LH favorite Arika Okrent has a nice Mental Floss post about “weird things that languages can do with number words,” from Oksapmin, with its base-27 counting system based on body parts (there’s a convenient diagram), to Nimbia, a dialect of the Gwandara language of Nigeria, which uses base 12 (143 is gume kwada ni kwada ‘eleven dozen and eleven,’ 144 is wo). Fun!


  1. Bukiyip wins. Counting different things in different bases, while apparently using the same words for the numbers* in both systems? How does that come about and how did they not give up ages ago?
    *the one number word quoted, anauwip, is 20-base 3 (6 base 10) and 20-base 6 (24 base 10).

  2. @s/o Is 20(6) not 12(10)?

  3. sorry, you’re right, i’ve done my math wrong

  4. I don’t even know how I got that, I misread as well as miscaluclated. The second system is base 4, not 6, so “anauwip” is 120 base 4. I’m even more impressed by the Bukiyip-speakers.

  5. I’m surprised that the description of the base-20 Yoruba system with subtraction didn’t mention the Roman numerals, that also use subtraction, as, for example, XIX means 10 – 1 + 10 = 19. However, I agree that the Bukiyip system is the most impressive. I don’t often need to count betel nuts, but the next time I do I’ll need to make sure I using the proper system.

  6. Jeffry House says

    Can anyone explain exactly how Danish came to have 20-based higher numbers, but Norwegian not? The Danish 60, 70, and so on look odd to me in a way that “tyve” for “tjue” does not, to say nothing of “bog” for “book”. The latter two could be found in Kjelland or Jonas Lie, but not the ones for 60, 70, and so on. (Or so my inner nordmann tells me.)

  7. Bill Walderman says

    The origin of the Danish number system: Somewhere I have a book that explains this, but I can’t find it. I think it was this:
    The Danes are firmly convinced that it is part of their heritage from the ancient Viking past, and cling to it tenaciously–efforts to promote a shift to the English-like system used by the other North Germanic languages such as Norwegian have been unsuccessful. (They do make a concession to the system prevalent elsewhere in Scandinavia in writing checks, though.)
    My recollection is that, contrary to the Danes’ belief, the system actually originated in the 19th century — it was how shopkeepers counted out change. Of course, the Danes don’t perform mental arithmetic as they count–they don’t think of halvtreds as halvtredssyndtyve and mentally multiply 20 by 2 1/2 to arrive at 50. They just think “halvtreds = 50”, just as English speakers don’t need to multiply 5 by 10 when confronted by “fifty”. The ordinal numerals 50 and above are so unwieldy that the Danes avoid using them, especially with the Germanic pre-position of the single digits: femoghalvtredssyndtyvende = 55th (five and two and a half times twenty-eth).

  8. Elizabeth Kendall says

    Dear Languagehat,
    What a terrific blog. I will follow it!
    Mainly, thank you for THE OTHER THING. I am infinitely grateful.

  9. You’re most welcome, and I’m glad you like the blog!
    (Note for bemused denizens of the Hattery: THE OTHER THING is an editing thing.)

  10. Skandinavians don’t write cheques. We use the giro.

  11. Trond Engen says

    We do use giro, but those long identification numbers are killing me. I’ll urge my brother, the internet bank system programmer, to add support for septemvigesimals.
    It’s another touch of weirdness that both hands are counted from thumb to pinky. And that reminds me: Could ‘pinky’ derive from *penkWe by some mysterious way?

  12. My grandfather used to say he was brought up to count in base five:
    ain, tain, tethery, fethery, pip
    ainpip, tainpip, tetherypip, fetherypip, dick,
    aindick, taindick, tetherydick, fetherydick, bumpit,
    ainbumpit, tainbumpit, tetherybumpit, fetherybumpit, gigit etc.
    This is a variant of the “Anglo-Cymric Score” as mentioned in, for example, THE COUNTING-OUT RHYMES OF CHILDREN by HENRY CARRINGTON BOLTON.

  13. Ah yes, “yan tan tethera”; we discussed it here.

  14. Nobody has gotten pinky past Dutch pinkje, though it also existed in Middle English in the form pink. No cognates, no etymology.

  15. The Danish system is attested in an authorization clause added to the Bylaws of Flensborg with a date of 1295 (“thet aar fra wors hærræ aar thusind tv hundrith oc half fæmpt sin tiygh oc fæm”).
    1295 is after the Viking age, but this is one of the earliest manuscripts in Danish — it will be hard to disprove that Danish Vikings counted like that as well.

  16. Bill Walderman says

    Thanks, Lars. I stand corrected. The Danish system doesn’t seem to have featured in Old Norse, at least not in the Western Norwegian/Old Icelandic variety, or in any other Germanic language. So when and how did this numeral system originate? Is there any evidence or even speculation on this point?

  17. Bill, the scholarly Dictionary of the Danish Language (Ordbog over det Danske Sprog) does not trace the 50-70-90 construction any further back than the citation I gave.
    It traces Danish ‘halvanden’ (one and a half) back to Old Norse ‘halfr annar’, so I assume they would have done so for 50-70-90 if they could.
    I did find ‘halfr þriði tugur’ in Landnamabok, but that is 2½ times ten = 25.
    (I don’t have access to Skautrup’s History of the Danish Language beyond the snippets of the index offered by Google, but it doesn’t seem to mention the words in the first volume, the one treating the oldest language stages).

  18. Trond Engen says

    ‘halfr þriði tugur’
    Samlagets Norrøn Ordbok (ON dictionary) says under ‘half-‘ that it’s used in compounds with numerical adjectives ending in ‘-tugr’ or ‘-ræðr’ to denote the lack of a half ten, especially as ‘halfþritugr’ “25”. Nothing about half twenties, so I think it’s safe to assume that it’s not in the corpus.

  19. William Walderman says

    Thanks, Lars and Trond. I wonder whether the origin of the Danish vigesimal system and perhaps the French, can be traced to the division of the pound–the monetary unit in use in medieval and early modern Europe and in Britain right up through the first two or three decades of my life–into 20 soldi (sous) or shillings. According to Wikipedia, this monetary system (with the sou divisible into 12 pence) was established by Charlemagne: http://en.wikipedia.org/wiki/French_livre ,
    but I wonder whether it goes back even further to the Dark Ages and maybe even to late antiquity. Can anyone enlighten me on the pre-decimal Danish monetary system?
    Incidentally, I found my copy of the Danish Grammar I referred to earlier. It seems to say that the decimal numeral system, with multiples of ten ending in -ti, had actually been in use in Denmark before the 19th century, but the older vigesimal system came to predominate during that period.

  20. Bill, librae et solidi et denarii silver coin designs go back at least to the early Roman Empire.

  21. Bill Walderman says

    John, yes, the libra was a Roman unit of weight, divided into 12 unciae (ounces), and you’re right that solidi and denarii were Roman monetary units. The question I was raising is whether the libra, divided into 20 units (solidi/soldi or shillings), was used as a monetary unit before Charlemagne instituted a standardized system.
    My thought is that the French or Danish vigesimal number systems might be traceable to the medieval vigesimal monetary system based on the division of the pound/libra into 20 units, which seems to have been first instituted by Charlemagne. I was wondering whether this system might go back even earlier to monetary systems in use in the Dark Ages or late antiquity, but I haven’t been able to find any evidence of that.
    Of course, it’s also possible that Charlemagne’s twenty soldi to the livre reflects a vigesimal way of thinking that is also reflected in the French and Danish languages. And the livre and even the solidus must have been very large amounts that wouldn’t be used in everyday transactions–they probably existed mostly as accounting units.
    Any information about the medieval Danish monetary system might shed some light on the subject, too.

  22. Bill, the oldest Danish monetary system went like this: 10 penning to the ørtug, 3 ørtug to the øre, 8 øre to the mark. As a metal weight, the mark was about 233g, or eight ounces, and indeed some people want to see the Latin “(uncia) aurea” in the word øre. This was soon (12-13th c) replaced by the Hanseatic (“lybske”) system of 12 pence to the shilling, 16 shilling to the mark. Not much there to inspire vigesimal counting.
    Later monetary history in Denmark (and Norway) didn’t see that system change much, but for long periods there were three parallel systems: Danish coinage, Lübeck coinage, generally at about twice the value for a given denomination, and the relatively stable Thaler/dollar/daler silver pieces, relative to which the others were always losing value.

  23. J.W. Brewer says

    Bill, via google books you can peruse the “Coins and Mints” section of “Medieval Scandinavia: An Encyclopedia,” which asserts that circa 1100 there were 240 pennies to a mark, but the next unit up wasn’t at that point a dozen pennies. At various other points in medieval Denmark per that source a mark was 288 pennies or 192 pennies (in England and Scotland at around the same time per other sources a mark was 160 pennies, i.e. 13 shillings & fourpence). For more recent Danish currency, wikipedia alleges that “The Danish currency system established in 1625 consisted of 12 penning = 1 skilling, 16 skilling [i.e. 192 pennies] = 1 mark, 6 mark = 1 rigsdaler and 8 mark = 1 krone.”

  24. Trond Engen says

    Isn’t it the English- and Swedish-style counting (twenty-two, tjugotvå) that is the recent accountancy innovation? It’s documented in Norwegian from the 17th century. I knew it had been in sporadic use, but not for how long, until I read the no.wiki article on the new counting system. It’s weak on sources, but I think it’s safe to assume that the 17th century usage is from the oldest known Norwegian arithmetics book, Tyge Hanssøn’s Arithmetica Danica from 1645.
    And still, as everybody knows, the switch to the new system in Norwegian was engineered by the telephone authorities, supported by a broad alliance of mathsteachers, and approved by parliamentary decision in 1950. It soon became hated by the conservative Riksmål movement, but it doesn’t really belong on a Nynorsk-Samnorsk-Bokmål axis at all. Anyway, it’s been a slow but steady success, and sociolinguistic surveys show that those who grow up now use only the new way of counting. I do maths in the new system but use the old one in free speech, which is typical of those a generation older than me.

    As much as I like the idea that counting in twenties is connected to weight- and monetary units, it would work better if the twenties were the smallest sub-fraction, not the largest.

  25. Bill Walderman says

    Thanks to everyone who set me straight. It occurs to me that we have a somewhat archaic vigesimal system in English — counting in scores — so perhaps the French and Danish systems aren’t as strange as they seem to language learners.

  26. I was reading the news about recent political developments in Mali and this led me to read a little bit about the Senufo languages, and that reminded me of this post and Okrent’s listicle, which includes the Supyire system. From Robert Carlson (1994) A Grammar of Supyire, p. 167, in a discussion of the etymology of some of the Supyire numbers:

    ŋ̀kùù ‘eighty’ also means ‘chicken’. The etymology is confirmed by the identical irregular plural ŋ̀kwuu ‘eighties, chickens’, although I could find no one who could give me an explanation of the semantic shift. I assume that it has something to do with the price of a chicken at some time in the past.

    I hope you can see the page here:


    I wonder if LH readers can think of other languages in which some number words have such concrete etymologies.

  27. January First-of-May says

    I did find ‘halfr þriði tugur’ in Landnamabok, but that is 2½ times ten = 25.

    See also (somewhat archaic, but still extant in 20th century) Russian полтораста “150”, isolated as a pair with the common полтора “1½” in the modern Russian system. The latter was definitely part of a now-missing pattern (полтретья etc) in pre-modern Russian; I’m not sure if the former was.

    other languages in which some number words have such concrete etymologies

    I am of course immediately reminded of the sporadic Russian сорок “40”, which is traditionally explained as a word for a kind of clothing (i.e. related to сорочка), though IIRC there are other competing proposals.
    OTOH it is not a major unit in the system the way Supyire 80 is; it’s just an unexpectedly suppletive word for 40 that otherwise functions just like any other tens number.

    [EDIT: which is to say, the effect is basically as if English counting went thirty-nine, smurf, smurf-one, …, smurf-nine, fifty – there’s no obvious reason why suddenly “smurf” instead of something that ends in -ty, but otherwise it fits right into the system.]

  28. (I hope this comment will make sense after the previous comment about “eighty” and “chicken” I made here is restored from the spam box)

    For example, from Indo-European, some possible examples of such concrete etymologies for numbers:

    5: https://en.wiktionary.org/wiki/Reconstruction:Proto-Indo-European/p%C3%A9nk%CA%B7e

    (A joke my advisor used to repeat in various forms many years ago, as a caution against excessive internal reconstruction: What is the etymology of *pénkʷe? It’s actually *pénkʷ-h₁e, dual of *penkʷ, “two and a half”.)

    8: https://en.wiktionary.org/wiki/Reconstruction:Proto-Indo-European/o%E1%B8%B1t%E1%B9%93w

    1000: https://en.wiktionary.org/wiki/Reconstruction:Proto-Indo-European/%C7%B5%CA%B0%C3%A9slom

    I wonder what other families and other regions have to offer.

  29. January First-of-May says

    For example, from Indo-European, some possible examples of such concrete etymologies for numbers

    I’m personally partial to Piotr Gąsiorowski’s theory for 4.

    EDIT: for 5 I’ve learned the “fist” option, which looks a lot more plausible in Russian (the words are пять and пясть respectively); the (Germanic-only?) “finger” word appears to be derived from the 5 word rather than vice versa.

  30. SFReader says

    I always thought that there has to be some etymological relationship between “two” and “thou” which goes back beyond PIE.

  31. Thank you for recalling сорок, January First-of-May. It also helped me learn that Slovak has archaic mera, for “40”, too, apparently from Hungarian mérő “measuring”?:


  32. January First-of-May says

    apparently from Hungarian mérő “measuring”?

    …um, mera for “measuring” is common Slavic. As far as I can tell the Hungarian word is supposed to be borrowed from Slavic rather than vice versa.

    I couldn’t find anything about 40 in your links though.

  33. David Eddyshaw says

    I wonder what other families and other regions have to offer.

    Oti-Volta throughout has what is surely the same stem for “five” and “hand”, e.g. Kusaal (a)nu “five”, nu’u(g) “hand”; *-nu- “five” is also found widely outside Oti-Volta, unlike *-nu- “hand.”

    I idly wondered here at one stage whether this meant that speakers of Oti-Volta languages brought the wonders of counting beyond “four” to their West African neighbours, but as DM (I think it was) rightly pointed out at the time, it seems a lot more likely that the Oti-Volta “hand” root derives from “five”, rather than vice versa.

  34. I couldn’t find anything about 40 in your links though.

    It wasn’t in his links; that’s why he was thanking you for reminding him of it.

  35. For the curious, Michał Németh (2008), Zapożyczenia węgierskie w gwarze orawskiej i drogi ich przenikania, p. 79ff, has a very full discussion of Orava dialectal meru “40”. It’s available here:


  36. Kusaal (a)nu “five”, nu’u(g) “hand”

    Thank you David Eddyshaw. It is nice to have a clear example of the nexus of “five” and “hand” (or “fist”) outside Indo-European.

    As for “chicken” in Sipyure, do you know if was there a small denomination that used to circulate in the region in colonial times, 80 of which would have been a reasonable price for a chicken?


    I should have written Slovak meru… (I’ve torn my supraspinatus muscle and I am typing with voice recognition.) I was looking at this article, p. 12, and was impressed by the similar discontinuity for “forty”.

    The Hungarian is in turn from the Slavic, I assume. There are some other typological parallels for “concrete” etymologies of numbers on page 12 too.

  37. David Marjanović says


    Wasn’t there something about 40 squirrel pelts…?

  38. PlasticPaddy says

    Maybe you are aware of this, but Irish has daichead (=”two” + “twenty(gen.or dual)”) for 40, where e.g., 14 is ceathardéag. Again thirty and fifty follow the expected pattern. My guess for sorok would be that a formation like *dvedvadtsati > *dvedtsati, which was abandoned because too similar to twenty. Of course this would not explain why sorok was the replacement word. Something similar happened in Romance with the teens, I think.

  39. In one of my previous comments, the link to Adam Fałowski “The East-Slavonic sorok ‘40’ revisited”, Studia Etymologica Cracoviensia 2011, vol. 16, no. 1, pages 7-15 was broken. LanguageHat readers can find multiple sites to download this article by googling the author and title. Naturally, you don’t have to believe Fałowski’s own proposal for сорок, but it is good to have a survey in any case. There are some typological parallels to сорок as “(forty) pelts” on page 12.

    My apologies for the problems. I haven’t been able to check and edit any posts lately, because the posts are being dropped by the spam filter.

  40. Somewhere I read that vigesimal counting in Danish and French is of Celtic origin and someplace else I read that some years ago a referendum was held in Denmark, Norway, and Sweden to simplify the numbering system (Swedes and Norwegians approved the changes but the Danes did not).

  41. In one of my previous comments, the link to Adam Fałowski “The East-Slavonic sorok ‘40’ revisited”, Studia Etymologica Cracoviensia 2011, vol. 16, no. 1, pages 7-15 was broken.

    I fixed it.

    I haven’t been able to check and edit any posts lately, because the posts are being dropped by the spam filter.

    Sorry about that; I have no idea why Akismet hates you.

  42. Sorry about that; I have no idea why Akismet hates you.

    No worries! Bu benim kısmetim.

  43. David Marjanović says

    “It’s my fate”?

  44. David Eddyshaw says

    do you know if was there a small denomination that used to circulate in the region in colonial times, 80 of which would have been a reasonable price for a chicken?

    Virtually any colonial coin I can think of, eighty of them would be far too much for a chicken. Cowries might be a possibility though; and the higher numbers were typically used for counting cowries more than anything else (Yoruba actually has a whole set of special “cowrie numbers” for, er, counting cowries.) The exchange rate for cowries in Nigeria when they finally ceased to be actual legal tender (I’m not sure when this was) was a phenomenal 80,000 to the pound, but I think that they were worth a lot more at one stage. (Also, there is a sort of gearing effect: the sort of things you might buy in pounds were expensive foreign items unlikely to find their way into the hands of subsistence farmers in any case.)

  45. Come to think of it, 20, 30. 40 and 50 in Turkic are also pretty irregular (yirmi, otuz, kırk, elli in Turkish).

  46. Cowries might be a possibility though

    That must be it! Thanks. From Robin Law, “Computing Domestic Prices in Precolonial West Africa: A Methodological Exercise from the Slave Coast”, History in Africa, vol. 18 (1991), p. 242:

    Prices in cowries present relatively few problems of interpretation. The system of cowry enumeration on the Slave Coast is both well-documented and well-known from earlier literature, including especially the work of Marion Johnson. Cowries on the Slave Coast were threaded together in units of 40, termed locally a “string,” but also called by Europeans a “tocky” (or “togee,” “toggy,” “toccy,” “tocque,” etc.), five tockies (200 cowries) in turn making a “galina” (from the Portuguese for “hen,” and 20 galinas a “cabess” or “grand cabess” (Portuguese cabeca, “head”) of 4,000 cowries. In the second half of the eighteenth century, a larger unit of 4 grand cabess, or 16,000 cowries, called the “ounce” (or “ounce trade”), was also recognized. In origin this referred to the price in cowries of an ounce of gold, but as the price of gold in West Africa appreciated it had become a conventional unit of account. The system was subject to minor modifications in the nineteenth century, when both the “galina” and the “ounce trade” went out of use, and a smaller “head” of 2,000 cowries replaced the grand cabess of 4,000. By the 1870s the size of the string had also been increased to 50 cowries, giving the “head” of 50 strings a value of 2,500.

    ( Available here on JSTOR here: https://www.jstor.org/stable/3172064 )

  47. From Francis Egbokhare (1995) “Socio-economic dynamics and the development of Emai counting system” in Owólabí, D. K. O. (ed.), Language in Nigeria: essays in honour of Ayọ Bamgboṣe:

    Early reports by Portuguese traders show that the name for a unit of forty cowries is galinhas which is a Portuguese word for “hen’. The Yoruba word for 400 (20×20) is Irínwó or Erínwó. Erin means elephant. It is clear from these that animal names were commonly employed as Mnemonic aids in the counting of cowries

  48. David Eddyshaw says

    It is nice to have a clear example of the nexus of “five” and “hand” (or “fist”) outside Indo-European.

    Nahuatl for “five” is probably etymologically related to “hand”, though the issue seems not to be certain:


    Inuit seems to be an instance:


    I’ve a feeling that Yimas does this too, but I’ll need to look that up when I go home.

  49. Turkish elli “50” (Old Turkic ellig), which juha mentioned, has also been suspected of being related to Turkish el “hand” (Old Turkic elig).

  50. David Eddyshaw says

    I was wrong about Yimas.

  51. I never thought I’d hear those words.

  52. The Tuvan бежен ’50’ is quite straightforward:

    From беш (beş) + он (on).

    20, 30, 50, etc are also formed in this way:

  53. I wonder what languages besides Hindi/Urdu and allies form numerals but subtracting 1, as in unniis ’19’ and biis ’20’.


    Off the bat, only kaheksa/kahdeksan ‘8’ and üheksa/yhdeksän ‘9’ come to mind.


  54. David Eddyshaw says

    Latin: undeviginti “nineteen”, duodeviginti “eighteen” … undetriginta “twenty-nine” …

    Incidentally, Tayap has ndaram nambar (literally “one hand”) for “five.”

  55. David Eddyshaw says

    Welsh has a vigesimal system, as every schoolboy knows, but also has deunaw (“twonine”) for “eighteen.”

    Hanner cant “half a hundred” is a respectable way of saying “fifty” in Welsh, too (though not the only way: you can also say pumdeg or, for the truly hardcore, deg a deugain “ten and two-twenties.”)

    This is quite nice:


  56. I wonder what languages besides Hindi/Urdu and allies form numerals but subtracting 1, as in unniis ’19’ and biis ’20’.

    Ainu comes to mind: tupesan “8”, sinepesan “9”, cf. sine “1”, tu “2”. Probably Harald Hammarstrom has discussed the global distribution of subtractive numerals somewhere.

    It is nice to have a clear example of the nexus of “five” and “hand” (or “fist”) outside Indo-European.

    In Arabic, “shake hands” (xamasa) is transparently related to xamsah “five”; but the direction of derivation is not so clear.

  57. Hanner cant “half a hundred” is a respectable way of saying “fifty” in Welsh
    In colloquial Russian, полсотня (polsótnya) “half-hundred” is also frequently used.

  58. Lameen, what’s the latest on the etymology of the words for ‘five’ and ‘shake hands’?

    Ed.: and Hebrew (presumably < Arabic) ḥamsa ‘a hand-shaped talisman’.

  59. ktschwarz says

    The CLICS database shows 57 languages using exactly the same word for FIVE and HAND, almost all of which are Polynesian, inherited from Proto-Austronesian *lima. (The same word also means ARM in some of them, e.g. Hawaiian, Samoan, Tongan.) However, many Austronesian languages have kept lima for FIVE but developed some other word for HAND.

  60. David Eddyshaw: In Garifuna, whose numerals are mostly borrowed from French, we find /dimi san/ for “fifty”, from “demi-cent”: the similarity to Welsh “Hanner cant” is rather striking. I wish I knew whether demi + cent is a specifically Garifuna creation made up of French borrowed morphemes or a feature of the variety of Colonial French in the West Indies which Garifuna must have borrowed its gallicisms from…

  61. David L. Gold says

    ” Ed.: and Hebrew (presumably < Arabic) ḥamsa ‘a hand-shaped talisman’".

    Hebrew < Arabic is right for the Hebrew word meaning 'hand-shaped talisman’ (no need for "presumably"), as shown by the fact that:

    1. the Hebrew word so meaning, חמסה , has, as its third consonant a samech, not, as the Hebrew for 'five' has, a shin.

    2. its irregular morphophonological structure: no Hebrew feminine noun derived from a Hebrew number word comes to mind that has the vowels that the Hebrew name of the talisman does (by contrast, the Hebrew noun has the same vowels as its Arabic etymon).

    3. its penultimate stress (identical to the stress of its Arabic etymon) would be unexplainable if we tried to derive it from the Hebrew for 'five'.

    If this has not already been mentioned, Proto-Semitic *ḫamš- ‘five’ and Proto-Semitic *yad- ‘idem’ are unrelated.

    “In Arabic, “shake hands” (xamasa) is transparently related to xamsah “five”; but the direction of derivation is not so clear.”

    Since people around the world were taking about five of this and five of that long before hand-shaking,’five’ < shake hands' is the relationship for those two Arabic words.

    The Arabic number word goes back all the way to Proto-Semitic. How far back can the verb be taken?

  62. David Eddyshaw says

    Since people around the world were taking about five of this and five of that long before hand-shaking

    I don’t think we know that: in fact, I’m not sure how we could know. Handshakes leave no archaeological record …

    In Niger-Congo, at any rate, “five” is not reconstructable.
    Handshaking is currently universal in West Africa, and undoubtedly antedates European arrival*, though admittedly, that by no means proves great antiquity. Still, there seems to be just as much evidence for that as for the antiquity of “five.”

    * One naturally wonders about Islamic influence, but the standard non-Islamic West African handshake differs from the way it’s done by Muslims, so I don’t think so.

  63. David Eddyshaw says


    Never really thought about the History of Handshaking. I expect there are theses about it out there somewhere.

    On the other hand, counting is probably a comparatively recent invention; and those of us who grew up with Indo-European or Semitic mother tongues tend not to appreciate that in a broader context number words are typically neither particularly resistant to borrowing nor reconstructable to especially deep levels. They get traded along with the goods they are used to count, and peoples make up their own systems ad hoc when the need arises, too.

    (In fact, Indo-European and Semitic are sufficiently atypical that I hereby create a new hypothesis: they reflect a culture of Number Worshippers, who used their superior accountancy skills to conquer their neighbours. It explains everything.)

  64. By “presumably” I meant that I didn’t know the immediate Arabic precedent. WP now clarifies it.

  65. Is there a mathematical reason why a appendage for gripping (say tree branches) must have a shape that makes it very convenient to hold a symmetrical appendage?

    “Handshakes leave no archaeological record …”
    Sounds aphoristic.

  66. David Eddyshaw says

    Aha! an evolutionary biology argument! Human beings have evolved to shake hands. That explains even more than my Theory of Weaponised Accountancy …

  67. very convenient to hold a symmetrical appendage?

    What do you mean by symmetrical? It’s the same-named hand that is being usually shaked, no? And of course there is a reason, it’s called “chirality”.

  68. David Eddyshaw says

    I think that that is actually drasvi’s point: successful shaking implies that the Other’s appendage be compatible shakingwise with the Self’s same-chirality appendage, which implies some degree of symmetry in the appendage in question (as we do indeed observe.)

    [This reminds me of the ancient conundrum: Why do mirrors reverse left and right, but not up and down? There was once an entire learned correspondence about this in Scientific American.]

  69. When I was five, I occasionally accompanied my mother to her organic chemistry classes. I don’t remember much, except that for one lecture, the topic was chirality, and the professor demonstrated with a student volunteer that you can’t shake a left hand with a right.

  70. I meant two people facing each other and their hands symmetric wiht respect to rotation.

    Letters Я and R have different chirality.
    Letters я and ʁ have the same chirality (one can obtained from the other by rotation).
    R and R too.

    It does not mean that you can combine two R’s in such a way, that they look as if they once where two parts of a whole that someone cut in halves. Hands do that. Among random 3d blots how many do that, at least approximately, when they are soft and flexible?

  71. I am going to contradict myself. L and R are not all that crucial for a handshake. After all, romantic walks with people holding hands are not very painful. In other words, if people are facing each other it is RR or LL, but if they are facing the same direction, it is LR, but involves a little bit of arm twist.Hands are really little marvels, aren’t they.

    And, of cousre, you can connect я and ʁ, but then being planar, they don’t really have chirality.

  72. January First-of-May says

    In Arabic, “shake hands” (xamasa) is transparently related to xamsah “five”; but the direction of derivation is not so clear.

    I’d personally have just guessed that handshakes were the Arabic version of “giving a five”. (Дай пять!)
    I wonder if there are West African antecedents of that gesture/phrase…

  73. Stu Clayton says

    Human beings have evolved to shake hands.

    But also to resist attempts to shake their convictions. Resistance is especially strong when the chiralities are different, for example when left-wing and right-wing convictions meet.

  74. David Eddyshaw says

    True. I myself am firmly of the opinion that only left is right, and right is wrong.

  75. And I am he as you are he as you are me and we are all together.

  76. (“I was the Walrus – Paul wasn’t the Walrus! I was just saying that to be nice, but I was actually the Walrus!”)

  77. Jen in Edinburgh says

    Why do mirrors reverse left and right, but not up and down?

    Surely what a mirror – in front of you – actually does is reverses front and back. Your right hand and your reflection’s right hand are still on the same side – I mean, to an observer standing behind you they’re both on the right hand side.

    A mirror above or below you does reverse up and down. Just don’t ask me what it does to left and right.

  78. David Eddyshaw says

    Surely what a mirror – in front of you – actually does is reverses front and back.

    This was effectively the explanation eventually given in Scientific American: in fact, mirrors don’t reverse left and right at all, but because most of us are bilaterally symmetrical (more or less) and used to pivoting about our long axes in everyday life, we mentally imagine ourselves standing in the position of the mirror image, turning ourselves 180 degrees in the process, but this wouldn’t actually achieve the desired effect – hence the illusion of a left-right reverse. If we were, instead, top-bottom symmetrical and accustomed to proceed by rolling along, it would seem to us that mirrors did indeed reverse up and down. (The explanation as actually given involved a thought experiment in which you put a paper bag over one hand before looking in the mirror; even now thinking about this undermines my grip on reality.)

  79. DE, the sinistral wags here say that only the left is right, but only the right is left.

  80. Athel Cornish-Bowden says

    A more interesting concept (to me, but then, I’m a biochemist) is prochirality, which was not well understood until surprisingly late, around 1948, when Sandy Ogston explained how a prochiral molecule, like citrate, can behave as achiral (symmetrical) when in an achiral environment but as chiral when in a chiral environment, like the surface of an enzyme. Thus the enzyme aconitase can distinguish two supposedly identical groups on citrate and produce a chiral product. Nowadays I suspect that many biochemists still don’t understand that, but before 1948 hardly anyone did: quite distinguished scientists claimed that despite ample experimental evidence citrate could not be an intermediate in the tricarboxylate (Krebs) cycle.

    An ordinary coffee mug with one handle is an everyday example of a prochiral object. If you immerse it in a corrosive liquid the left- and right-hand sides will be attacked equally. If you hold it in your right hand you’ll find it easy to drink out of the left side, and very difficult to drink out of the right side.

    Sandy Ogston was, incidentally, the admissions tutor at Cambridge who convinced Richard Dawkins that he was better suited to zoology than to biochemistry. The rest is history — where would the world be without The Selfish Gene, The God Delusion, Climbing Mount Improbable, and many others?

  81. David Eddyshaw says

    can behave as achiral (symmetrical) when in an achiral environment but as chiral when in a chiral environment

    The Liberal Democrats of the biochemistry world!

  82. Stu Clayton says

    where would the world be without The Selfish Gene, The God Delusion, Climbing Mount Improbable, and many others?

    At least we thus were spared The Selfish Enzyme.

  83. DE, I do not know if it is related to our symmetry or our sense of “top”.

    Take a clock from the wall in your hands and look in a mirror.

    Where do you expect to find 12? On top.
    Where do you expect to find 9? On the left.

    Where do you find these digits in the mirror image? Depends on how you hold the clock.

    – If you have turned it around the vertical axis (preserving the top) then:
    – – the symbol 9 of your (actual) clock is against the right side of your (actual) chest, while 12 is under your chin.
    – – the symblo 9 of your mirror clock is to the right too.

    – If you have turned it around the horizontal axis (thus preserving its left and right)
    – – 6 is under your chin, 9 is to your left, both in the mirror and in the actual clock.

    – If you did not turn the clock at all, and it is still facing you, then all the digits are at their rightfull places (maybe the clock is transparent).

  84. The clock, because this object does not have bilateral symmetry of its own.
    My point is: what affects the outcome is how you turned it. Most likely 12 is up, this is how people usually carry objects. I tend to blame our sense of “up” rather than our bilateral symmetry.

    P.S. if you want to undermine your gripof reality, imagine yourself doing a somersault (the view from inside your head:))

  85. @Athel Cornish-Bowden: When I took biochemistry (the version of the class taught by the biologists, rather than the chemists), we talked about the puzzle of how the citric acid cycle managed to avert racemization. Once the problem had been posed, the solution (that the citrate is never unbound from the chiral enzyme complex) occurred to me immediately. However, the lecturer seemed to regard the solution as practically black magic. Moreover, while we were supposed to learn every example in which a reaction we studied involved a covalent bond between enzyme and substrate, that piece of information about the Krebs cycle may have been the only time we talked about any other aspect of how enzymes held smaller chemicals in place; it was certainly the only time the three-dimensional structure of enzymatic binding was discussed.

    Maybe I just got lucky to see a solution intuitively. However, I think I was probably assisted by not thinking the way chemists often do, of reactions as discrete events with free, undisturbed molecules in between the steps of a reaction sequence. (What I know was not responsible for my seeing the solution was a strong feel for three-dimensional geometry, which I do not possess.)

  86. DE/Lameen, thank you very much!
    Still, it seems that it’s only Hindustani and allies that have irregular numerals from 1 to 99.

  87. David Eddyshaw says

    Well, they did invent the Arabic numeral system.


    They’re entitled to their fun after that.

  88. Jen in Edinburgh says

    I did wonder if it had to do with the fact that we have our own personal right and left but a shared up and down (except maybe for people in Australia, but they’re not likely to be looking in the same mirror as me…)

    Where I got really mindboggled was trying to work out if a mirror to the right or left of you would genuinely reverse right and left – I feel like it must, by analogy, but it was too difficult to work out where the viewpoint would be.

    It was thinking about that that made me realise that the usual idea of the mirror-in-front reversal was based on two different viewpoints at 180 degrees to each other, though. Would we think mirrors did something different if our eyes were in the side of our heads?

  89. John Cowan says

    irregular numerals from 1 to 99

    They’re not irregular in the sense of suppletion or even unlevelled sound change or the opposite: they are simply Sandhi Run Wild! (Whether you call it internal or external is up to you.)

  90. January First-of-May says

    Maybe a good way to think of Hindustani numerals is that they’re basically like French 11-16 except they don’t stop at 16.

    (Why does this sort of thing stop at 16 in French, anyway? How come it’s 16 specifically and not beyond? That if anything seems weirder than going with it all the way.
    For what it’s worth, my naive guess is that after the relevant sound changes 17 turned out to be homophonous with 16, which impeded further progress.)

  91. I got really mindboggled was trying to work out if a mirror to the right or left of you would genuinely reverse right and left

    I am not 100% sure what you mean by genuinely, but any single mirror does LR, that’s the idea of chirality.

  92. Jen in Edinburgh says

    Why does this sort of thing stop at 16 in French, anyway? How come it’s 16 specifically and not beyond?

    15 in Spanish…

  93. The teens are frequently weird. In French and Italian (and, according to the Web, Romansh), they follow a pattern through 16, then settle into the regular system for two-digit numbers. Spanish and Portuguese do the same, except make the switch one place earlier, after 15. The exception among the Romance languages is apparently Romanian, which is regular all through the teens. In Germanic languages (all of them, I think) 11 and 12 have an opaque pattern to them, then the remainder of the teens are all formed in the same fashion—but not by the same rules the languages use to form numbers above 20. I imagine that some other languages may have other variations on this behavior pattern.

    As to why the changes come after 12, 15, and 16, than again at 20: I suspect it just contingency and that those are all relatively “round” numbers. Certainly, 12 and 16, with their many factors, occur more commonly in practice than their prime successors 13 and 17. (How much more commonly is the relevant linguistic contexts is, of course, harder to say.) Fifteen is also fairly round, being three hands’ worth (and the only odd number in the teens that is not prime).

  94. Jen in Edinburgh says

    I am not 100% sure what you mean by genuinely

    A mirror in front of you (or behind you, but it’s harder to look into!) will reverse front and back. Your right hand’s reflection is still on your right hand side, and on the right hand side of an observer standing behind and parallel to you.
    To reverse left and right, you have to imagine yourself rotated by 180 degrees to stand in your reflection’s place (and doing that rotation would always swap left and right).

    A mirror above (or below) you will reverse top and bottom. Your right hand’s reflection is still on your right hand side, and on the right hand side of the observer crouched at your feet.
    To reverse left and right, you have to imagine yourself rotated by 180 degrees (vertically, this time!) into your reflection’s place.

    By analogy with the above, as I said, a mirror directly to the side of you *should* swap left and right.

    I think it does in a way – assuming the mirror is on your right, your right hand’s reflection is now closer to your left hand (and to the left hand of an observer standing parallel to you) than your left hand’s reflection is.
    To take your reflection’s place, you no longer have to rotate 180 degrees, just to step sideways – and then you would find the reflection of your right hand where your left hand would usually be, and the reflection of your left hand where your right hand would usually be.

    That’s my ‘genuine’ – a L/R reversal which doesn’t rely on a 180 rotation of the viewpoint to create it.

    However, it does rely on defining left and right by where your hands are, not by where your eyes are looking, which I think is why my head starts to go round and round if I think about it too much…

  95. Thanks, Jen, I’ve got it now. Your right and left hands are “defined” not by their location, but by their intrinsic property, how the direction into your open palm, stretched fingers, and angled away thumb are related. I find the best viewing position not from inside my head, but from an out of body ghost.

  96. Lars Mathiesen says

    It’s the sign of your trivector innit.

    It’s a physical fact that you can always tell a left hand from a right hand. Also we have chosen to formulate the mathematics of Euclidean space to reflect that, so that (an ordered list of) three vectors will define a ‘negative’ or ‘positive’ chirality — it’s a convention which is which, but if you hold the index finger of your right hand straight and bend the middle finger, then vectors along thumb, index and middle in that order have positive chirality. The cross product of two vectors is defined so that it has positive chirality relative to the inputs, and that is why you will see fledgling physicists perform that gesture when faced with induction loops and such where the equations are full of cross products.

    (I’m not sure if Lorentzian fourspace necessarily is the limit of quantized spacetime models, because I haven’t found a treatment that I could understand while not glossing over such things. Maybe I’m asking for the impossible).

  97. Right-hand rule. (In Russian it’s the правило буравчика ‘rule of the auger/gimlet.’)

  98. Lars Mathiesen says

    I learned the Russian version, it seems — the English article goes index, middle, thumb, but of course that defines the same convention. (An even permutation of vectors preserves chirality. How could it not!)

  99. Athel Cornish-Bowden says

    In recent years I nearly always notice if someone — in real life or on television — is left-handed, but there has been one striking exception, the woman who cuts my hair: it took about five years before I noticed that she was left-handed. Why the exception? Probably because I mostly see her in the mirror, and her image in the mirror is right-handed.

  100. I have taught every electricity and magnetism class that my department offers, and when it comes time to start working with cross products for magnetic calculations, I always give the students one particular piece of advice. When they start trying to find the direction of a magnetic field or Lorentz force, they should stick their left hands firmly in their pockets. As the calculations become more complicated, requiring the invocation of the right-hand rule more than once, there is an incredibly common tendency to bring in the left hand—which, of course, gives wrong answers!

  101. правило буравчика ‘rule of the auger/gimlet.’

    As it happens, I completely internalized that rule. I find sticking my hand and fingers in various directions awkward and uncomfortable, but making a clockwise rotation while moving my hand forward (or counterclockwise/backward) is just fine.

  102. Lars Mathiesen says

    @Brett, that is also excellent advice if you go inside a substation. Both hands, preferrably.

  103. Possibly another striking example of a “concrete etymology” for a numeral, as discussed above. Balinese (as far as I can gather)…

    39 telung dasa sia
    40 petung dasa
    41 petang dasa besik
    42 petang dasa dua
    43 petang dasa telu
    44 petang dasa empat
    45 setiman
    46 setiman besik
    47 setiman dadua

    Why does it reset at 45 with setiman? I found this explanation out there:

    Opium was imported from China wrapped in small packets covered with metal foil, probably lead foil. Each packet sold for 45 pis bolong (or possibly 45 Dutch sen, it is not clear which). The Balinese word for metal is timan. So one package of metal was call se- (one) timahan (package of metal)—abbreviated setiman. And so the common word for 45 became the word for the package of opium wrapped in metal foil that cost 45 pis bolong or sen.

    Fred B. Eiseman, ‎et al. (1989) Bali: Sekala & Niskala II, p. 165f.

  104. Wonderful!

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