Background credit, thank you mattz from shadertoy.
You read the webpage link, you read the header, let's talk nomeclature. Dozen two (or fourteen if you prefer) is a funky lil base. Not particularly good, but in a fun charming way. "Tetradecimal" is the common name for it, and it means:
Tetra - 4
Dec - 14
Imal - Numbering System
This requires you to memorize the factors of every number, and think additively. For an example like this, it's fine, but what about a larger base like hexavigesimal? If you're like me, you probably thought six times twenty. Anyway, even if you happen to know that 42 is two times dozen one, that's a bad system for forcing you to memorize it.
The obvious solution is to use factorizations, as those are what matters for a base. With the example from the header, "biseptimal" is an obvious choice, meaning two times seven. This whole webpage is basically a ripoff of the seximal page, but I plan to do some stuff differently!! Mainly that for bases larger than 100, you use a prime factorization scheme rather than a naming one. Also, I'm going to be more consistent about prefixes, and fix an issue with the misalian system.
My goals for this aren't to make something that I would reasonably use, but something base neutral that makes good sounding names, and is relatively comprehensible. I'll only use the prefixes 1-12 to make things more base neutral. Kind of silly but I think it makes things a little more regular. Well, not kind of silly. I wrote the first half of this thinking it was going to be serious, but after my solution for naming primes I'm just fully embracing that this is a silly system.
| Base | Name |
| 1 | Unary |
| 2 | Binary |
| 3 | Trinary |
| 4 | Quaternary |
| 5 | Quinary |
| 10 | Seximal (or heximal if you can't handle the word sex) |
| 11 | Septimal |
| 12 | Octimal |
| 13 | Hyperternimal |
| 14 | Bipentimal |
| 15 | Unbipentimal |
| 20 | Biheximal |
| 21 | Unbiheximal |
| 22 | Biseptimal |
| 23 | Tripentimal |
| 24 | Hypertetrimal |
| 25 | Unhypertetrimal |
| 30 | Triheximal |
| 31 | Untriheximal |
| 32 | Tetrapentimal |
| 33 | Triseptimal |
| 34 | Biunbipentimal |
| 35 | Unbiunbipentimal |
| 40 | Tetraheximal |
| 41 | Hyperpentimal |
| 42 | Biunbiheximal |
| 43 | Ultraternimal |
| 44 | Tetraseptimal |
| 45 | Untetraseptimal |
| 50 | Pentaheximal |
| 51 | Unpentaheximal |
| 52 | Tetroctimal |
| 53 | Triunbipentimal |
| 54 | Biunhypertetrimal |
| 55 | Pentaseptimal |
| 100 | Hyperheximal |
Small bases <10 get the -ary suffix, while larger bases get the -imal suffix. With base eleven, you can see the perfect vs usable difference. I like elevenary better as a name, but eleven does mean ten one left over, and it ends with ary, so I gotta cut it for consistency. I use the un- prefix to mean plus one.
As you can see with base twelve, when seximal is multiplied, it becomes heximal. This is just to distinguish them a bit more. This also happens with other base names, like triquinary becoming tripentimal. I made some prefixes for exponents. Hyper means squared, ultra means cubed, and supra means to the fourth.
Getting to the larger ones, especially base 35, you can notice a problem. It becomes an incomprehensible string of uns and bis. How do we fix this? We could add dec back in, but that seems lazy. The real problem is the damn primes, giving us all these "uns". To solve this, let's name some primes. I will have to concede on my goal of keeping things base neutral for a lot of these names.
Wait, no I don't, I'll use the periodic table! It's super stupid, but think about it. The elements cover every integer up to 314 without gaps, and already have truncations to prefix like letter combos. It's perfect. I should still call eleven elev though, for simplicity. If these go before another prefix, add -i to the end.
| 15 | Elev |
| 21 | Al |
| 25 | Chlor |
| 31 | Potass |
| 35 | Van |
| 45 | Balt |
| 51 | Gall |
| 101 | Rubid |
These are obviously stupid, but I actually kind of love them? Why didn't I think of this atomic trick earlier! Well, it does kind of defeat the whole "understandable" goal. Oh well. People want to learn systems to make sure they understand them anyway. It's not like you can know what biker's dozenal is without knowing the misalian system. For the first use of the prime, we'll use the standard name to make things less confusing. But when using the prime in multiplication, we'll include the atomic name. Except elevimal, it gets to use it's new prefix. Let's take a look at the table again.
| Base | Name |
| 1 | Unary |
| 2 | Binary |
| 3 | Trinary |
| 4 | Quaternary |
| 5 | Quinary |
| 10 | Seximal |
| 11 | Septimal |
| 12 | Octimal |
| 13 | Hyperternimal |
| 14 | Bipentimal |
| 15 | Elevimal |
| 20 | Biheximal |
| 21 | Unbiheximal |
| 22 | Biseptimal |
| 23 | Tripentimal |
| 24 | Hypertetrimal |
| 25 | Unhypertetrimal |
| 30 | Triheximal |
| 31 | Untriheximal |
| 32 | Tetrapentimal |
| 33 | Triseptimal |
| 34 | Bielevimal |
| 35 | Unbielivmal |
| 40 | Tetraheximal |
| 41 | Hyperpentimal |
| 42 | Bialimal |
| 43 | Ultraternimal |
| 44 | Tetraseptimal |
| 45 | Untetraseptimal |
| 50 | Pentaheximal |
| 51 | Unpentaheximal |
| 52 | Tetroctimal |
| 53 | Trielevimal |
| 54 | Bichlorimal |
| 55 | Pentaseptimal |
| 100 | Hyperheximal |
| 101 | Unhyperheximal |
Wow, judging purely by pronounceability and character count these are certainly better! So to solve the whole "what's the factorization of b-1" issue, I'm gonna do something kinda lazy. You just append "over [base name of b-1]" to it. So like, base 42 or bialimal's full name is bialimal over hyperpentimal. Kind of cheating, but it works and is simple, what more could you want?! Of course, these aren't necessary with certain prime bases.
What about bases larger than nif one? Actually, let's expand that table. Nif one isn't a fair stopping point. I won't expand past rubid, since I think 101 is the largest prime I can reasonably ask people to memorize. Also, at base 113, I ran into a problem. It's equal to 3 * 3 * 5, so I want to say "hypertern", but then how could you know if it was 32 * 5 or (3 * 5)2? Well, for simple cases like this, I can just call it "pentahyperternimal", but for more complicated cases I will use "kra" as the closing parenthesis particle. So you could also call base 113 "hyperternkrapentimal", but that would mean 13 * 5, which the smaller number always goes first, so it's invalid.
| Base | Name |
| 102 | Bipotassimal |
| 103 | Trialimal |
| 104 | Pentoctimal |
| 105 | Unpentoctimal |
| 110 | Hexaseptimal |
| 111 | Unhexaseptimal |
| 112 | Tetralevimal |
| 113 | Pentahyperternimal |
| 114 | Bivanimal |
| 115 | Unbivantimal |
| 120 | Hexoctimal |
| 121 | Hyperseptimal |
| 122 | Bihyperpentimal |
| 123 | Trichlorimal |
| 124 | Tetralimal |
| 125 | Untetralimal |
| 130 | Biultraternimal |
| 131 | Pentelevimal |
| 132 | Heptoctimal |
| 133 | Tripotassimal |
| 134 | Bibaltimal |
| 135 | Unbibaltimal |
| 140 | Tritetrapentimal |
| 141 | Untritetrapentimal |
| 142 | Bigallimal |
| 143 | Heptahyperternimal |
| 144 | Hyperoctimal |
| 145 | Pentalimal |
| 150 | Hexelevimal |
| 151 | Unhexelevimal |
| 152 | Tetrachlorimal |
| 153 | Trivanimal (funny number base) |
| 154 | Bipentelevimal |
| 155 | Unbipentelevimal |
| 200 | Bihyperheximal/Octohyperternimal |
Hmmph. Okay. I thought writing the next nif names, they'd quickly become unmanageable. Some were kind of unmanageable, but overall they were all still pretty simple. So, there's no rule. You can use the large base naming system whenever you want. Though the convention will be, hyperheximal or lower use the standard names, unhyperheximal and above use the large base name system.
So, the system will work based on prime factorizations. The symbols are 2, 3, 5, and 7 for the first four primes. Then we use the alphabet. So A is the fifth prime, B is the sixth prime, and so on. So a base like 22 would be called "base 27", and a base like 154 would be called "base 25A". That gets you up to three nif five, which is a pretty large prime. Obviously, this system is very inconvienent. It requires a lot of memorization. Let me try again...
Okay new system idea. Every prime greater than 3 is one less or one more than a multiple of six. So take the prime factors of a number, sort them lowest to greatest, and then for primes larger than 7 you prefix them with a u or d, and then the multiple of six they're nearest to, in seximal. Basically, a number like 25 is one less than 30, so it's written as d3, because it's down from the third multiple of six.
So. Base 154. It's equal to 2 * 5 * 15. 15 is one less than the second multiple of six, so it's called "Base 25d2" in this system. Very silly, but more practical. What about base 2^12, supratetrimal? That would be base 22222222, which is bad. Let's use exponents. Again using seximal as the standard. So it would be "Base 2^12". Very simple, not really a name, but hey, it's still a system for naming large bases that's pretty helpful since it makes the factors obvious!
On second thought, using the prefix "base" for both of these systems is very confusing. Use a lowercase b as the prefix instead. So something like base 3,5025,2530 would be called b2335d3d11u25. Perhaps commas would help. So it would be b2,3,3,5,d3,d11,u25. Much more readable.
Thank you for reading this super silly article.