# What follows next in the sequence “unary, binary, ternary…”?

I looked on Oxford's online dictionary and was able to find the names identifying orders of a given degree:

1. primary
2. secondary
3. tertiary
4. quaternary
5. quinary
6. senary
7. septenary
8. octonary
9. nonary
10. denary
11. -- no term for 11th degree??
12. duodenary

I am curious as to what would be the sequence of terms regarding a set of 'n' items? I have up to four:

1. unary
2. binary
3. ternary
4. quaternion

but, I cannot seem to find anything beyond that. Anyone know where this list may be?

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What do you mean "set"? A "set" of two is a pair, of three a trio, of four a quartet, of five a quintet... – Hot Licks May 4 '15 at 11:57
@HotLicks: Not in all spheres of thought. In mathematics, a "set" is defined as a finite or infinite collection of objects in which order has no significance, and multiplicity is generally also ignored (unlike a list or multiset). (Source: Wolfram MathWorld, bit.ly/1U8iU0Q) In other words, a set is any number of things, as long as the order (and generally the number of elements) are ignored. – SarahofGaia Aug 23 '15 at 17:52
@HotLicks: Besides, that hardly contributes toward an answer for the question. All it does is waste time nitpicking. – SarahofGaia Aug 23 '15 at 17:53
Also, Will: Undenary might be a good candidate. :) As the Wikipedia page "List of Numeral Systems" (bit.ly/1U8j5t9) points out, the word for base-11 is undecimal; and as @Cerberus pointed out below, after 3 the suffixes change from -us- to -nary. So one could adapt the numeral system term for list purposes in this way. :) – SarahofGaia Aug 23 '15 at 18:01

The problem is that English uses two different kinds of adjectives to mean "first, second, etc". The ones in -ary without the -n- come from the Latin ordinals, "first, second, etc."; but they are different after 3.

1. Primus — primary "first"
2. Secundus — secondary "second"
3. Tertius — tertiary
4. Quartus — *quartary
5. Quintus — *quintary

...

The -arius suffix is also used in Latin with ordinals, and secundarius means something like "second, pertaining to two, second in rank", though it often comes very close to the simple ordinal secundus. It usually adds some connotation of ranks and order in a grand system. There is also secundanus, which I believe isn't much different.

The -n- ones come from Latin distributive adjectives, "one each, two each, etc."; they are always used in plural. They were sometimes also used in a sense roughly similar to the ordinals, which is probably why English uses them in an odd way.

1. (Singuli — single "one each")
2. Bini — binary "two each"
3. Terni/trini — ternary/*trinary
4. Quaterni — quaternary
5. Quini — quinary
6. Seni — senary
7. Septeni — septenary
8. Octoni — octonary
9. Noveni — *novenary
10. Deni — denary
11. Undeni — *undenary
12. Duodeni — duodenary
13. Terni/trini deni — *ternidenary/*tridenary

...

I believe the ones derived from ordinals were originally used to mean "second [in order]" in English, and the distributive -n- ones to mean "of two parts", or "characterised by the number 2". But then, because these meanings are related and often overlap, they got mixed up, resulting in the current defective lists, where the -n- forms serve both senses from 4 up.

The number one is the strangest exception of all, where a new word unary was made up, though no Latin equivalent exists (there is only unus, "one", but that is like using *duary from duo, "two"). Nonary is odd as well.

These are the Latin cardinal numbers for reference:

1. Unus/una/unum/etc. (depending on gender and case) — "one"
2. Duo/duorum/duarum/etc. (depending on case and gender) — "two"
3. Tres/trium/etc. (depending on case)
4. Quattuor
5. Quinque
6. Sex
7. Septem
8. Octo
9. Novem
10. Decem
11. Undecim
12. Duodecim
13. Tredecim

...

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ah, that's why I couldn't find a separate listing ... that's fine then for my purposes; the terms were going to be used in naming functions/classes in a library I am developing, and I was trying to be as grammatically accurate as I could. That's not usually a big deal for us programmers, but I tend to be more finicky about certain things than others. – Will May 13 '11 at 12:51
@Will: An excellent attitude! Users of the future with literary tastes will appreciate it in your software! – Cerberus May 13 '11 at 12:56
while coming up for short names for number systems and possible name-sharing, could we call base 0, "imaginary" ? (and i don't mean in the math sense, back off trig calculus!) – osirisgothra Sep 5 '14 at 15:26
@osirisgothra: Hah, I think Trig calculus would poison your food and seduce your wife! But, really, what would "base 0" even mean? You can't really base a system on zero. – Cerberus Sep 8 '14 at 13:08

The arity of a function or operation is the number of arguments or operands that the function takes.

N-ary:

• Nullary means 0-ary.

• Unary means 1-ary.

• Binary means 2-ary.

• Ternary means 3-ary.

• Quaternary means 4-ary.

• Quinary means 5-ary.

• Senary means 6-ary.

• Septenary means 7-ary.

• Octary means 8-ary.

• Nonary means 9-ary.

Hope this helps.

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...and for 10... decimal. – Mitch May 13 '11 at 1:04
...and alternatively base n. – Mitch May 13 '11 at 1:05
@Mitch No, bases are a different meaning. The words "binary" and "ternary" are used for both, but "octary" and "octal" are different. – aschepler May 13 '11 at 2:29
@aschepler: oops, you're right. what would the sequence be then for base n? – Mitch May 13 '11 at 2:32

I know I'm a little late here, but I thought it might be worth mentioning that Wikipedia has a great list of base systems, which goes all the way up to 16 (Hexadecimal, of course) without holes, and then on to 85 (Pentaoxagesimal). Here's a quick reproduction of part of it:

1. unary (not actually on the main list, but listed farther down as being used in tally marks)
2. binary
3. ternary
4. quarternary
5. quinary
6. senary
7. septenary (used in weeks)
8. octal
9. nonary
10. decimal (everybody's favorite!)
11. undecimal
12. duodecimal (used in hours, months)
13. tridecimal

18 is octodecimal

20 is vigesimal

It's interesting to note that even our method of naming these systems reflects our attachment to the decimal system, as we begin to add prefixes after decimal. Also, if you want to form a higher number, it appears that you can use the following formula:

prefix for 2nd digit + prefix for 1st digit + gesimal

So, 27 is septemvigesimal. I invented this formula in answer to this question, but it appears to fit every case on the list.

Wikipedia also lists −2 as negabinary and −3 as negaternary. Theoretically, you can add the nega- prefix to anything, but I have no idea what you would use it for.

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As Cerberus notes, the first list you give is a combination of two lists, both on Latin roots: ordinals for 1–3, then arities (from distributive numbers). To keep these straight, and include the corresponding words from Ancient Greek, I’ve written two Wiktionary appendices:

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