15

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".

  • 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 '12 at 13:52
  • 4
    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 '12 at 15:19
  • 4
    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 '12 at 15:20
  • 3
    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? – Mark Beadles Oct 5 '12 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 '12 at 6:47
18

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 (Kwomtari): binary system (base 2)
  • Pazeh (Austronesian): quinary (base 5)
  • Amis (Austronesian): decimal (base 10)
  • Ron (West Chadic): duodecimal (base 12)
  • Barapasi (East Geelvink Bay): vigesimal (base 20)
  • Meyah (East Bird Head-Sentani): mixed-quinary/vigesimal
  • 1
    From that link, Fas isn't really binary; it seems to be a doubled unary (non-positional) system. Would have been cool. – Mechanical snail Jan 22 '13 at 0:15
  • 1
    I regret to report that all your links are dead 404. – Wilson Apr 25 at 11:44
  • http://lingweb.eva.mpg.de/numeral/https://mpi-lingweb.shh.mpg.de/numeral/ – meisterluk Apr 25 at 22:09
  • Thank you @meisterluk for fixing those broken links! – Gaston Ümlaut Apr 25 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).

  • Thanks you very much for your post. The WALS article is very interesting :) – meisterluk Oct 7 '12 at 12:09
  • 5
    I don't think anybody would seriously suggest Pirahã to have a base-10 number system. – user3503 Jun 1 '14 at 11:48
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.

  • Yucatec Maya has completely adopted Spanish numbers, except when talking about the Long Count, where the vigesimal terms are used. – jlawler Jan 22 '17 at 21:08
5

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?

Even in Danish numbers have a similar mechanism:

Number — Danish shorthand spelling — Danish full spelling — English literal translation — Not so literal English translation

50 — halvtreds — halvtredje-sinde-tyve — half-three times twenty — half-to-tree times twenty

60 — tres — tresindtyve (tre-sinde-tyve) — three times twenty

70 — halvfjerds — halvfjerde-sinde-tyve — half-four times twenty — half-to-four times twenty

80 — firs — firsindetyve (fire-sinde-tyve) — four times twenty

90 — halvfems — halvfem-sinde-tyve — half-five times twenty — half-to-five times twenty

Note that the sinde word is obsolete.

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!

  • What sort of German says "vier-zwozich" ? – fdb Sep 6 '14 at 22:28
  • @fdb: corrected. – sergiol Sep 6 '14 at 22:57
  • You have "Four-twentys-ten" twice. – curiousdannii Sep 6 '14 at 23:17
  • @curiousdannii: corrected. – sergiol Sep 7 '14 at 23:36
4

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).

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