This nine-minute video features linguist Tom Scott on numbers and linguistics; Lars Mathiesen, who sent it to me, says it has “a good explanation of otteoghalvtreds. Also septemvigesimal gestural numbers.” Fun pull-quote: “Hindi is so irregular that for the numbers from 1 to 100 you essentially have a hundred different words.” Scott says “Danish is astonishing” and adduces the number Lars mentions, which represents 58 and where the halvtreds part represents “half thrice times twenty” = 3 – 1/2 x 20 = 50. He ends with a peroration that warmed my heart, about how sf writers — who he had assumed would blow his mind with their wild and crazy alien number systems — are tame compared to the number systems of, say, the South Pacific. And it’s worth having the captions on so you can enjoy things like “dot is the freebie action” = “that is the abbreviation.” Thanks, Lars! (Oh, and if you look at the frame of the video and think “That’s almost ten minutes, why does he call it a nine-minute video?”: the final minute is an ad for a hosting service, which you should feel free to skip. There is no bonus Easter egg at the end.)
I am confused as promised … Surely it must be “twenty times half to three” = (3 – 1/2) x 20 to yield 50?
Or else, how does one parse “half thrice times twenty” to avoid (1/2) x 3 x 20 = 30?
“Thrice”? I thought “the third [20]”?
Anyway:
(3 − 1/2) × 20 = 50
3 − 1/2 × 20 = −7
Sinde, given as ‘times,’ is cognate with E send (formally OE sið) and meant things like ‘journey’ in ODa. So it makes sense that you have ‘gone the twenty’ half the third time. (It’s obsolete but has been replaced by gange (cognate ‘go’, also a frozen dative), and English ‘times’ is the same idea).
Halvtredje is obsolete, but halvanden for 3/2 is very much alive. (Though 30 tredive is from ‘three tens’).
Also the Danish system was common Nordic in the 11th/12th, but if you ask a Swede or Norwegian now they think it is something we invented recently ago to confuse the others. Having the units before the tens may be later Low German influence, though, I don’t remember.
Also also the tens are opaque to modern speakers, learned by rote. When you go to the beach as a kid and the water is a bit cold (i.e., every time), you make a circle and count ti, tyve, tredive, …, HUNDREDE and everybody ducks under.
@lars
In German anderthalb for 3/2 is parseable as “halfway to the other (formerly second)”. BTW “recently ago” does not work for me, I would have to say “not long ago” or “in the recent past”.
recently ago — Bad edit, read recently. I had something about X centuries ago and decided that was too specific.
@lars
It was interesting for me, because “gone away recently” is OK for me and i suppose ago is some sort of fossilised verbal adjective from go. But it only takes a noun timespan (possibly indefinite, as a long time) as preceding modifier
Having the units before the tens may be later Low German influence, though, I don’t remember.
English had it too until very recently, as in “four-and-twenty blackbirds baked in a pie”, so it’s at least Common West Germanic. What is more, it seems that Old Norse was happy with either tuttugu ok einn or einn ok tuttugu, and even with ellifu tigir ok sjau ‘eleventy and seven’ or sjau ok ellifu tigir ‘seven-and-eleventy’.
Maddeningly, not a single Gothic numeral greater than ten is recorded except for multiples of ten. Even more maddeningly, searching for Gothic numeral finds a whole bunch of confounds for blackletter-style digits and for sans-serif ones, as Gothic/gothic can mean either in the font context.
At the Linguistic Society of Papua New Guinea meeting that I attended in September 2019, an old-timer, Syd Gould, gave a talk on counting and number in Huli, a Highlands language that uses base-15 very regularly. You count on two hands, then one foot. There are quite a few Austronesian languages in the Pacific that have subtractive numerals to render 6-9 (e.g., 10-4, 10-3, 10-2, 10-1). Yapese 7 translates ‘and-3’, 8 is ‘and-2’, and 9 is ‘and-1’. That pattern is also found in the Admiralties (Manus).
Ainu also has a very interesting system. 78 is rendered as something like ‘(2 from 10 =) 8 excess, 10 away, 4 score’.
subtractive numerals
kahdeksan
From Proto-Finnic *kakteksa, i.e *kakt-e-ksa, consisting of an earlier form of kaksi + a form of the negation verb ei + ksa, which has the same origin as the endings in the reflexive forms found in dialects, c.f. Kalevala forms such as istuikse = istuutuu (“he/she sits”), so that the combination means “two are not there”, and “of ten” was implied or originally said. Compare yhdeksän (“nine”), which developed the same way.
The original nominative kahdeksa was replaced by the genitive due to kymmen ending in -n (which is also the genitive ending), but is attested in older texts, particularly in compounds.
https://en.wiktionary.org/wiki/kahdeksan
I should think it would be easier to count on one hand and both feet, but what do I know? (Last night I found out just how hard it is to take a picture of one’s own foot with a laptop: it took both of us and all our patience, and still turned out crap, but the doctor saw enough to be concerned and asked me to come in, foot problems being Very Bad for diabetics. Treatment is underway and I’ll come in again on Thursday.)
I think the language with the largest number of body locations used to count has 53.
I think I find halvtreds particularly odd because of Gaelic leth-cheud (?sp), half-hundred, an optional exception to a base-20 system. I obviously can’t cope with the same number being half of such different things! 🙂
Ainu also has a very interesting system. 78 is rendered as something like ‘(2 from 10 =) 8 excess, 10 away, 4 score’.
This reminds me a bit of an algorithm for finding roots. You close in on the number by approximating from above, then correcting down, then correcting up. Except they do it back’ards.
The reverse Newton(-Raphson) method using differences instead of derivatives.
@jc 10.09
http://www.wulfila.be cites following from streitberg’s 1910 dictionary:
ainlif Num.2 elf
twalif Num.2 zwölf
fidwortaihun Indeclinable vierzehn
fimftaihun Num.2
fimftataihunda Adj.a fünfzehnter
so I suppose you mean no numerals over twenty except the tens. But the bible had
Nehemiah 7:21
D sunaus Azeiris, sunaus Aizaikeiins, niuntehund jah ·h· (98)
Which to me suggests gothic “ninety and eight” but I agree you can say that W is used to writing n-digit numbers as e.g. letter1 (hundreds) letter2 (tens) letter3 (units), so he may write the above even if he would say it as “eight and ninety”.
Now that sounds more like a kind of Roman numeral system. 98 = IIXC = (2 from 10 =) 8 excess, 100. The standard Roman way is LXXXXVIII, or not ? Or is IIC allowed ?
What’s confusing to the modern reader is the combination of additive and subtractive, depending on whether you’re reading from the left or the right. You need an LALR(4) parser in your head – left-to-right with four symbol lookahead and backtracking.
You count on two hands, then one foot.
To avoid toppling over when counting past 15?
@Stu Clayton: The standard Roman numeral for 98 is XCVIII. You should never need more than three of a single symbol in a row. (Ideally, the system could have been designed to never need more than two in a row, with 3 = IIV, 8 = IIX, 80 = XXC, etc.; however, that’s not what happened.) Forms like IIC for 98 are certainly attested, but, like forms such as 4 = IIII, it is not clear how widely they were used, either among native Romans or later Latin speakers.
David L: Yes, that’s exactly how Syd Gould introduced the Huli base-15 system!
And that’s limited geographically nowadays.
Not much stranger to me than the English practice of counting in hundreds well beyond ten of them. Rendering 4500 as “forty-five hundred” seems unthinkable in German, and even 1100 is “eleven hundred” only when it’s a year – otherwise it’s tausendhundert for me and eintausendeinhundert for the more profligate.
He ends with a peroration that warmed my heart, about how sf writers — who he had assumed would blow his mind with their wild and crazy alien number systems — are tame compared to the number systems of, say, the South Pacific.
Alas, it is true. Somewhere I read a comment that hardly anyone describes an extraterrestrial race whose patterns of thought and behavior are stranger to us (i.e. Americans) than the Japanese.
Somtow Sucharitkul aka S. P. Somtow’s post-apocalyptic first novel Starship and Haiku (1981) is about how the Japanese are in fact the spiritual offspring of whales (“human-shaped and whale-minded”), who consider dying to be the greatest of the arts. When the Japanese learn this, they are so shamed by having killed their ancestors generation after generation that the Triumvirate ruling Japan orders them all to commit honorable suicide (anyone who refuses is simply murdered out of hand).
Somtow was raised in both Thailand and the UK; the P stands for Papinian.
I’ve certainly read a lot of 16th-century stuff where 4 was iiij. On the other hand, I’ve also seen 40 from that period as xl. I always had the impression that subtractive notation didn’t become The Normal Way till the 17th century, after Arabic numerals became usual in everyday texts.
I always had the impression that subtractive notation didn’t become The Normal Way till the 17th century
…though attested since at least the late 5th century, when the Vandals of Carthage issued coins valued at XLII = 42 nummi.
(IIRC there are even earlier sporadic examples.)
My telephone number looks as if it was designed to give foreigners practice in French numbers. It isn’t
Zero four, four twenties eleven, sixty ten seven, zero five, four twenties six
but it’s similar enough to that to make the point. While I’m thinking of it there is a place near Toulon called Six Fours. I think of it as 24.
I don’t think I could cope with Ainu.
David M.: Not much stranger to me than the English practice of counting in hundreds well beyond ten of them. Rendering 4500 as “forty-five hundred” seems unthinkable in German, and even 1100 is “eleven hundred” only when it’s a year – otherwise it’s tausendhundert for me and eintausendeinhundert for the more profligate.
We count in hundreds up to twenty of them:
åttehundre, nihundre, tusen, ellevehundre, […], attenhundre, nittenhundre, totusen, totusenetthundre.
School’s been trying to to get us out of that habit for … a century, probably, and that works as those things do.
Older german and even Croatian, presumably from germn influence, used to count in hundreds beyond 1000. Croatian migrants in Australia have also adopted the practice.
To go along with the English-language practice of continuing to count numbers above one thousand in hundreds, there is the practice of omitting a comma (which would be a space in continental Europe*) in four-digit numerals. In the Heath mathematics textbooks that I used from second to fifth grades, the second chapter was always “Place Value,” and those books asserted that the comma was optional in four-digit numbers and only required for five-digit and larger numbers. To me, at least, “4,500” looks definitely like “four thousand five hundred,” while “4500” could either be four thousand five hundred” or “forty-five hundred.”**
However, other pedagogical sources for children sometimes stated that leaving out the comma in a four-digit numeral was an error, unless the number was a year. According to this prescription, the numeral “1776” could only be the year “seventeen seventy-six,” not the number “one thousand seven hundred (and) seventy-six.” Obviously, for numbers well over two thousand, the possible ambiguity is lessened in practice (to the limited extent that it ever might exist in practice—as in the joke that “1986 pennies are worth almost twenty dollars.”); “4500” is unlikely to be mistaken for a year, even though it is written with no comma.
*Europeans also seem to use the spaces on the other side of the decimal point, separating groups of three numerals the same way. While it seems like a good idea for readability, I don’t think I have ever seen an American version of this, with commas—that is, something like, “e is approximately 2.718,281,828,459,045….” (I just copied this value from Wikipedia, since I only know the first ten digits of e off the top of my head, and strangely, the Wikipedia article writes the numeral with spaces every five decimal digits.)
**Naturally enough, we have already discussed this here, as I discovered while tracking down the other link I included.
However, other pedagogical sources for children sometimes stated that leaving out the comma in a four-digit numeral was an error, unless the number was a year.
I wish more people would learn a distinction between error and bad practice. Writing no punctuation in numbers at all (except for decimal point/comma) cannot be an error; writing 1.34e5 or 1.34 10^5 instead of 1.34·10⁵ may offend one’s aesthetic sensibility, but it is not a mistake. Even writing the aforementioned number 13.4e4 should be considered (if done without a good reason) as unnecessarily weird, not wrong.
I am reluctant though to lay it on school teachers, who in general are overworked, underpaid, and tasked with herding cats simultaneously with nurturing precocious geniuses, that tasks when performed successfully is nothing short of a circus act.
the English-language practice of continuing to count numbers above one thousand in hundreds
It’s English-lang. nowadays but up until I’d say the mid-1960s, I think it was just American. This makes me think it originated either in another language and was brought to America, or that it started with the American style of enunciating phone numbers eg. 4800 being said as “forty-eight” (tiny pause) “hundred” rather than the English-style “four, eight, double-oh”.
That’s less common than using a point. The decimal point is replaced by a comma; in Austria that’s even practically the only usage of the word Komma.
The Swiss, apparently of all languages, mostly use apostrophes: 1’000.
Omitting the separator is also common, but there’s no sharp break at 4 digits; the more digits there are, the more likely the separator is used.
Even in 2007, Lynne Murphy said that saying twenty three hundred rather than two thousand three hundred was apt to cause momentary puzzlement in British listeners. I find that surprising, but she lives in England and I don’t.
David: That’s less common than using a point. The decimal point is replaced by a comma; in Austria that’s even practically the only usage of the word Komma.
Not here. The standard way is comma for the decimal point and (non-breaking) space for separation into sets of three digits.
apt to cause momentary puzzlement in British listeners. I find that surprising
So do I. There’s nothing puzzling.
but she lives in England and I don’t.
That never stopped me. She doesn’t speak for everyone.
and strangely, the Wikipedia article writes the numeral with spaces every five decimal digits
I used to think that large numbers looked weird in the Volshebnyy Dvurog (1st edition, 1948 – the favorite popular math book of my childhood), because the integer part had spaces every three digits, but the fractional part had spaces every five digits. (I don’t recall whether they had any numbers long enough to feature both.)
I didn’t realize that it was more common than that, though in retrospect I had seen it elsewhere.
That said, I just checked what Wikipedia has on Ramanujan’s constant (probably the most famous number with lots of important digits on both sides), and sure enough, both parts have spaces every three digits. I guess the different standards could’ve been confusing.
A very instructive thread on a surprising number of issues!
IIRC, the standard log-and-sine-table book wheniwasalad (Erlang C — yes, Stu, that Erlang) — had 5-digit groups after the decimal comma. But most tables only went to 5 digits anyway, only constants like e and pi got more.