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Today I've heard a talk about division rules. The lecturer stated that base 12 has a lot of division rules and was therefore commonly used in trade.

English and German name their numbers like they count (with 11 and 12 as exception), but not French:

  # |   English |           German | French
-----------------------------------------------
  0 |      zero |            null  | zero
  1 |       one |            eins  | un
  2 |       two |            zwei  | deux
  3 |     three |            drei  | trois
  4 |      four |            vier  | quatre
  5 |      five |            fünf  | cinq
  6 |       six |           sechs  | six
  7 |     seven |          sieben  | sept
  8 |     eight |            acht  | huit
  9 |      nine |            neun  | neuf
 10 |       ten |            zehn  | dix
 11 |    eleven |             elf  | onze
 12 |    twelve |           zwölf  | douze
 13 | thir|teen |       drei|zehn  | treize
 14 | four|teen |       vier|zehn  | quatorze
 15 |  fif|teen |       fünf|zehn  | quinze
 16 |  six|teen |       sech|zehn  | seize
 17 |seven|teen |       sieb|zehn  | dix-sept
            18 and 19 are "regular"
 20 |    twenty |          zwanzig | vingt
 21 |twenty-one |  ein|und|zwanzig | vingt et un
 22 |twenty-two | zwei|und|zwanzig | vingt-deux
            23 - 69 are "regular"
 70 |  seven|ty |         sieb|zig | soixante-dix = 60 + 10
              ....
 80 |   eigh|ty |         acht|zig | quatre-vingts = 4*20 ?!?!
 81 |eighty-one |  ein|und|achtzig | quatre-vingt-un = 4*20 + 1
      ...

So my question is:

Why do French count so strangely after 79?

(Are there other languages that count similar? What's the historic / mathematical reason for this system?)

Related Questions

Martin Thoma
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    Note that there exist words ("huitante", "nonante") used in some variants of French for 80, 90 ([source](http://french.about.com/od/vocabulary/ss/numbers_4.htm)). – Alfonso Fernandez May 14 '13 at 10:14
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    I wouldn't say that from 23 to 79 they are regular: what about *soixante-quinze* for 75? Some varieties of French have *septante*, but *soixante-dix* is, I'd say, more common. – egreg May 14 '13 at 11:47
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    French.SE is probably a better fit for this question? – Najib Idrissi May 14 '13 at 11:54
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    For perspective, a Chinese person would wonder why we count so strange in English. There was an article once which hypothesized that Chinese kids progress faster in maths than English kids because English kids end up wasting a lot of time/mental power to internalize the *teens. – Supr May 14 '13 at 12:35
  • Why do German and English count so strangely? Estonian and Finnish treat the range 11-19 in a uniform way without exceptions at 11, 12. Archaic Finnish treats 21-29, 31-39,..., analogously to 11-19 range in the sense that if you interpret, say fourteen, literally to mean `four of the second (group of ten)`, then `two of the third` "obviously" means 22 :-) – Jyrki Lahtonen May 14 '13 at 14:01
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    @AlfonsoFernandez: I think that the Belgians and the Swiss use the french words except 'septante', 'octante' (or 'huitante'), 'nonante' (being more logical than we are !). In fact it was used earlier by the French too ! [Ref](http://www.langue-fr.net/spip.php?article202) in french of course... – Raymond Manzoni May 14 '13 at 14:03
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    of the limited number of languages I know, Japanese seems to be the most straight-forward for counting, with the translations being essentially `one, two, ..., nine, ten, ten one, ten two, ..., ten nine, two ten, two ten one, two ten two, ..., ..., nine ten nine, hundred, hundred one`. It makes learning to count in japanese, as simple as learning `0-10`, `100`, `1,000`, and `10,000`. – zzzzBov May 14 '13 at 14:29
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    I think questions of the format "Why do the French ______ so strangely?" would make popular questions on any forum. – rschwieb May 14 '13 at 15:12
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    [Here's an article](http://larrycheng.com/2009/10/07/how-language-and-math-intersect-chinese-v-english/) expanding on what @Supr said. – Bobson May 14 '13 at 15:21
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    @Supr I thought a very similar thing, though about Japanese rather than Chinese, when I saw the question. Once you can count to ten, you can count to 99 without having to learn a new word. Then just one new word for hundred, then you can go to 999. etc. So to be able to count from 1 to 999, you only need to learn 11 words where as in English, you need 29 words. – Kevin May 14 '13 at 15:49
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    Why do English and German count so strangely after ten? Why eleven and twelve, and not firsteen, seconteen, thirteen, ... – Kaz May 14 '13 at 16:06
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    Japanese also has the most logical way of writing dates (year-month-day) – JoelFan May 14 '13 at 16:14
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    @zzzBov Japanese may be simple for counting, but you have two entire systems for counting from 1 to 10; the "native" Japanese and the Sino-Japanese: ichi, ni, san, .... hitotsu, futatsu, mittsu, ... then there are different suffixes for counting different things: ichi dai, ni dai, ... ippiki, nihiki, ... quite a number of them! then in the calendar, the first day of the month is tsuitachi. The second through tenth are futsuka, mikka, ... touka. The next? Juu-ichi-nichi, and so on. But the twentieth day is not juu-ni-nichi, but hatsuka! Overall, counting is a pretty large area of Japanese. – Kaz May 14 '13 at 16:19
  • I very much doubt that our way of describing the numbers is any kind of drawback in maths since the decimal system is expressed numerically and actually only requires you to know 0-9. The label to describe a given number is nice to know, but not a requirement of understanding. – JamesRyan May 14 '13 at 17:33
  • @Kaz, please see http://linguistics.stackexchange.com/a/563/225 for Japanese number+classifier words. The linked video is both funny and insightful. – rajah9 May 14 '13 at 18:13
  • By the way, I meant "the twentieth dai is not juu-ni-nichi, but .." – Kaz May 14 '13 at 18:14
  • Hindi number words are famously irregular; you might be interested to compare. – MJD May 14 '13 at 18:15
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    I lived in a French-speaking country at the Fin de siècle. While "Windows 98" takes 5 syllables and alternating stressed and unstressed syllables, my manager would always refer to that operating system as "Windows Quatre-vingt-dix-huit." – rajah9 May 14 '13 at 18:28
  • ahah yes I also wondered about the 4 x 20 when I was young and asked in class... it was that or soixante vingt... – jokoon May 15 '13 at 00:06
  • French is not the weirdest — see http://www.sf.airnet.ne.jp/ts/language/number.html . Also see http://linguistics.stackexchange.com/questions/1949 . – 200_success May 15 '13 at 03:45
  • @Kevin, it may indeed have been Japanese I originally read about. I forgot which and ended up referencing Chinese in this case :) – Supr May 15 '13 at 07:34
  • @JamesRyan How do you count in your mind without labels? Do you actually visualize the glyphs 0-9 or something? – Supr May 15 '13 at 07:40
  • @supr for small numbers of things you don't count them, you just see how many there are, plus 0-9 are standard anyway. For large numbers of things something like a tally chart is far more reliable than numerical counting. For addition, multiplication, division or anything more complicated you are following a process based on digits that doesn't involve any counting. Eg. 16 + 3 is 6+3 with 1 on the front, not seventeen, eighteen, nineteen. – JamesRyan May 15 '13 at 11:55
  • @JamesRyan, what I mean is how do you keep the number 16 or 19 in your mind without expressing it as "nineteen" or something? Suppose you calculated `16 + 3` as `6+3 with 1 on the front`, then how do you remember/use that result later? Do you picture `9 with 1 on the front` in your mind? I'm actually starting to think that this is what we do when dealing with numbers "unconsciously", but that requires us to have developed that "unconscious" ability first. It's as if we develop an internal math-processor capable of storing the values directly instead of as words. – Supr May 15 '13 at 12:14
  • @supr well that is an interesting question. I picture the numbers not words, but then I am rubbish at mental arithmatic so maybe not the best example. :) Would be interesting to know how other people think of them. – JamesRyan May 16 '13 at 12:04
  • We have a linguistics SE; this is better there. – MathematicsStudent1122 Jul 30 '16 at 06:37

8 Answers8

65

Many languages have (at least relicts of) non-decimal counting, very often vigesimal (because we have 20 fingers plus toes), but also many other systems. I recommend an old Gutenberg project of mine, The Number Concept

Note for example that the Danish word for 55 is femoghalvtreds "five more than half the third twenty-block"

Hagen von Eitzen
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    So femoghalftres is 5+20*2+20/2 = 5+40+10=55? Thanks for mentioning this! I just took a look at this [danish counting table](http://www.ucl.ac.uk/atlas/danish/numbers.html). Very interesting how different languages count. – Martin Thoma May 14 '13 at 10:14
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    Another suggestion I've heard is that some cultures count both above and below the larger knuckle on each finger to end up with base 20. Others also use gaps between the fingers in conjunction with various parts of each finger to get some strange bases. – mdp May 14 '13 at 10:18
  • @Hagen If I was to just say *femtifem*, people would know what I meant, right?! Femoghalftres looks too much like six and a half to me! – mdp May 14 '13 at 10:21
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    @Matt: Yes, but we would take you for a Swede. – hmakholm left over Monica May 14 '13 at 10:38
  • @moose: It is more $5+(3-\tfrac 12)\cdot 20$. (Note that this derivation is not something native speakers actively keep in mind while using the numbers -- it's a relic of constructions that are not linguistically productive anymore, and the part that meant "20" has almost entirely disappeared in the modern word. We just remember that _halvtreds_ is how one pronounces $50$). – hmakholm left over Monica May 14 '13 at 10:48
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    @HenningMakholm Even if I don't pronounce any of the consonants? ;-) – mdp May 14 '13 at 10:48
  • @MattPressland: No, in that case I don't think people would know what you meant. – hmakholm left over Monica May 14 '13 at 10:52
  • @HenningMakholm Fair enough! Apologies for my cheap shot based on trying to speak Swedish in Denmark! – mdp May 14 '13 at 10:56
  • Why do they change systems from base 10 to base 20, though? – Ryan Amos May 14 '13 at 15:21
  • I was given to understand that the Swedes changed their numbering system because they didn't understand Danish numbers. ;) I've always been slightly amused that if something should cost 100 Crowns the Danish price will be 95 Crowns (femoghalvfems) but the equivalent in England would be reducing something from £10 to £9.99 (nine ninety nine). So the Danish numbering system saves you 5 Crowns every time you visit a supermarket. – Tim May 14 '13 at 17:10
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    In bases vigesimal, binary and decimal, I am the very model of a modern Major General. – Kaz May 14 '13 at 18:17
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    I can even count the ordinals and numbers infinitesimal. – Tim May 14 '13 at 18:20
  • This is not how I've understood the Danish system. 1. The number 60 is _tres_, short for _tresindtyve_, literally three-times-twenty. 2. The number 2.5 is _halvtredje_, literally half-third (i.e. subtract a half from the third). 3. The number 50 is originally _halvtredjesindtyve_ (2.5-times-twenty), shortened to _halvtreds_. 4. Finally, 55 is _femoghalvtreds_ = five-and-fifty. – Sverre May 26 '13 at 18:25
  • I don't remember why I read this, but sometime after, I noticed your name on the Gutenberg title page, so thank you! – MJD Apr 29 '14 at 16:25
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Actually, if you go back in time a bit in English, you'll realise that English was 'strange' too:

Four score and seven years ago our fathers brought forth on this continent a new nation, conceived in liberty, and dedicated to the proposition that all men are created equal.

(The Gettysberg Address, 1863)

Now if you were to translate that into French in 1863, you'd get pretty much what you'd get today:

Il y a quatre-vingt sept ans, nos pères donnèrent naissance sur ce continent à une nouvelle nation conçue dans la liberté et vouée à la thèse selon laquelle tous les hommes sont créés égaux.

Why this has remained the case in French but has changed in English is probably more of a question for French.SE or English.SE.

Rich
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  • OT: the latest book by Douglas Hofstadter and Emmanuel Sander has a passage in which the authors explain that such a translation would be correct at a basic level but wrong at a higher one: but this really belongs to another SE. – mau May 14 '13 at 11:38
  • @mau With the answers/discussion spanning multiple languages; unless there's a generic linguistics SE I'm not sure where it would go. – Dan Is Fiddling By Firelight May 14 '13 at 12:41
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    [Linguistics](http://linguistics.stackexchange.com/)? – TRiG May 14 '13 at 13:07
  • @TRiG: I guess my question would be more appropriate there. I didn't know that this one existed. But I've got my answer (and quite a lot of interesting additional information) – Martin Thoma May 14 '13 at 13:58
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    I don't think "fourscore and seven" was actually common in English in 1863; Lincoln was using unusually ornate language for rhetorical effect. – MJD May 14 '13 at 18:17
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    According to [Google Ngram viewer](http://books.google.com/ngrams/graph?content=fourscore%2Ceighty&year_start=1500&year_end=2008&corpus=15&smoothing=10&share=), "eighty" has been more common than "fourscore" since at least the 1600s. – Dan May 15 '13 at 01:52
  • Fourscore isn't a common phrase in 1863, it is just those ornate fancy words like use perchance instead of perhaps – user29418 Feb 15 '18 at 09:05
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actually numbers from 11 to 16 are quite regular in French (and in Italian) too: they just are a derivation from Latin.

        | French    | Italian       | Latin 
un      | on·ze     | un·dici       | un·decim
deux    | dou·ze    | do·dici       | duo·decim
trois   | trei·ze   | tre·dici      | tre·decim
quatre  | quator·ze | quattor·dici  | quattuor·decim
cinq    | quin·ze   | quin·dici     | quin·decim
six     | sei·ze    | se·dici       | se·decim
sept    | dix-sept  | dici(as)sette | septem·decim
huit    | dix-huit  | dici-otto     | duo·de·viginti
neuf    | dix-neuf  | dici(an)nove  | un·de·viginti

(18 and 19 in Latin are computed as 20-2 and 20-1). Each language has its own way to cope with small numbers.

mau
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Ya, right, so strange, so how about to use Chinese count? Maybe a little convenient. :D

  # |   English |           Chinese
----------------------------------------------
  0 |      zero |            零
  1 |       one |            一
  2 |       two |            二
  3 |     three |            三
  4 |      four |            四
  5 |      five |            五
  6 |       six |            六
  7 |     seven |            七
  8 |     eight |            八
  9 |      nine |            九
 10 |       ten |            十
 11 |    eleven |            十|一
 12 |    twelve |            十|二
 13 | thir|teen |            十|三
 14 | four|teen |            十|四
 15 |  fif|teen |            十|五
 16 |  six|teen |            十|六
 17 |seven|teen |            十|七
                             十|八
                             十|九
 20 |    twenty |            二|十
 21 |twenty-one |            二|十|一
 22 |twenty-two |            二|十|二

100,000,000 | 1 hundred million  |  一|亿 (1 followed by eight zeros)
OK, let me show you the 999,999,999.   九亿九千九百九十九万九千九百九十九

Just imagine that what if we use this to do math. ;-P

Hang Pan
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    Funny thing that the most crowded ideogram is that of "zero". – Andrea Mori May 14 '13 at 17:30
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    @AndreaMori It seems to leave zero whitespace ... – Hagen von Eitzen May 14 '13 at 17:45
  • @AndreaMori maybe because, as in most societies I guess, zero is quite an advanced concept, so it came later and all the "simple" ideograms were already used? – Olivier Dulac May 14 '13 at 17:51
  • @hang-pan: then I believe chinese counts as : "one ... nine, ten, ten one, ten two, ... ten nine, 2 ten, 2 ten one, 2 ten two, ....... 9 ten 9, hundred, hundred one, .... hundred nine, hundred ten one, .... ? seems faily regular (but not easy to read/write ^^) – Olivier Dulac May 14 '13 at 17:53
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    For larger calculations Chinese historically used abacuses, and recorded the numbers using [rod numerals](http://enwp.org/rod_numeral), which are superbly uniform and simple. Your complaint is analogous to an Anglophone saying "let me show you 999,999,999: nine hundred ninety-nine million, nine hundred ninety-nine thousand, nine hundred and ninety nine. Just imagine if we used this to do math." The point is, we didn't and don't, and neither did the Chinese. – MJD May 14 '13 at 18:19
  • @AndreaMori, there are two characters for each number in Chinese, one simple and one more complex (used to write checks for instance). Hang Pan here put the complex "zero" character but usually it's just written "O". – laurent May 15 '13 at 07:24
  • @MJD wow, you're cool, you know "rod numerals", which is a kind of counting method in ancient China. And no complaint here, but really wanted to have fun with moose on the "strange" count. :) – Hang Pan May 15 '13 at 08:11
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    @Laurent ya, thanks for clarification, sometimes, we can use simplified "〇" to substitute "零" in Chinese, but not in all cases, and notice that it is not Arabic numeral 0, or letter 'o' or 'O', it only exists in Chinese numeric phrase. – Hang Pan May 15 '13 at 08:19
  • The thing to take away is that Chinese names for numerals are a lot more faithful to the decimal meaning than English. I always found it interesting that Chinese counting later becomes cyclic at multiples of 10,000, not 1,000 as in English :) – rschwieb May 15 '13 at 20:31
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Actually, the French system was designed regularly, i.e. with "septante" for 70, "octante" for 80 and "nonante" for 90. But this system was not accepted at the time (roughly the 16th century), because people were used to the old system ("quatre-vingt", etc.). But the old system remained in Belgium (for "septante" and "nonante") and Switzerland (for "septante, the slightly modified "huitante" in Vaud canton, and "nonante").

deltux
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This is a bit of a guess but I think this has a lot to do with whether the numbering system was rationalised at any point.

If you tell someone from England that you're having trouble sleeping he'll probably suggest you count sheep. I won't attempt a weak joke about having a bedroom full of farm animals. The phrase comes from an old numbering system

 1  |  yan
 2  |  tan
 3  |  tether  
 4  |  mether
 5  |  pip  
 6  |  nether
 7  |  aether
 8  |  oevro
 9  |  cuevro
10  |  dick
11  |  yanadick
12  |  tanadick
13  |  tetherdick
14  |  methradick
15  |  bumfit
16  |  yanabum
17  |  tanabum
18  |  tetherbum
19  |  metherbum
20  |  jiggit

This died out a long time ago but shepherds kept it up for longer. So to count sheep is to If you look on Wikipedia there are loads of different ones people counted differently in different areas of the country. (It seems the one my great-granddad taught me isn't in there.)

At some point this was changed by introducing a new numbering system. Similarly The swedes and Norwegians use a base 10 numbering system today, but they used to use the Danish system. Similarly German (I think) has undergone several standardizations to keep it as one language. It wouldn't surprise me if there used to be less rational German counting systems that died out.

If the French tried to rationalise their numbering is would have been during the revolution. But as they tried to rationalise everything, including a ten hour day and a decimal calender (Today is Carp the 25th of Flower) A lot of things didn't take off. So I would guess the numbers were one of the things that failed.

Tim
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These numberphile videos explain the irregularity quite well:

Problem with french numbers

In english 1-12 have individual names and after that we have the -teens (4 and 10, 5 and 10, etc.) similar break point happens at the 17th number in the French number system. The major problem which occurs is at 70s. *The French have no word for 70, they call is 60 and 10 - soixante-dix.*So on, 71 is 60 and 11 till 79.

Similarly, 80 is also not invented, it is 4 20s, quatre-vingt. 81 is 4 20s 1, quatre-vingt-un. 90 is, 4 20s 10, quatre-vingt-dix. 91 is 4 20s 11, quatre-vignt-onze.

Finally, they have a word for 100, ceut.

It looks similar to roman numbers Roman number system, where you repeat the letters to combine the values. X is 10 XX is 20 and so on.


Another peculiar thing in french grading system, 19 out of 20

rk.
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    It is not true that there is no word for 70, really, not for either 80 and 90: there are there perfectly good words *septante* and so on. They are just local variations. In English you have the same thing, and a famous speech starts «four score and seven years ago...». – Mariano Suárez-Álvarez May 14 '13 at 17:52
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    About the 19/20 video, it is amazing how a deep tone of voice and a lot of self-confidence can help make believe pure nonsense: the summary of the video reads *In French culture, it is traditional for all grades to be out of 20 - and many teachers will NEVER give full marks!* and this is indeed what the video says. Well, while the first part is true, the second part is plain false. – Did May 26 '13 at 09:22
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As former Chinese then French colonists we Vietnamese dropped the Chinese script and took on the Roman alphabets and the Arabic numbers as our own, thank to France. But we didn't take on the French counting system. Ours is very straight forward, and like Chinese, is mostly a monosyllabic language.

một (one),
hai (two),
ba (three),
Bốn (four),
năm (five),
sáu (six),
bảy (seven),
tám (eight),
chín (nine),
mười (ten),
mười một (ten one = eleven),
mười hai (ten two = twelve),
hai mươi (two tens = 2(10) = 20),
hai mươi một (two tens one = 2(10) + 1 = 21),
ba mươi (three tens = 3(10) = 30),
ba mươi một (three tens one = 3(10) + 1 = 31),
một trăm (one hundred),
một trăm một (one hundred one = 1(100) + 1 = 101),
hai trăm (two hundreds).