decadozenal: a base that's just as good as seximal only thirsy two times larger

seximal is the best base. let's get that out there before I start complimenting this other base, which I also like.

decadozenal (base ten times twelve; base three nif twelve) is good for most of the same reasons seximal is good. obviously, it doesn't have the finger counting advantage, but other than that it's got it all covered.

six and three nif dozen are superior highly composite numbers, which means that they have more factors than every number smaller than them, relative to the size of the numbers themselves. this is all fine and dandy, but it's not all about what factors the base number has. you also need to look at the factors of the numbers adjacent to the base number. for example, even though twelve is a superior highly composite number, its neighbors (eleven and dozen one) are both large primes, so dozenal ends up not being very good at handling fives and sevens. six has fewer factors than twelve, but it's directly between two small primes (five and seven), so it ends up being very good at handling halves, thirds, fifths, and sevenths.

right from the gecko, 320 has a bunch of factors. it's divisible by two, three, four, five, six, eight, ten, twelve, dozen three, thirsy two, foursy, fifsy, nif four, and nif foursy. including one and itself, that's dozen four factors! however, that alone wouldn't be enough to make decadozenal worth it. after all, looking at prime factors, a base like hexagesimal (base nif foursy) has the same ones (two, three, and five) at half the "price". the advantage with decadozenal comes in when you look at its neighbors.

315 is seven times dozen five, and 321 is eleven squared. that means that, including neighbor factors, decadozenal is good at handling twos, threes, fives, sevens, elevens, and dozen fives. that's some pretty good prime coverage, all things considered. now, yes, you might notice that dozen one is missing here, but it's not really that much of an issue.

now hold on just a moment, I get how you can use the Latin alphabet to extend Arabic numerals enough to make hexaseximal work, but where am I going to find another two nif twelve digits??

simple, you don't have to! all you need to do is a little trick called "mixed radix", which is where each digit requires two numerals. you can really just think of decadozenal as being a combination of decimal and dozenal.

okay, here's an example of what I'm talking about. the number SEX5555 is written as X7E in decadozenal. (NOTE: I don't have a good TLA to use for this base so from here on out all numbers are in decadozenal unless otherwise specified)

the ones place works like dozenal, so it can be 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, X, or E. I'm using X and E for the digits ten and eleven, so the E at the end means "eleven ones".

the "twelves place" is really just an extension of the ones place. it works like decimal, so it can be 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9, and it CAN'T be X or E. the number after 9E is 100. the 7 here means "seven twelves", or SEX220. adding that to the eleven ones, you get SEX235.

to the left of the "twelves place" is the actual next digit, the "three nif twelves place". the X here means "ten times three nif twelve", or SEX5320. adding that to SEX235, we get SEX5555, our original number.

so, remember, each pair of digits has one decimal digit followed by one dozenal digit. having the dozenal digit second makes it so you can take advantage of how twelve has more factors than ten. however, after the decadozenal point, the order switches, so the dozenal digit goes first. this might sound arbitrary, but it makes things more symmetrical. uh, here's a quickly drawn mspaint chart.

achart.png

the places highlighted in gray can have values up to eleven, and the other places can have values up to nine. anyone wanna see some rational numbers?

seximal decadozenal
a half .3 .6
a third .2 .4
a fourth .13 .3
a fifth .1 .24
a sixth .1 .2
a seventh .05 .17
an eighth .043 .15
a ninth .04 .134
a tenth .03 .12
an eleventh .0313465421 .10X9
a twelfth .03 .1

how do I say these numbers with my mouth

you can really just read off the pairs of digits as though they were numbers in dozenal, counting from 1 (one) to 9E (nine dozen eleven). next, we just need a good word for three nif twelve. I propose "twelfty", because it's twelve times ten. our example from earlier, X7E, is read as "ten twelfty seven dozen eleven". it's kinda a mouthful, but remember that this number is actually "fifsy five nif fifsy five".

twelfty times twelfty can be called something like "gross hundred" if we want, but I don't think we do want that. I'm not quite sure what the best solution for this is, largely because I can't think of a good -illion ish suffix that feels like twelfty in the same way -exian feels like six. I'm just going to leave this question open. if you have any ideas for what to call powers of twelfty, let me know.

sidenote: other decadozenals

the version of decadozenal I've been describing is what I'm going to call "symmetrical decadozenal", which means that 100 - .01 = 9E.E9. there's three other main ways to do this base, and here they are in order of how good they are:

  • dozagesimal (100 - .01 = E9.E9) is just as good at writing expansions of rational numbers, but makes it a bit harder to do divisibility tests. it also has the advantage of making it very easy to think of things in terms of twelfty.
  • asymmetrical decadozenal (100 - .01 = 9E.9E) is like dozagesimal, but instead of perserving expansions, it preserves the simple divisibility tests.
  • symmetrical dozagesimal (100 - .01 = E9.9E) is the worst of both worlds.