Generally all number systems use addition or multiplication to express numbers like - '12*3 + 6 for 42 in a base 12 system', '2 on the way to 50' or 'even 10+10+10+10+2 in some'. But, are there number systems which use subtraction like 10*5-8 (for an exemplar base 10 system), I know Ainu is close but not what I want to know about.

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    @Suresh Colloquially English has "dozen" and "gross "(a square dozen). I'm pretty sure these were common in spoken English before widespread literacy and education, but I couldn't give a citation. Anecdotally, older people even today are more likely to estimate groups of objects in dozens than younger people (my grandfather would happily have referred to a crowd of "about 4 dozen people" for example) Doesn't really qualify but it's worth noting.
    – Some_Guy
    May 4 '17 at 13:41
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    Also, in some parts of the world people conventionally count on their fingers not to 10, but to 144. By using the thumb to count along finger segments, you can mark up to twelve with one hand, and keep track of dozens on the other.
    – Some_Guy
    May 4 '17 at 13:45
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    @Suresh Languages using duodecimal number systems are uncommon. Languages in the Nigerian Middle Belt such as Janji, Gbiri-Niragu (Gure-Kahugu), Piti, and the Nimbia dialect of Gwandara;[5] the Chepang language of Nepal[6] and the Mahl language of Minicoy Island in India are known to use duodecimal numerals. (en.wikipedia.org/wiki/Duodecimal) May 4 '17 at 13:51
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    @some-guy, wondering in English why does the teen numbers start after 12
    – Suresh
    May 4 '17 at 13:51
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    @Suresh yeah , but I am looking for a fuller system, not just nine words, Hindi also has un+chas for 49 (50-1) and pachas for 50 (please ignore the way I represent sounds, very unprofessional, I know) May 5 '17 at 3:40

It's somewhat curious that both the Roman calendar and the Roman numerals used subtraction quite extensively; for neither is there any modern equivalent. For example, even the Advent sundays (which are defined as the four sundays before Christmas) are counted upwards; surely one could call them the fourth, third, second, first sunday (before Christmas) in that order instead, but that is not being done.

By contrast, the Roman calendar used subtractive counting throughout, being entirely based on months and 'signal post' days within each month (kalends, nones, ides) that are being counted 'towards'.

Likewise, in numerals, I, II, III, IV, V stand for the numbers one to five, with IV replacing the more straightforward IIII (which form is still used on clock faces, though). This is repeated with higher numbers, viz IX = 9, XC = 90, CM = 900 and so on. Be it said that usage appears to have been far from uniform, though; also, the Wikipedia article has many contradictory examples, and most are not from early (classical and pre-classical) times, so it's not immediately clear what the earliest forms were.

In both cases, the downward counting is so counter-intuitive to a modern observer that one is compelled to think there should have been a strong motivation for it to occur; maybe that motivation was in the number words of the early language; and, indeed @Yellow Sky cites duodētrīgintā ('2 from 30', 28 = 30 – 2) and ūndētrīgintā ('1 from 30', 29 = 30 - 1), which are quite naturally written out as IIXXX and IXXX in Roman numerals.

One could venture a wild hypothesis that in ancient Latium, there was a people with a language that used (partially) subtractive counting, vestiges of which system are to be found in their calendar and written number systems, akin to the 'twelve' vs 'thirteen', 'dozen' and 'gross', 'quatre-vingt-dix' of modern times that betray earlier duodecimal and vigesimal systems. The German Wikipedia article on Roman numerals somewhat rejects the link between written and spoken forms: "Die subtraktive Schreibung [...] stimmt aber mit [den Schreibweisen] nicht überein. Bei den lateinischen Zahlwörtern werden die Wörter für 1 und 2, aber nicht auch die für 10 und 100 subtraktiv verwendet und hierbei dann in der Regel auch nur den Vielfachen der 10 ab 20 (duodeviginti = 18, undeviginti = 19)"; OTOH, it admits that 'undecentum' = 99 (= IC) does sometimes appear in ancient sources.

https://en.wikipedia.org/wiki/Roman_calendar#Months https://de.wikipedia.org/wiki/R%C3%B6mische_Zahlschrift

  • Ok so tell me this, does the Roman system do it all the way down from 19 to 11? (every number except first ten and the tens expressed in this form), because what we learned was this I II III IV V VI VII VIII IX X, Only the bold ones being perfectly subtractive. May 6 '17 at 0:51
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    The Wikipedia articles I quoted do list some non-standard forms; they seem to be more common with bigger numbers. For example, 1999 could conceivably be written as MCMXCIX, MCMIC, CMMIC, MIM or IMM. Not all of these forms have been used. To spell 11 subtractively, one would have to write 9 to the left of 20: VIVXX(?). I have never seen that. May 6 '17 at 17:52
  • So yeah that's what, since one can't subtractively use all the numbers thus it comes into the overcounting category again. May 7 '17 at 2:21

The Yoruba language has a rather elaborate vigesimal (base-20) numeral system that involves both addition and subtraction and multiplication.

There are words for each of the decades; units in 1–4 are created by adding to these, while units in 5–9 are created by subtracting from the next decade. The odd decades are created by subtracting ten from the next even decade:

14=10+4; 15=20-5; 16=20-4; 50=20×3−10, 700=200×4−100; etc.

Also, Latin used subtraction in numerals. Among the numbers from 18 to 98, those that ended in 8 or 9 were formed by subtracting 2 and 1 from the following decade:

27: septem et vīgintī ('7 and 20')

28: duodētrīgintā ('2 from 30')

29: ūndētrīgintā ('1 from 30')

  • Perfect answer I agree, but are these languages not overcounting examples? like the one I mentioned in my example (8 on the 50). I trying to find a language which says 50-8 and not 2 on the way to 50, that too all the way i.e if the base is 12: 1 -> 12-11, 2 -> 12-10, 3 -> 12-9, 4 -> 12-8, 5 -> 12-7, 6 -> 12-6, 7 -> 12-5, 8 -> 12-4, 9 -> 12-3, 10 -> 12-2, 11 -> 12-1. May 4 '17 at 15:33
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    @Gusto - I'm sure every natural language in the world has the numerals for at least 1 and 2 as simple words, without any subtraction.
    – Yellow Sky
    May 4 '17 at 15:38
  • I was just trying to give an example, make it 13 -> 12*2-11, 14 -> 12*2-10, 15 -> 12*2-9, 16 -> 12*2-8, 17 -> 12*2-7, 18 -> 12*2-6, 19 -> 12*2-5, 20 -> 12*2-4, 21 -> 12*2-3, 22 -> 12*2-2, 23 -> 12*2-1, I know 1 and 2 being the most basic numbers would have their names of own, also my example being base 12, there will be individual names till 12. May 5 '17 at 3:36

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