Numerals

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HoskhMatriarch
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Numerals

Post by HoskhMatriarch »

OK, I keep getting pestered by Gornec (even when I haven't made a new language) and I'd like to give him numbers, but none of my languages have numeral systems, except I think Spraka might have a few numbers (although Spraka is a fauxlang with most of the words copied from English and German and occasionally words from other Germanic languages so who cares). But, I really want to make some interesting systems. The first idea I had for a numeral system was basically Danish but with 11 and 12 as oneteen and twoteen and numbers greater than 10 that are 8 or 9 after being 1 or 2 before (so, twenty eight would be two from thirty) like Latin. That's not particularly creative. My second idea is almost completely normal except that 5-9 are one-five, two-five, three-five, four-five in "backwards" order even though twenty-one would be twenty-one in that order, and I don't like that idea much either, even if it's more creative and natural than Latin meets Danish. So, what are some of the most interesting numbers you all have seen or made?
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Re: Numerals

Post by Sumelic »

Well, there's Piraha (there's always Piraha...) Actually, I've been thinking about this as well; do you mind if I add one more question? Does anyone know of a natlang where the default position for compound numbers is around the modified noun, in the following way:
one dog = one dog.sing
two dogs = two dog.plur
twenty dogs = twenty dog.gen
twenty-one dogs = one dog and twenty

I guess it could work around classifiers as well, something like the following:
one dog = dog one [animal-CLASS]
two dogs = one dog = dog two [animal-CLASS]
twenty dogs = dog [animal-CLASS] twenty
twenty-one dogs = dog one [animal-CLASS] twenty
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druneragarsh
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Re: Numerals

Post by druneragarsh »

Finnish has base numbers 1-7, forms 8 and 9 as "2 away from 10" and "1 away from 10" (though using the PIE dek- root), and has 10 be derived from "palm, hand".

For Proto-Ṭelö, I formed the numbers like this:
1 = finger, ḷirä
2 = pair, tẹsa (ordinal ḳora)
3 = 4 without 1 (4 1-PRIV), kärä ḷiräti
4 = palm, hand, kärä
5 = other hand, ḳora kärä
6 = hand and pair (hand pair-COM), kärä tẹsaka
7 = 8 without 1, ṭelö ḷiräti
8 = person, ṭelö
9 = 8 1-ASSOC, ṭelö ḷiräŋy
10 =8 2-ASSOC, ṭelö tẹsaŋu
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HoskhMatriarch
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Re: Numerals

Post by HoskhMatriarch »

Wow, these are some long numerals... https://en.wikipedia.org/wiki/Alyutor_language#Numerals
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Re: Numerals

Post by Micamo »

Tazaric numerals:

bún̪ wááŋ gá
little eye one.F
one little eye

bùn̪ wáàŋ d̪ó̤ó̤
PL\little PL\eye two.F
two little eyes
My pronouns are <xe> [ziː] / <xym> [zɪm] / <xys> [zɪz]

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Keenir
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Re: Numerals

Post by Keenir »

HoskhMatriarch wrote:OK, I keep getting pestered by Gornec (even when I haven't made a new language)
that's unlike him.

could he have gotten the wrong impression, that you were making more conlangs?
and I'd like to give him numbers, but none of my languages have numeral systems,
just say, in your post or in a reply to him, either
"Haven't gotten around to the numbers yet, sorry."
or
"Speakers of this conlang, just use the numbers of their neighbors."
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sangi39
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Re: Numerals

Post by sangi39 »

HoskhMatriarch wrote:Wow, these are some long numerals... https://en.wikipedia.org/wiki/Alyutor_language#Numerals
Meh, one through to five are basic, being two or three syllables long, which isn't too unusual and you can find these number with similar lengths in a number of languages (although it looks like three and four might be derived in Alyutor, but it's not mentioned, and some of the endings seem to drop in combined forms), and then ten and twenty are base numbers too. After that it's a case of multiplying and adding numbers together, which is pretty universal when it comes to forming higher and higher numbers (although oddly enough, twenty isn't used in forming higher numbers at all).

So 99 is basically ((4+5)x10)+(4+5) as opposed to the English (9x10)+9 or the French (4x20)+9 or the German 9+(9x10).

In the Chumashan languages, apparently, 99 would have been something like ((2+4)x16)+3 since they used base-4 and base-16, while Komnzo, using base-6 would have it as (2x36)+(4x6)+3 while 72 would simply be (2x36) as opposed to the French (6x10)+12 or the English (7x10)+2.

I've used base-8, base-10 and base-5 so far in my conlangs, trying to shift the base-8 one into base-10 and base-12 and found that number length is pretty relative to the language, the base system they use and what number you're actually trying to use. Base numbers tend to be shorter than derived/combined forms and then the system kind of just starts over with a few extra syllables thrown in. In counting, they might find short cuts, such as only counting in units until they hit the next base and then only saying the full number at regular intervals (I tend to do this at work, e.g. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, ...., 8, 9, 20, 1, 2, ... and so on) so that large numbers don't become a problem during counting. Specific numbers will likely turn up in set circumstances, while the majority of the time the bases will do, in a similar manner to English users saying "hundreds of people" or "thousands of flies" except Komnzo speakers will say "216s of people" or "1296s of flies".
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eldin raigmore
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Re: Numerals

Post by eldin raigmore »

How do speakers of your conlangs — and/or natlangs if you like — say and/or write the following eighteen numerals?
  1. 16
  2. 81
  3. 256
  4. 512
  5. 625
  6. 6,561
  7. 19,683
  8. 65,536
  9. 262,144
  10. 390,625
  11. 1, 953,125
  12. 33, 554,432
  13. 43, 046,721
  14. 134, 217,728
  15. 4,294, 967,296
  16. 152,587, 890,625
  17. 847,288, 609,443
  18. 7, 625,597, 484,987
In English these are
  1. sixteen
  2. eighty-one
  3. two hundred fifty-six
  4. five hundred twelve
  5. six hundred twenty-five
  6. six thousand five hundred sixty-one
  7. nineteen thousand six hundred eighty-three
  8. sixty-five thousand five hundred thirty-six
  9. two hundred sixty-two thousand one hundred forty-four
  10. three hundred ninety thousand six hundred twenty-five
  11. one million nine hundred fifty-three thousand one hundred twenty-five
  12. thirty-three million five hundred fifty-four thousand four hundred thirty-two
  13. forty-three million forty-six thousand seven hundred twenty-one
  14. one hundred thirty-four million two hundred seventeen thousand seven hundred twenty-eight
  15. four billion two hundred ninety-four million nine hundred sixty-seven thousand two hundred ninety-six
  16. one hundred fifty-two billion five hundred eighty-seven million eight hundred ninety thousand six hundred twenty-five
  17. eight hundred forty-seven billion two hundred eighty-eight million six hundred nine thousand four hundred forty-three
  18. seven trillion six hundred twenty-five billion five hundred ninety-seven million four hundred eighty-four thousand nine hundred eighty-seven
In case you’re wondering (you probably are), these numbers aren’t random.
They’re the values of A^(B^C) where each of A and B and C varies through the whole numbers from 2 to 5, except I didn’t include values greater than 3^(3^3). I also left out duplicates such as 2^(2^5) = 4^(2^4) = 4^(4^2).

I’m interested how languages, especially your languages, handle very large numbers.
Or which ones they don’t handle; I think European languages didn’t handle numbers as great as one million before the First Crusade.

If your language’s numeral system has a base other than ten, of course that will be interesting.
But even with base ten, your language might use the equivalents of myriads or crores or lakhs instead of millions and billions and trillions.
And if they use billions and trillions, maybe a billion is a million million and a trillion is a million billion, and a thousand million is a milliard.
And so on.
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Reyzadren
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Re: Numerals

Post by Reyzadren »

^My conlang would just say the digits out individually, amongst several methods. Easy.

Base 10, but it doesn't use thousand/million, it uses tenthousand instead.
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eldin raigmore
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Re: Numerals

Post by eldin raigmore »

Reyzadren wrote: 27 Sep 2020 23:56 ^My conlang would just say the digits out individually, amongst several methods. Easy.

Base 10, but it doesn't use thousand/million, it uses tenthousand instead.
If your base B is ten or fewer, it might be sensible to use the sequence
B^(2^n)
for a base-sequence, rather than
B^n.

OTOH if your base B is ten or greater, it might be more sensible to use the
B^n sequence, than the B^(2^n) sequence.

....

The sequence in which each is the square of the previous one, starting at ten, is
Ten 10
Hundred 100
Myriad 10000 (I think this is what you meant by “ten thousand”?)
Myriad-myriad or hundred-million 100,000,000
10^16 or ten quadrillion
10^32 or hundred nonillion
10^64 or ten vigintillion

I don’t see any point in going greater than 10^80.
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Reyzadren
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Re: Numerals

Post by Reyzadren »

eldin raigmore wrote: 28 Sep 2020 01:42
Reyzadren wrote: 27 Sep 2020 23:56 ^My conlang would just say the digits out individually, amongst several methods. Easy.

Base 10, but it doesn't use thousand/million, it uses tenthousand instead.
Myriad 10000 (I think this is what you meant by “ten thousand”?)
Yes, ten thousand = 10000
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Re: Numerals

Post by Iyionaku »

Reyzadren wrote: 27 Sep 2020 23:56 ^My conlang would just say the digits out individually, amongst several methods. Easy.

Base 10, but it doesn't use thousand/million, it uses tenthousand instead.
That's interesting. Are there any natlangs who express big numbers like this?
I'm not saying you cannot absolutely do this - it's your conlang, after all. But I lay awake last night for a while thinking about it, and I (as a linguistic amateur, mind you) can see many practical problems with it.

Mostly, you have no idea how big a number is until it's said completely. For instance, take the simple question "how big was the turnover of this company last year?" And now somebody answers "well, it was a total of two four five eight zero zero zero zero zero Dollars."... and you need to count how many digits were said in your head. So was it four zeros? Or five? What exactly was the first number again?
Also, you don't have any redundancy in case you couldn't the speaker properly. In English or Chinese (among all other natlangs I know) you can still get how big the number was at all, even though you didn't completely understand the number. "It's [inaudible] thous-[inaudible] forty-five" - you still know it's in the 4-digits, or maybe 5. In griuskant, someone might just hear "it's [inaudible] four [inaudible]", leaving the listeners completely clueless.

These are just my thoughts and I might be completely wrong here. Maybe some of the more adept or professional linguists in this board can share their thoughts on that.
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Reyzadren
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Re: Numerals

Post by Reyzadren »

^Eldin raigmore posted very specific large numbers, without trailing zeroes. In general conversation with approximation, like your example, they would also use place values. In griuskant, this can be expressed with the standard form as usual, which is coefficient x 10^magnitude.

:con: griuskant (without script here)

zhe pludsaenae degnastae sloukekson az voe? 2.458 x 10^8.
/'ʒə 'pludsene 'dəgnaste 'slɔukəksɔn az vɯ? 'tʃaus gəz 'hiɔdliugθuan 'kəru 'θuan/
this group-PL-POSS-PASS-POSS back-year-POSS trade-plus-EB-PASS is howmany? 2 decimal 458 10magnitude 8
How big was the turnover of this company last year? Well, it was a total of 245800000 dollars.

Of course, having 1 convenient place value at the end would still require one to wait, but the opposite also has problems: Every intermediate place value is repeated, and at the end, I might have forgotten the initial magnitude [:O]
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Re: Numerals

Post by lsd »

I do the same in sajátnyelvföld...
and all languages do it in writing without the need to count the digits...
you just have to write/pronounce them in short, regular groups...
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eldin raigmore
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Re: Numerals

Post by eldin raigmore »

It seems to me that some natlangs group them in twos and fours and maybe eights (not sixteens unless the base is lower than ten.)
Some natlangs group them in threes and sixes; not aware of any that group them in twelves.
I will be surprised if that exhausts the possibilities.

The numbers I asked about are all powers of 2 or 3 or 4 or 5. That’s why none of them have trailing zeroes in decimal representation.

If you would like to express, for instance, all the B^(2^n) where B is your base and n runs from, say, zero to 9, go ahead!
That would be particularly interesting if B is one of 4 or 5 or ten or twelve or twenty. Ten and twenty seem to be the most common exponential bases in WALS.info. Twelve is popular among conlangers. And when Greenberg’s students studied them, those five bases — four, five, ten, twelve, and twenty — were found to be most common among the languages in their sample.

2^(2^9) is probably going to be bigger than your computer can express exactly.
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eldin raigmore
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Non-Somatic Choices of Systems of Bases for Numerals in NonHuman Conlang

Post by eldin raigmore »

How would you structure your conlang’s numeral system of counting numbers, if your conspeakers didn’t have four limbs, and/or didn’t have five digits per limb, and/or didn’t have three phalanges per digit? In other words, weren’t shaped like humans?
….
First let me get this out of the way.
I’m just going to declare that any system in which some base is not a real number is unrealistic; and any system in which some base is not a natural number is unnaturalistic.
So at least in this post, I’m only going to be talking about multiplicative bases, or systems/sequences of multiplicative bases, that are natural numbers greater than one.
….
There are some few to several lists of desiderata for a base or a sequence of bases to have; and some few to several interpretations of some of those lists.

For instance, “a base should have a lot of divisors”.

If this means “more divisors than any smaller base”, that’s what the OEIS.org calls “highly composite numbers”.
Some candidates might be:
2 (two) has 2 divisors
4 (four) has 3 divisors
6 (six) has 4 divisors
12 (twelve) has 6 divisors
24 (twenty-four) has 8 divisors
36 (thirty-six) has 9 divisors
48 (forty-eight) has 10 divisors
60 (sixty) has 12 divisors
120 has 16 divisors
180 has 18 divisors
240 has 20 divisors
360 has 24 divisors
720 has 30 divisors
….
I’d assume the larger numbers, if one of them appears as a base, appears as a superbase that is a multiple of some smaller base.

….

But maybe it just means “has at least as many divisors as any smaller number”.
That’s the sequence OEIS.org calls “largely composite numbers”.
It includes all the numbers in the above set, as well as more.

2 and 3 (two & three) each have 2 divisors;
4 (four) has 3 divisors;
6 & 8 & 10 (six, eight, & ten) each have 4 divisors;
12 & 18 & 20 (twelve, eighteen, and twenty) each have 6 divisors;
24 & 30 (twenty-four & thirty) each have 8 divisors;
36 (thirty-six) has 9 divisors;
48 (forty-eight) has 10 divisors;
60, 72, 84, 90, 96, & 108 (sixty, seventy-two, eighty-four, ninety, ninety-six, and 108) each have 12 divisors;
120 and 168 each have 16 divisors;
180 has 18 divisors;
240 and 336 each have 20 divisors;
360, 420, 480, 504, 540, 600, 630, 660, 672 each have 24 divisors;
720 has 30 divisors.

….

But maybe we’d like to ensure the base is divisible by every positive whole number up to and including some number.
In other words the Least Common Multiple of all numbers up to some given number.
2 is LCM(2)
6 is LCM(2,3)
12 is LCM(6,4)
60 is LCM(12,5)
60 is LCM(60,6)
420 is LCM(60,7)
840 is LCM(420,8)
2520 is LCM(840,9)
2520 is LCM(2520,10)
27,720 is LCM(2520,11)
27,720 is LCM(27720,12)
360,360 is LCM(27720,13)
360,360 is LCM(360360,14)
360,360 is LCM(360360,15)
720,720 is LCM(360360,16)
12,252,240 is LCM(720720,17)
12,252,240 is LCM(12252240,18)
….
I think for aliens with minds like humans’ minds, even if their bodies weren’t like human bodies, 420 or 840 might already be too many. 2520 is close to the full everyday-lexicon size for some ‘lects of natlangs; 27720 is close to some expert lexicon-size for some ‘lects of some natlangs. 360360 is more than twice as many roots as are in the OED, and 720720 is almost as many as words in the OED when different senses of homonyms/homophones/homographs are counted separately.

….

But maybe we only care how many and which prime factors the base has, so the base might as well be square-free (not divisible by any square number greater than 1).
Maybe we want it to be divisible by the first n prime numbers for some n.
This is the sequence OEIS.org calls “primorials”.
The first eight are:
2, 6, 30, 210, 2310, 30030, 510510, 9699690
6 is 2*3
30 is 6*5
210 is 30*7
2310 is 210*11
30030 is 2310*13
510510 is 30030*17
9699690 is 510510*19
….
Unless their minds are as alien as their bodies, I think they’d find 2310 and 30030 and 510510 too large. At any rate your readers or game players would probably have difficulty following your counting if you used bases that big.

=====…..=====…..=====…..=====…..=====

So much for “having lots of divisors”.

I have one more idea.

….

WALS.info has several examples of languages whose numeral systems have a base and a subbase and a superbase.
The subbase is a divisor of the base — the base is a multiple of the subbase — but the base is not necessarily a power of the subbase; indeed, the base may not be a divisor of any power of the subbase. In other words, the base might have a prime divisor that doesn’t divide the subbase.
For instance what if the subbase is three and the base is six?
Or the subbase is four and the base is twelve?
Or the subbase is five and the base is twenty?
Or the subbase is six and the base is thirty?
And so on.

If the subbase is A >=3 and the base is B = (A-1)*A, then the base B is guaranteed to have divisors that don’t divide A.
The A=4 B=12 and the A=5 B=20 examples are, IIANM, attested rather frequently in natlangs.
The A=3 B=6 and the A=6 B=30 are attested at least once at least in Conlangs.

. . . . . . . . . .

The same sort of thing can happen between a base B and a superbase C.
B will divide C and C will be a multiple of B. But C needn’t be a power of B. In fact perhaps C doesn’t divide any power of B, because some prime divisor of C is not a divisor of B.
If B is an even number >=8 and C=(B-2)*B then C will probably have at least one prime divisor that doesn’t divide B.

Let’s take these ideas and put them together.

Some possibilities are;
Base 6, subbase 3, superbase 24; (9*24 = 6^3, so this is an exception);
Base 12, subbase 4, superbase 120; (5 divides 120 but not 12);
Base 20, subbase 5, superbase 360 (brings to mind the Mayan calendar); (3 divides 360 but not 20);
Base 30, subbase 6, superbase 840; (7 divides 840 but not 30);
Base 42, subbase 7, superbase 1680; (5 divides 1680 but not 42);
Base 56, subbase 8, superbase 3024; (3 divides 3024 but not 56);
Base 72, subbase 9, superbase 5040 (Plato’s favorite number); (5 divides 5040 but not 72).

That may be enough.
But if we want to have a superbase divisible by 11, we could use
base 90, subbase 10, superbase 7920 (11 divides 7920 but not 90)
Or
Base 110, subbase 11, superbase 11880 (3 divides 11880 but not 110)
Or
Base 132, subbase 12, superbase 17160 (check my arithmetic?) (5 and 13 both divide 17160 but neither divides 132).

But in any of those cases we’d have gained divisibility by 11 (and maybe 13) at the expense of losing divisibility by 7.

…..

======———======———======———======———======———======

So, which of these ideas have you used or started to use or considered using in one or some of your Conlangs?
Which ones have you seen in other’s conlangs?
Which have you seen in natlangs?
Which one is your favorite?
Which one is your least favorite?
Which one is your second-favorite?
Which one is your second-least favorite?

…. …. …. ….

Have you seen other ideas about how to construct a sequence of multiplicative bases, that you think I should know about, or should have asked about, but didn’t?

….

I admit my favorites are Srinivasa Ramanujan’s “largely composite numbers”.
That’s the one that starts off 2, 3, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 48, 60, 72, 84, 90, 96, 108, …
My Adpihi and Reptigan are (going to be?) dozenal.
I haven’t decided yet about Arpien but I think I'm leaning toward vigesimal (base twenty) with a subbase of five.
Last edited by eldin raigmore on 03 Dec 2021 16:03, edited 1 time in total.
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Re: Numerals

Post by tikoo »

I'm content with nothing, few, many and extreme. Feel no need to measure with exactness . A language of accountants and traders do this too well. Just listen to the news that reports data as numbers and every day. It is their value system and defined a language system . I'm not dissing mathematics. The relationship of pi is certain. Digitalizing it is hopeless. The evolution of language may resolve this.
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