Number types

[Sasquatch]

I've been searching the internet for hours. I know somewhere I've seen a list explaining the various linguistic purposes of numbers, ie cardinal, ordinal, ranking, etc. I've tried searching for "linguistic numerals", "types of numbers", and several more permutations with zero luck.

I'm really curious as to what options I may be missing. I don't want to include every type of quantifier imaginable, but I do want a more involved number system than what I'm accustomed to in the natlangs I'm familiar with. Mainly because I am looking to use the numbers and math functions as the derivational base for some of the lexical and syntactic bits of the language.

Does anybody know where I can find such a list? Or maybe even just list the types you can think of. I have fourteen different types of numbers already in my notes. If nothing else, they serve as motivation to figure out how to convey the information they represent.

[Creyeditor]

I know about:

• cardinal (amounts: one, two, three, four, ...)
  ordinal (rankings: first, second, third, ...)
  distributive (distribution: one by one, two by two, three by three, ...; one each, two each, three each)

There are also fractions, but I never heard of them as a seperate category.

[clawgrip]

There are also fractions, but I never heard of them as a seperate category.

English has "half" and "quarter" which fall within this category. Of course, every other fraction coincides with ordinals.

Responding to the OP, there are of course classifiers (measure words). These can lead to a lot of irregularity and interesting things. They can be extensive, like in Chinese, Japanese, Korean, etc., or they can be less common. Khmer, for example, only requires a measure word when counting people, but nothing else.

[DesEsseintes]

Some languages have separate series of collective numerals for counting pairs or sets.

Russian has them, see here.

In Icelandic, this system only goes up to four. So when counting trousers, you use the numerals "einar, tvennar, þrennar, fernar" instead of "ein, tvær, þrjár, fjórar". Basically, Icelandic cardinal numbers up to four exist in all three genders, normal and collective. That gives you 24 different words for the first four cardinal numbers.

By the way, I for one would be interested in seeing fourteen different kinds of numbers. Why don't you post the types you have in your notes already?

[eldin raigmore]

This is a good question. I don't have "the" answer.

Adpihi can make three fraction-words out of almost any of its first twelve (it's a base-twelve numeral system) cardinal number words; and also out of the words for one-hundred-forty-four (twelve-squared; a gross) and 20,736 (a gross-squared) and other monomorphemic numbers.

These are:
1/n
(n-1)/n = 1 - (1/n)
(n+1)/n = 1 + (1/n).

(Of course if n=1 no such word is necessary; and if n=2 then 1/2 and 1-(1/2) are the same.)

There are also corresponding verbs; to divide by n, to reduce by an nth, to augment by an nth.

---

Cardinal numbers also probably should correspond to verbs;
as "two" to "double", and "three" to "triple", and so on.

In some natlangs, I'm sure, they do so, at least up to a point.

[Sasquatch]

By the way, I for one would be interested in seeing fourteen different kinds of numbers. Why don't you post the types you have in your notes already?

I should have done that at the start. My apologies.

I don't know the actual names for some of the types. So I just listed examples.

Cardinal; 1, 2, 3, …
Ordinal; 1st, 2nd, 3rd, …
Ranking; primary, secondary, tertiary, …
Distributive; one each, two apiece, three each, …
Composite; unitary, binary, trinary, … to represent how many “ingredients” in a compound
Multiplicative; once, twice, thrice, …
Duplicative; single, double, triple, …
Series(?); think “trilogy”
Collective; solo, duo, trio, … / pair, triad, ...
Fractional; whole, half, third, …
?; every other, every third, every fourth, …
?; singly, in pairs, in threes, …
?; side-by-side, three-wide, four-wide, … (also deep and high?); English "three abreast"
Ratio(?); “five percent”, “eighteen pergross”.

It's kind of interesting to see how many different forms English has already. Even if some of the forms are limited in range.

I am interested in measure words as well. I actually want to devise the math side of my language first and then see how much of the grammar I can derive from the math. Things like using "plus" as the base for conjunctions. I realize not all mathematical functions will map well to words. But it seems like an intriguing idea.

[Salmoneus]

Most of those aren't 'types of numbers', they're just numbers modifying different things. "Five percent" tells you how many percent. "Three abreast" tells you how many things abreast there are.

You could have a 'type of number' that modifies elephants, if you really want - one elephant, two elephants, three elephants. Or a 'for every six words spoken by President Carter' number series.

[Sasquatch]

Most of those aren't 'types of numbers', they're just numbers modifying different things. "Five percent" tells you how many percent. "Three abreast" tells you how many things abreast there are.

You could have a 'type of number' that modifies elephants, if you really want - one elephant, two elephants, three elephants. Or a 'for every six words spoken by President Carter' number series.

And Ordinal just tells you how many things there are in a row. That doesn't mean it isn't a different type of number from Cardinal, Multiplicative, or any of the other documented types of number.

Your elephant example is just Cardinal numbers being used in their expected fashion. Unless you mean to decline the number specifically to indicate it is counting elephants.

"Three abreast" tells you more than how many things are abreast. It would be used in the context of "Twelve-hundred riders crossed the bridge three-abreast." It isn't a simple count of things. It is a definition of the arrangement of things.

In the "percent" example you ignored the second part of the example, "eighteen pergross". The "five" in "five percent" is just a Cardinal number counting how many "per-cent" there are, as you said. But "per-cent" is the type of number being discussed in that example. Just like "per-gross" in the second half of the example. Another way of writing it would be "five one-hundredths" and "eighteen one-hundred-and-forty-fourths". The "-ths" in English denotes a fractional number. That makes it a distinct type of number compared to "one, two, three...".

[clawgrip]

Khmer has a limited set of numbers specifically for counting fruits and vegetables (4, 40, 80, 400) and the roots for these numbers are unrelated to the regular counting numbers. English has the word dozen, which is the same type of word. And this comes from French douzaine, and as far as I know French can apply this -aine suffix to many numbers (though I imagine most of them are not particularly common).

[Sasquatch]

Khmer has a limited set of numbers specifically for counting fruits and vegetables (4, 40, 80, 400) and the roots for these numbers are unrelated to the regular counting numbers. English has the word dozen, which is the same type of word. And this comes from French douzaine, and as far as I know French can apply this -aine suffix to many numbers (though I imagine most of them are not particularly common).

"Dozen" fits in one of the groups I listed above. It's basically "in groups of twelve". So you could, hypothetically, have "duzen, trizen, quazen, quizen,..." for groups of 2, 3, 4, 5,...

Actually, that form seems very useful; "A quizen geese floated across the pond" has a fairly poetic tone to it. But you could still say "five" geese. So the "-zen" suffix could become the "herd" marker so to speak. It just needs an indefinite root number word, say "som". Then "somzen" means "a group of unspecified number". That's a subtle difference from just "an unspecified number" in that it implies a cohesive group rather than just a collection of something. Just like English "herd".

I wonder how important fruits are to the Khmer that they need a special set of words to count them? Odd. I definitely want to keep my number words derivative so they are easy to parse.

[pyrrhic]

Korean has two sets of numerals, native Korean and Sino-Korean. The native Korean is used more as an ordinal, while the Sino-Korean as a cardinal (although the distinction is not black and white).

In some instances they are used together, as on the telling of time (native Korean for the hour, Sino-Korean for the minutes).

Sometimes they may be interchangeable depending on context, such as when telling age. For someone's age you would use native Korean numbers, but for the age of a building you would use the Sino-Korean numbers.

http://www.omniglot.com/language/numbers/korean.htm

[clawgrip]

Japanese is the same way. Also, while it is the Sino-Japanese numbers that are generally used in combination with classifiers, certain classifiers require or optionally allow the use of native Japanese numbers. For example, when counting long, thin items or cups of liquid, you will use Sino-Japanese numbers from 1 and up, but while counting, say, train stations or people, you will use native Japanese numbers for 1, 2, and 4, and optionally 7 (and 1 and 2 are irregular for people), but Sino-Japanese numbers for everything else. Days of the month use native Japanese numbers from one to ten, but Sino-Japanese numbers for everything else...except the 14th and the 24th, which begin with Sino-Japanese numbers, but express the 4 using a native Japanese number. I assume Korean is probably just as confusing.

[Sasquatch]

I thought of a number type I'd overlooked, exponential. The equivalent of "squared" and "cubed". English stops there because we run out of geometric dimensions to refer to. But there's no reason a more advanced language couldn't have a numeric declination to replace clumsy phrases like "to the eighth power".

[Khemehekis]

I thought of a number type I'd overlooked, exponential. The equivalent of "squared" and "cubed". English stops there because we run out of geometric dimensions to refer to. But there's no reason a more advanced language couldn't have a numeric declination to replace clumsy phrases like "to the eighth power".

Zenzizenzizenzic, maybe?

Oh, and we could use "tesseracted" to mean "to the fourth power".

[Sasquatch]

I came across this page today and figured it was relevant to this thread. I hadn't even thought about the various geometric numbers, anniversary numbers, and other sundry ways to use numbers.

So now I need to add things like -gon (octagon, hexagon...) and -ennial (bicentennial, *triennial...) to my list of numerical affixes.

[Ànradh]

Wouldn't bicentenary etc. be in the same category as decade?

[Sasquatch]

Zenzizenzizenzic, maybe?

Oh, and we could use "tesseracted" to mean "to the fourth power".

I have to admit, when I first read this post I thought it was meant to be mocking. It wasn't until I found the page I posted in the previous post that I saw that zenzizenzizenzic was a "real" word. It's a might overcomplicated though. I started another thread to discuss math operations in conlangs. Somebody pointed out there that some natlangs and conlangs use noun cases to mark math functions. That idea could supplant some of the number types discussed here. For example, 28 could be denoted as 2-REFL 8-ADV. Where REFL is a Reflective case and ADV is an Adverbial case. So "2 itself 8 times". That thread has me now looking for other ways case tags can affect numbers.

"Tesseracted" seems to be the right form. "Tesseract" would be the verb form, "raise to the fourth power". "Tesseracted" would then be an adjective. Of course, the precise lexicon would be different.

I also noticed that there only seem to be two, at most three, actual forms of the number itself. The cardinal form; one, two, three... is seldom used in any other number with the exception of -some; twosome, threesome.... "Two" comes in the most forms; /di-/, /du-/, /bi-/ and possibly more that I can't think of. But there doesn't seem to be any rationale as to which is used when. The rest of each number type is defined by the suffix which is applied.

I find this Latin/Greek/Anglo system appealing. It fits well with the structure of my language. But it does need to be regularized. I'll have to sort out what case tags can be used with numbers. I'll have to create additional tags specific to numbers for things that have no grammatical analog. At least I can't think of any reason the /-gon/ in /octagon/ would be used as a case tag on a regular noun.

And then I'll have to decide how many forms of the numbers to use, and why. I think at least two forms are necessary, one for cardinal numbers and one for the others. If for no other reason than aesthetics; I'm not one of those loglangers who thinks adding adjectives to C++ creates a viable human language. So the idea of "three-ad" and "two-nary" instead of "triad" and "binary" is rather appalling to me. Maybe one form for adjective numbers and another for adverb numbers. Basically what English does with stress in the word "duplicate". I could even see more than two forms possibly being helpful.

Wouldn't bicentenary etc. be in the same category as decade?

Not exactly. "Decade" means a period of ten years. "Bicentenary" means once every 200 years. Sort of the same difference as "five inches long" vs "five inches apart". It is a small, but important, distinction.

[eldin raigmore]

I bet those of you who have never heard of "Ackermann's function G" can look it up.
It is defined recursively as follows;
For any ordered triple (x,y,z) of positive whole numbers;

Code: Select all

G(1,y,z)=yz
G(x,1,z)=z
G(x+1,y+1,z)=G(x,G(x+1,y,z),z)

So G(2,y,z) = zy

---

There's a mathematical operation called "tetration".
z tetrated y times, or z tetrated to the y, is G(3,y,z).

So
z tetrated 2 times is zz or z^z
z tetrated 3 times is z^(z^z) or zz^z
z tetrated 4 times is z^(z^(z^z)) or zz^(z^z)
and so on.

G(3,3,3) is 3 tetrated 3 times which is G(4,2,3)
which is 3^(3^3) = 3^27 = 7, 625,597, 484,987

G(3,4,4) is 4 tetrated 4 times which is G(4,2,4)
which is 4^(4^(4^4)) = 4^(4^256)
which is about 4^1.3408E+154, too big a number for the average P.C. to handle.

G(3,1,1)=1=1
G(3,1,2)=2=2
G(3,1,3)=3=3
G(3,1,4)=4=4
G(3,2,1)=1^1=1
G(3,2,2)=2^2=4
G(3,2,3)=3^3=27
G(3,2,4)=4^4=256
G(3,3,1)=1^(1^1)=1
G(3,3,2)=2^(2^2)=16
G(3,3,3)=3^(3^3)=7, 625,597, 484,987
G(3,3,4)=4^(4^4)=1.34078079299426E+154
G(3,4,1)=1^(1^(1^1))=1
G(3,4,2)=2^(2^(2^2))=65,536
G(3,4,3)=3^(3^(3^3))=3^7625597484987
G(3,4,4)=4^(4^(4^4))=4^1.34078079299426E+154

G(4,1,1)=1=1
G(4,1,2)=2=2
G(4,1,3)=3=3
G(4,1,4)=4=4
G(4,2,1)=G(3,1,1)=1
G(4,2,2)=G(3,2,2)=4
G(4,2,3)=G(3,3,3)=3^(3^3)=3^27=7625597484987
G(4,2,4)=G(3,4,4)=4^(4^(4^4))=4^1.34078079299426E+154
G(4,3,1)=1
G(4,3,2)=G(3,G(4,2,2)=G(3,4,2)=2^(2^(2^2))=65536
G(4,3,3)=G(3,G(4,2,3),3)=G(3,G(3,3,3),3)=G(3,7625597484987,3)=3 tetrated 7,625,597,484,987 times
G(4,3,4)=G(3,G(4,2,4,4)=G(3,G(3,4,4),4)=G(3,4^1.34078079299426E+154,4)=4 tetrated 4^1.34078079299426E+154 times
G(4,4,1)=1
G(4,4,2)=G(3,G(4,3,2),2)=G(3,65536,2)=2 tetrated 65,536 times

G(4,4,3) and G(4,4,4) are too big to even write down in any conventional format.
Trying to even calculate G(4,4,4) will probably blow the memory stack and the memory heap of any computer ever built to date.

[Sasquatch]

Proof positive that mathematicians have the best drinking games.
Why would anybody even come up with something like that?

Although, Zz seems like an idea that really should have its own lexical entry.

[eldin raigmore]

In Archimedes's "The Sand Reckoner", he tries to calculate how many grains of sand it would take to fill a sphere one Astronomical Unit (about 93,000,000 miles) in radius. (That is, the sphere's radius is the same as the radius of the Earth's orbit around the Sun.)

(Naturally he ignored the fact that if you put that much sand into such a sphere it would ignite under its own gravity. Since it would mostly consist of elements as heavy as or heavier than silicon it would also proceed to supernova pretty quickly. I recommend we forgive him.)

Archimedes had to invent some "orders" of numbers to give an idea of what his answer was.

Most people who think about languages' numeral systems are aware that it's convenient to have a sequence of bases each of which is the square of the next lower base.

For Archimedes's Greek, the "basic base" was ten.
The sequence consisted of:
ten;
a hundred (ten tens);
a myriad (a hundred hundreds, that is, ten thousand to us);
and "M" (a myriad myriads, that is, a hundred million to us).

That is,
10^(2^0) = 10^1 = 10
10^(2^1) = 10^2 = 100
10^(2^2) = 10^4 = 10,000
10^(2^3) = 10^8 = 100,000,000

He then (if I remember correctly -- do I?) defined the "first order" of counting numbers or natural numbers (that is, positive whole numbers) as all those less than or equal to M^M = 100,000,000 ^ 100,000,000.

You can readily see that's a lot bigger than a "googol" (10^100); in fact it's about the eight-millionth power of a googol.
However, it is a lot less than a "googlplex" (10^googol = 10^(10^100) ), since a googol is 10^92 times as big as M = 100,000,000.

If I remember correctly he defined the "second order" based on the first order; and the "third order" based on the second; and so on.
But I don't remember those definitions.

---

I once read that the total number of elementary subatomic particles in the visible universe is around 10^80.

Suppose your language has a numeral-base "B", a positive integer greater than 1.
You could have a sequence of superbases formed by successive squaring;
B^(2^0) = B^1 = B
B^2 = B^(2^1) = B^2 = B^2
(B^2)^2 = B^(2^2) = B^4
(B^4)^2 = B^(2^3) = B^8
(B^8)^2 = B^(2^4) = B^16
(B^16)^2 = B^(2^5) = B^32
(B^32)^2 = B^(2^6) = B^64
(B^64)^2 = B^(2^7) = B^128
(B^128)^2 = B^(2^8) = B^256
(B^256)^2 = B^(2^9) = B^512
(B^512)^2 = B^(2^10) = B^1024
and so on.

If B is ten or more, B^(2^7) is more than 10^80, and there aren't that many things to count.

Suppose B is just 2.
Then still 2^(2^9) = 2^512 = 2^2 * 2^510 = 4 * (2^10)^51 > 4 * (1,000)^51 = 4 * 10^153.
Still a uselessly large number. (Excel says it's about 1.3408E+154)


But 2^(2^8) is more than 64 * 10^75 (Excel says it's about 1.15792E+77), which is smaller than 10^80; there are that many things to count, though I don't imagine any mortal could ever count them.

I think we could divide natural numbers into "orders" ourselves.
"zeroth order" -- every natural number less than 2^(2^0) (that is, "zeroth order" consists of 1 and that's it).
first order -- every natural number greater than or equal to 2^(2^0) but less than 2^(2^1) (that is, 2 and 3).
second order -- every natural number greater than or equal to 2^(2^1) but less than 2^(2^2) (that is, 4 thru 15 inclusive).
third order -- every natural number greater than or equal to 2^(2^2) but less than 2^(2^3) (that is, 16 thru 255 inclusive).
fourth order -- every natural number greater than or equal to 2^(2^3) but less than 2^(2^4) (that is, 256 thru 65,535 inclusive).
fifth order -- every natural number greater than or equal to 2^(2^4) but less than 2^(2^5) (that is, 65,536 thru 4,294,967,295 inclusive).
sixth order -- every natural number greater than or equal to 2^(2^5) but less than 2^(2^6) (that is, 4,294,967,296 thru (2^64)-1 -- about 1.84467440737096E+19 -- inclusive).
seventh sixth order -- every natural number greater than or equal to 2^(2^6) but less than 2^(2^7) (that is, 2^64 thru (2^128)-1 -- about 1.84467440737096E+19 -- inclusive).
And so on.