# What comes after the ducentiquinquagintasexions?

Hypercomplex numbers that use the Cayley-Dickson construction seem to follow a Latin naming convention related to the size of the algebra (which is always a power of two). As an English.SE question, I'm interested in the larger names that I haven't been able to find on the web. Here is what I know:

• Quaternions (4-ions)
• Octonions (8-ions)
• Sedenions (16-ions)
• Sexagintaquatronions (64-ions)
• Centumduodetrigintanions (128-ions)
• Ducentiquinquagintasexions (256-ions)

What would the next higher orders be following this naming scheme?

• ??? (512-ions)
• ??? (1024-ions)
• ??? (2048-ions)
• ??? (4096-ions)
• Any of these beyond octonions are so rarely used that,even if they have names, those names would not be used in mathematical writing about them. Mar 19, 2015 at 23:54
• @Hooked: It is mentioned that they simplified some of the terms and it is explained that it is unnecessary to go beyond a certain point. Do you really need this information or is it just trivia? If you really need terms for these, simplified versions would be easier to read :) (and the next question: 8192, 16384, 32768 ...) Mar 20, 2015 at 2:30
• @GEdgar while I defer to your mathematical expertise (and I agree that without usage it is unlikely to warrant a permanent name) I thought that the question of the naming itself was an interesting, if academic, exercise. The motivation stemmed from a small project I made last night to visualize the multiplication tables. I felt that calling them 512-ions was just so pedestrian. Mar 20, 2015 at 2:32
• I think we should call them `absurd`, `preposterous`, `ludicrous`, and `insane`. Seriously. By the time we get to sedenions, the Algebra isn't even associative, let alone commutative. Just useless. And this from a pure mathematician. Mar 20, 2015 at 6:14
• I’m voting to close this question because it does not relate remotely to everyday English; such arcane usages might be acceptable on Maths if the question were closed and resubmitted there. Apr 13, 2020 at 18:18

I'm afraid the words you mention were already formed incorrectly and inconsistently.

I see some were formed from distributive numbers + an unknown suffix -ion (like quaternion).

Others were formed from cardinal numbers + -nion (like trigintaduonions)

Others again were formed from a cardinal number + something that doesn't look like a Latin word + a suffix that looks like -onion (sexagintaquatronions).

The ones that would be several separate words in Latin are the most problematic. Consider the cardinal number triginta duo, "thirty-two". Normally in Latin, if you wanted to turn that into a distributive number, you'd change both words into their distributive forms: trigeni bini "thirty-two each". Then if you want to add -ion to the stem of the last word and turn them into a single word, you will get trigeni-bin-ion.

The way some of the words in the list were formed, on the other hand, is by changing only the last word into a distributive number and keeping the other one(s) cardinal: triginta-bin-ion.

The Romans would normally write larger numbers as separate words. But, if we ignore that, using something based on all distributive numbers would seem closer to the Roman way, like trigeni-bin-ion above. I will provide both options; first I give [all distributive + -ion], then [cardinal(s) + distributive + -ion]:

Quaternions (4-ions)
Octonions (8-ions)
Sedenions (16-ions)
Tricenibinions / trigintabinions (32-ions)
Sexageniquaternions / sexagintaquaternions (64-ions)
Centeniduodetricenions / centumduodetricenions (128-ions)
Duceniquinquagenisenions / ducentiquinquagintasenions (256-ions)

Quingeniduodenions / quingentiduodenions (512-ions)
Miliaviceniquaternions / millevigintiquaternions (1024-ions)
Quaternamilianonagenisenions / quattuormilianonagintasenions (4096-ions)

Note that distributive forms of thousand in Latin are a bit different from lower numbers, and I couldn't find any examples of their use combined with lower numbers: I only found passages with "three thousand each" in the HP Latin corpus, not e.g. "three thousand two hundred each". Note also that in Latin some words have two (or more) alternative forms, like triceni/trigeni "thirty each"; I have used the commonest alternative in each case.

• thank you for the info! With this in mind, can you formulate all of the terms, from the 2-ions to the 4096-ions, using the rule of "distributive numbers + unknown suffix -ion" like the quaternion? Mar 20, 2015 at 2:20
• `something that doesn't look like Latin word` I beg your pardon? "sexāgintā" IS latin Mar 20, 2015 at 8:29
• @Federico: Sexaginta is a Latin word, but not if you add quatr-something to it. (Of course I ignored the fact that the Romans would never have stuck large numbers together to form single words in the first place: I conceded that to English liberty. But then the Latin words on which it is based should at least be in the right form.) Mar 20, 2015 at 15:56
• @Hooked: OK done! Mar 20, 2015 at 17:31
• @Cerberus This is great, thank you! It is also part of the impetus for why I asked the question. Without knowing it, it highlighted a problem with the current naming scheme and gave me a little Latin lesson. Also, for posterity, it's worth noting that all of your new coinages are completely new to Google. Mar 20, 2015 at 17:42

Following the Latin prefix tables from phrontistery's reference, they would probably be:

512-ion: Quincentumduodecion
1024-ion: Millevigequaternion

• I have no idea what you're talking about here, but +1 for the effort. Mar 19, 2015 at 23:53
• I'm afraid those look even less like English words formed based on proper Latin numerals, but bravo for daring to try... Mar 20, 2015 at 0:26

I propose an alternative convention: since each level is achieved by applying the Cayley-Dickson Construction to the previous, use numerical prefixes logarithmically and add the contraction -cadi-. (Use Greek prefixes, to further avoid confusion with the original system which uses Latin.) An n-cadinion will have 2n dimensions. So a Quaternion, could also be called a Dicadinion and so forth:

• Complex = Monocadinion (21 = 2 dimensions)
• Quaternion = Dicadinion (22 = 4 dimensions)
• Octonion = Tricadinion (23 = 8 dimensions)
• Sedenion = Tetracadinion (24 = 16 dimensions)