind.

once you get to fives, things take a turn for the worse for dozenal. decimal wins at fives, because ten is two fives, with seximal in second, for being one more than five. how does dozenal treat fives?

what's that? fives are on the BOTTOM of dozenal's column of this chart? yes, that's right, dozenal cares about fives the same amount seximal cares about nif ones, ie. not very much at all!

you might say, "okay, so dozenal is bad at dealing with fives. whatever, at least it's good at dealing with fours. at most that puts it on equal footing with seximal. certainly that alone isn't enough to make seximal better." and you're right. we can keep going, naturally, to sixes. seximal and dozenal are equally good at sixes, and decimal isn't very good at sixes.

next are sevens. looking at the chart, you'll notice that there's only one seven, and it's a "B tier factor" in seximal. yes, decimal and dozenal are both absolutely garbage with sevens. this is easiest to see with how they write fractions.

seximal | decimal | dozenal | |
---|---|---|---|

a half | .3 | .5 | .6 |

a third | .2 | .3 | .4 |

a fourth | .13 | .25 | .3 |

a fifth | .1 | .2 | .2497 |

a sixth | .1 | .16 | .2 |

a seventh | .05 | .142857 | .186X35 |

an eighth | .043 | .125 | .16 |

a ninth | .04 | .1 | .14 |

a tenth | .03 | .1 | .12497 |

(that X technically should be a ↊, the dozenal digit for ten, and the fact that that's almost certainly displaying as a square for you is why it's just an X instead.)

just from looking at these, it's clear to see that even though dozenal is the best at writing fourths, on average, it's worse than seximal AND decimal! plus, look at how simple every single one of those seximal numbers looks! though, to be fair, I somewhat gerrymandered these results to exclude seximal's biggest weakness, which is that it writes 1/15 as .0313452421, which is, unfortunate. yet, even taking that into consideration, seximal is unreasonably good for its size at writing any given rational number.