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Rationale: While writing a document about foundations of computer science and describing that a number is a sequence of digits, I was wondering about our relation to the decimal system.

In English counting goes like this: "one", "two" and "three". If we create a number with 2 digits, the number might be called "forty-two". "Forty" is an alias for 4‧101 and "two" adds 2‧100. So it refers explicitly to the decimal system, because neither "forty-two" nor "ten" exists in binary1. The language uses decimal-based aliases. "Dreizehn" (3+10 = 13) and "quatre-vingt trois" (4*20+3 = 83) are instances of the same concept in German and French.

Cerberus has pointed out that Aztecs and Babylonians have used different number systems from the very beginning which of course became part of the spoken language (radix 20 and 60). A nice hint, but I am interested in modern, spoken languages.

My question: Which modern, spoken languages do not use the decimal system2? I would be happy to be able to find 2 to that kind.

A related question is How do you pronounce numbers written in different bases?


1: It might be a valid point that "ten" is an alias for 1‧101 + 0‧100 and therefore "ten" can also be applied to the binary system with 102 = 210, but that is pretty uncommon and I am not sure about it. Correct me if I am wrong. See related link.

2: With "language without the implication" I mean "the pronunciation of a number is only a concatenation of the pronunciation of the individual digits".

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  • 2
    Whether or not a language describes numbers in decimal terms depends on whether the culture that it belongs to uses decimal numbers. There have been many cultures that use or used other systems, like the Aztecs and the Babylonians, I think, and no doubt they did not describe their numbers in decimal terms.
    – Cerberus
    Oct 5, 2012 at 13:52
  • 5
    Languages in the Nigerian Middle Belt such as Janji, Gbiri-Niragu (Kahugu), the Nimbia dialect of Gwandara; the Chepang language of Nepal and the Mahl language of Minicoy Island in India are known to use duodecimal numerals. / The Yuki language in California and the Pamean languages in Mexico have octal systems because the speakers count using the spaces between their fingers rather than the fingers themselves.
    – Cerberus
    Oct 5, 2012 at 15:19
  • 5
    It is possible for people to count on their fingers to 12 using one hand only, with the thumb pointing to each finger bone on the four fingers in turn. A traditional counting system still in use in many regions of Asia works in this way, and could help to explain the occurrence of numeral systems based on 12 and 60 besides those based on 10, 20 and 5. Wiki on "Radix - In numeral systems" - Follow links from there.
    – Cerberus
    Oct 5, 2012 at 15:20
  • 6
    In your question you mention French and even write that ""quatre-vingt trois" (4*20+3 = 83)"...which is base 20, not base 10 and seems to already fit the bill. So what are you looking for exactly? Oct 5, 2012 at 19:20
  • 5
    Any language which uses second (e.g. of time or of arc) to mean a sixtieth of the previous unit (such as minute) which is a sixtieth of the unit above that (hour or degree) is implicitly working in a form of base 60.
    – Henry
    Oct 9, 2012 at 6:47

7 Answers 7

20

Modern Englishes, and many other well-known languages, use base 10 (ie decimal) systems of numerals. But there are many languages in the world that currently use different systems.

Eugene Chan has been studying the numeral systems of the world's languages for many years and his website Numeral Systems of the World's Languages contains a vast amount of information on this topic. I think a quote from Eugene's introduction is useful:

The surviving thousands of the world's ethnic groups use a variety of different numeral systems: duodecimal systems, decimal systems, quinary systems, quaternary systems, ternary systems, binary systems, incomplete decimal systems, mixed systems, body-part tally systems and so on. Certain South American indigenous languages even only distinguish the numbers "one" and "many".

Some examples:

  • Orokaiva (Trans–New Guinea): digit-tally system
  • Fas (Fas): binary system (base 2)
  • Pazeh (Austronesian): quinary (base 5)
  • Amis (Austronesian): decimal (base 10)
  • Ron (Afro-Asiatic, Chadic): duodecimal (base 12)
  • Barapasi (East Geelvink Bay): vigesimal (base 20)
  • Meyah (East Bird Head-Sentani): mixed-quinary/vigesimal

There do not appear to be any true binary systems occurring naturally as the primary numeral system for any language, although Chan claims Fas (above), Malol (Austronesian) and Malak Malak (Northern Daly) as examples.

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  • 1
    From that link, Fas isn't really binary; it seems to be a doubled unary (non-positional) system. Would have been cool. Jan 22, 2013 at 0:15
  • 1
    I regret to report that all your links are dead 404.
    – OmarL
    Apr 25, 2019 at 11:44
  • http://lingweb.eva.mpg.de/numeral/https://mpi-lingweb.shh.mpg.de/numeral/
    – meisterluk
    Apr 25, 2019 at 22:09
  • 1
    Thank you @meisterluk for fixing those broken links! Apr 25, 2019 at 22:58
12

WALS has a chapter on numeral bases. It appears most languages use either a decimal or a vigesimal (base 20) system (or some mixture thereof, as in Basque), with other patterns being relatively rare. There are also some language that cannot actually express all possible numbers but only a limited set. Pirahã is even claimed not to have any numerals at all (but every claim about Pirahã should be taken cum grano salis).

2
  • Thanks you very much for your post. The WALS article is very interesting :)
    – meisterluk
    Oct 7, 2012 at 12:09
  • 5
    I don't think anybody would seriously suggest Pirahã to have a base-10 number system.
    – user3503
    Jun 1, 2014 at 11:48
6

In some nowadays Indo-European languages you can still see rests of base-20 numeral systems.

For example in France French:

Number — French spelling — English translation — Expansion (where possible)

70 — Soixante-dix — Sixty-ten

71 — Soixante-onze — Sixty-eleven

72 — Soixante-douze — Sixty-twelve

73 — Soixante-treize — Sixty-thirteen

74 — Soixante-quatorze — Sixty-fourteen

75 — Soixante-quinze — Sixty-fifteen

76 — Soixante-seize — Sixty-sixteen

77 — Soixante-dix-sept — Sixty-seventeen — Sixty-ten-seven

78 — Soixante-dix-huit — Sixty-eighteen — Sixty-ten-eight

79 — Soixante-dix-neuf — Sixty-nineteen — Sixty-ten-nine

80 — Quatre-vingts — Four-twentys

81 — Quatre-vingt-un — Four-twentys-one

82 — Quatre-vingt-deux — Four-twentys-two

83 — Quatre-vingt-trois — Four-twentys-three

84 — Quatre-vingt-quatre — Four-twentys-four

85 — Quatre-vingt-cinq — Four-twentys-five

86 — Quatre-vingt-six — Four-twentys-six

87 — Quatre-vingt-sept — Four-twentys-seven

88 — Quatre-vingt-huit — Four-twentys-eight

89 — Quatre-vingt-neuf — Four-twentys-nine

90 — Quatre-vingt-dix — Four-twentys-ten

91 — Quatre-vingt-onze — Four-twentys-eleven

92 — Quatre-vingt-douze — Four-twentys-twelve

93 — Quatre-vingt-treize — Four-twentys-thirteen

94 — Quatre-vingt-quatorze — Four-twentys-fourteen

95 — Quatre-vingt-quinze — Four-twentys-fifteen

96 — Quatre-vingt-seize — Four-twentys-sixteen

97 — Quatre-vingt-dix-sept — Four-twentys-seventeen — Four-twentys-ten-seven

98 — Quatre-vingt-dix-huit — Four-twentys-eighteen — Four-twentys-ten-eight

99 — Quatre-vingt-dix-neuf — Four-twentys-nineteen — Four-twentys-ten-nine

Note: There is a process of elision of the s on Quatre-vingts when it has a number after it.

This happens because the Gauls, a celtic people, had a base-20 counting system. In my personal opinion, it is strange that the forms septante, huitante and nonante, used in Swiss French sound outdated in almost all the French territory. Shouldn't things evolve exactly in the opposite sense, towards simplicity?


Danish numbers employ a similar, though slightly more complex, mechanism:

Number — Danish normal form — Danish full, archaic form (deconstructed origin of full form) — English literal translation — Loose English translation

50 — halvtreds — halvtredsindstyve (halv-tredje sinds tyve) — half-third times twenty — 2.5 times twenty

60 — tres — tresindstyve (tre sinds tyve) — three times twenty

70 — halvfjerds — halvfjerdsindstyve (halv-fjerde sinds tyve) — half-fourth times twenty — 3.5 times twenty

80 — firs — firsindstyve (fire sinds tyve) — four times twenty

90 — halvfems — halvfemsindstyve (halv-femte sinds tyve) — half-fifth times twenty — 4.5 times twenty

Notes: (1) Outside numbers, sinde is obsolete as a multiplicative. (2) Outside the full number forms, this ‘subtractive’way of forming ‘half numbers’ by prepending ‘half’ before the next ordinal number (half-third = two whole + half of the third) is obsolete except for halvanden ‘one and a half’. (3) The short forms are the normal forms for cardinals; ordinals are officially based on the full forms (halvtredsindstyvende ‘fiftieth’, etc.), but especially among younger people may be based on the short forms colloquially (halvtredsende).


We can see also that in other languages, like Portuguese and English that the twenty numbers between 11 and 19 are spelt on a different form than 21 to 99, which may be some hint of a past vigesimal system

Number — Portuguese — English

11 — onze — eleven

12 — doze — twelve

13 — treze — thirteen

14 — catorze — fourteen

15 — quinze — fifteen

16 — dezasseis — sixteen

17 — dezassete — seventeen

18 — dezoito — eighteen

19 — dezanove — nineteen

As you can see, in Portuguese, you say 24 as vinte e quatro , literally twenty and four, but you don't say 14 as unte e quatro or deste e quatro (in Portuguese: 1=um; 10=dez)

The same reasoning goes for English: you say twenty-four for 24, but you don't say onety-four or tenty-four for 14.

And, in German, the numbers between 21 and 99 have their position changed when spelt. So for 24 you say vier-und-zwanzig, literally four-and-twenty. This also does not happen between 11 and 19, the system also seems the previous ones. This can also indicate a historic link to a 20-base numeral system.

By the way, spelling 24 in German, seems spelling 80 in French!

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  • What sort of German says "vier-zwozich" ?
    – fdb
    Sep 6, 2014 at 22:28
  • @fdb: corrected.
    – sergiol
    Sep 6, 2014 at 22:57
  • You have "Four-twentys-ten" twice.
    – curiousdannii
    Sep 6, 2014 at 23:17
  • @curiousdannii: corrected.
    – sergiol
    Sep 7, 2014 at 23:36
  • 1
    The Danish examples are all miswritten. The word sind (plural sinde) in the sense ‘time(s)’ is not exactly obsolete (it’s part of the very common phrase nogen sinde / nogensinde ‘ever, at any time’). In the numerals, it’s used in a sort of absolute genitive–like form which is otherwise obsolete – but it is there in the numbers, and all your instances of sind(e) should be sinds. Jun 9 at 18:11
5

The classical, pure versions of Mesoamerican languages used for the most part a base-20 system. If you go to some indigenous towns in rural Mexico, you may still listen to old people using base-20 numbers in modern versions of Maya, Nahuatl and a handful of other languages.

For the most part, though, young people are adopting a base-10 numbering system similar to that of Spanish (when not preferring Spanish altogether), some other indigenous languages dropped the base-20 earlier. And then there's a very small, isolated indigenous language in northern Mexico which uses base-8. It's disappearing as well, so you'd better hurry and record number samples before it goes away.

1
  • Yucatec Maya has completely adopted Spanish numbers, except when talking about the Long Count, where the vigesimal terms are used.
    – jlawler
    Jan 22, 2017 at 21:08
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Some certain New Guinea languages use base-4: Rawo, Nafri, Vanimo, Bukiyip (which has base-3 and base-4, depending on what is being counted) and, in Papua New Guinea, Wiru and Kewa, along with others. Another base used in New Guinea and some parts of Africa is 6 (senary) in languages such as Ndom, Khimaghama, and Riantana, much like some California Native American counting systems. Some of the above have base-5 and base-20 as secondary.

Huku (Uganda) and Nimbia (Nigeria) are two instances of base-12, arguably the best base in general (for math especially).

3

By "modern" do you mean a language that's still spoken by living people nowadays?

The most unusual modern example is the Telefol and Oksapmin languages, which use a base-27 counting system

Telefol counting starts with the fingers of the left hand (1 being the pinky), progresses from the thumb (5) to the wrist, lower arm, elbow, upper arm and shoulder (6--10), the side of the neck, ear and eye (11--13) and thence through the nose (14) and right eye (15) to the right pinky finger (27). The Telefol idea of a very large number is kakkat=14*27=378.

http://www.math.bas.bg/~iad/serafin.html#Telefol

And here's the Oksapmin 27-body part counting system

Oksapmin counting system

In fact there are many other languages that use a body-part counting system like this

Many other cultures, mostly indigenous/ethnic ones, do also count in other non-decimal bases such as 4, 5, 6, 8, 12, 15, 16, 20, 24, 32, 36, 60... although the most common non-decimal bases nowadays are base 20, 12 and 60

Some cultures do, or did, use other bases of numbers.

  • Pre-Columbian Mesoamerican cultures such as the Maya used a base-20 system (perhaps based on using all twenty fingers and toes).
  • The Yuki language in California and the Pamean languages in Mexico have octal (base-8) systems because the speakers count using the spaces between their fingers rather than the fingers themselves.
  • The existence of a non-decimal base in the earliest traces of the Germanic languages is attested by the presence of words and glosses meaning that the count is in decimal (cognates to "ten-count" or "tenty-wise"); such would be expected if normal counting is not decimal, and unusual if it were. Where this counting system is known, it is based on the "long hundred" = 120, and a "long thousand" of 1200. The descriptions like "long" only appear after the "small hundred" of 100 appeared with the Christians. Gordon's Introduction to Old Norse p. 293, gives number names that belong to this system. An expression cognate to 'one hundred and eighty' translates to 200, and the cognate to 'two hundred' translates to 240. Goodare details the use of the long hundred in Scotland in the Middle Ages, giving examples such as calculations where the carry implies i C (i.e. one hundred) as 120, etc. That the general population were not alarmed to encounter such numbers suggests common enough use. It is also possible to avoid hundred-like numbers by using intermediate units, such as stones and pounds, rather than a long count of pounds. Goodare gives examples of numbers like vii score, where one avoids the hundred by using extended scores. There is also a paper by W.H. Stevenson, on 'Long Hundred and its uses in England'.
  • Many or all of the Chumashan languages originally used a base-4 counting system, in which the names for numbers were structured according to multiples of 4 and 16.
  • Many languages use quinary (base-5) number systems, including Gumatj, Nunggubuyu, Kuurn Kopan Noot and Saraveca. Of these, Gumatj is the only true 5–25 language known, in which 25 is the higher group of 5.
  • Some Nigerians use duodecimal systems. So did some small communities in India and Nepal, as indicated by their languages.
  • The Huli language of Papua New Guinea is reported to have base-15 numbers. Ngui means 15, ngui ki means 15 × 2 = 30, and ngui ngui means 15 × 15 = 225.
  • Umbu-Ungu, also known as Kakoli, is reported to have base-24 numbers. Tokapu means 24, tokapu talu means 24 × 2 = 48, and tokapu tokapu means 24 × 24 = 576.
  • Ngiti is reported to have a base-32 number system with base-4 cycles.
  • The Ndom language of Papua New Guinea is reported to have base-6 numerals. Mer means 6, mer an thef means 6 × 2 = 12, nif means 36, and nif thef means 36×2 = 72.

https://en.wikipedia.org/wiki/Decimal#Other_bases

-3

The only spoken language I found is French, which instead uses base 20. (I agree, because 8421 becomes 'eight times one thousand plus four times one hundred plus one times twenty plus one' instead of 'eight times one thousand plus four times one hundred plus two times ten plus one') Translations: huit mille quatre cent vingt-et-un (eight thousand, four hundred, one twenty, and one)

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1
  • The fact that there’s a word for ‘twenty’ that doesn’t transparently mean ‘two tens’ does not make French base-20. The same is true of nearly all Indo-European languages, and you might as well argue that French is base-100 or base-centillion. Also, vingt does originally mean ‘two tens’, it’s just not transparent anymore. Jun 30 at 23:15

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