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)
  • Trigintaduonions (32-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)
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    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. – GEdgar Mar 19 '15 at 23:54
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    @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 ...) – ermanen Mar 20 '15 at 2:30
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    @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. – Hooked Mar 20 '15 at 2:32
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    @ermanen Even if they are cyclic at 1024, this still leaves the first two terms listed in the question unnamed. From the comment by GEdgar, you'll note that this is a bit of "trivia" if you will -- but I thought the question was interesting enough to learn some Latin from a few commonly accepted terms already (like quaternions and octonions) and thus enough merit for a question here at English.SE. – Hooked Mar 20 '15 at 2:36
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    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. – imallett Mar 20 '15 at 6:14
up vote 5 down vote accepted

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)
Binamiliaduodequinquagenions / duomiliaduodequinquagenions (2048-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.

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    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? – Hooked Mar 20 '15 at 2:20
  • something that doesn't look like Latin word I beg your pardon? "sexāgintā" IS latin – Federico Mar 20 '15 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.) – Cerberus Mar 20 '15 at 15:56
  • @Hooked: OK done! – Cerberus Mar 20 '15 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. – Hooked Mar 20 '15 at 17:42

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

512-ion: Quincentumduodecion
1024-ion: Millevigequaternion
2048-ion: Duomillequadragoction
4096-ion: Quadrimillenonagesextion

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    I have no idea what you're talking about here, but +1 for the effort. – Oldbag Mar 19 '15 at 23:53
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    I'm afraid those look even less like English words formed based on proper Latin numerals, but bravo for daring to try... – Cerberus Mar 20 '15 at 0:26

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