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TYPOLOGYOFNUMERALSYSTEMS

BernardComrieUniversityofCaliforniaSantaBarbara

<comrie@linguistics.ucsb.edu>Versionof2021May26Wherenosourceisgiven,Ihaveusedmyownknowledgeofthelanguageinquestion,supplementedwherenecessarybychecking(e.g.spelling)inastandarddictionary.A.Restrictedsystems,withlittleornointernalstructure(1) Nonumerals [Pirahã;Everett2005:623-627](2) 1 (ŋa)wumbawa 2 ŋabaranwa 3 ŋabaḷawa [Mangarrayi;Merlan1982:92](3) 1 guman 2 jambul 3 dagul 4 yunggan.gunyjiiormugungabi 5 mala ‘faceofhand’ [Yidiny;Dixon1991:224] Note:Theformsfor4andabovemaynotbefullyconventionalized.(4) 1 püg(ma) 2 xepxep 3 ebapüg 2+1 4 ebadipdip 2+1+1 5 pügpögbi ‘onehand’ [Mundurukú;Picaetal.2004;seealsosectionB](5) 1 towenyxa 2 asako 3 osorwawo 4 towtɨnkerye ‘itsbrothertwiceover’ 5 kamorɨɨrakay(o)me ‘halfofourhands’

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10 kamothɨrɨtkatxehkaxero ‘ourhandscompletely’ [Hixkaryána;Derbyshire1979:154-155] Note:Theformsfor4andaboveare“non-basic”and“arealsousedwithoutprecision”.(6) subitize Subitizingistherapid,accurate,andconfidentjudgmentofnumbers performedforsmallnumbersofitems withacut-offpointaround4 B.Simplesystemswithadditiononly(7) 1 paŋ 2 mös 3 möspaŋ 2+1 4 mösmös 2+2 [Haruai;Comrie:Ownfieldwork](8) 1 añɨ(nöbö) 2 möhöp 3 möhaunögaŋ 4 möhaumöhau [Kobon;Davies1981:208](9) 1 bits 2 iruʔ 3 iruʔdabits 2+1 4 iruʔdairuʔ 2+2 5 iruʔdairuʔdabits 2+2+1 6 iruʔdairuʔdairuʔ 2+2+2 etc. 10 dzibangimarafaindadzibangimarafainsib ‘myhandhalfandmyhandhalfcompleted’ [Adzera;Lean&Owens2018:299;partiallycheckedagainstSmith(1988:66),

Holzknecht(1989:127–128).]C.Morecomplexsystemsusingmultiplicationandadditionappliedtoabase(10) Generalpattern: Forbaseb:(nxb)+m wherem<b andusuallyn<b,orn≤b

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Decimal(base10)(11) wǔ-shí sì five-ten four 54[50+4] [MandarinChinese]Vigesimal(base20)(12) kəlgən-qlekken məngətken ŋireq parol fifteen-twenty ten two left 312[(15x20)+(10+2)] [Chukchi;Skorik1961:390]Base60(13) èna ma gàati dàimita mutò one and ten and sixty 71[60+(10+1)] (14) muto wìi sixty four 240[4x60] [Ekari;Drabbe1952:30]Base32(15) yfɔ wǎdhy four thirtytwo 128[4x32] [Ngiti;KutschLojenga1994:355-358]Base12(16) ba-kuru ba-ba ná CL-ā CL-bā PL-twelve PL-two plus CL-this CL-two 26[(2x12)+2] [Berom;Bouquiaux1970:259]

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Note:Thelasttwoelements-āand-bārequireaclassprefixcorrespondingtotheclassoftheheadnoun.

Base8(17) karnuʔ tenhiuɲ rnuʔ three eight three 27[(3x8)+3] [NorthernPame;Avelino2006:45]Base6(18) 1 näbi 2 eda/yda 3 etha/ytho 4 asar 5 tabuthui 6 nibo 36 fta 6² 216 taruba 6³ 1296 damno 6⁴ 7776 wärämäkä 6⁵ 46656 wi 6⁶ 72 edafta 2x6² 73 edaftaanäbi (2x6²)+1 50 näbiftaaedaniboaeda (1x6²)+(2x6)+2 [Komnzo;Döhling218:93-94;Yam-countingvideo:

https://vimeo.com/54887315]NewGuineaHighlandbody-partcountingsystems(“bases”18–74)(19) wañɨgnöbö littlefinger 1 23 24 46 igwo ringfinger 2 22 25 45 igwoaŋnöbö middlefinger 3 21 26 44 igwomɨlö indexfinger 4 20 27 43 mamɨd thumb 5 19 28 42 kagoƚ wrist 6 18 29 41 mudun forearm 7 17 30 40 raleb insideofelbow 8 16 31 39 ajɨp biceps 9 15 32 38 siduŋ shoulder 10 14 33 37 agɨp collarbone 11 13 34 36 mögan holeabovebreastbone 12 35

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[Kobon;Davies1981:206-208]

Notes:a)Thesecondhalfofthebodyoneachpassacrossthebodyisdistinguishedbyböŋ‘onesideof’.b)Thesecondpassacrossthebodyisdistinguishedbyñinjuöladogda‘pullingoutthehand,giveback!’,subsequentpassesbymodifyingñinjuölwithanumeral.c)Thesystemis“symmetrical”,i.e.thebodypartsusedononesideofthebody(inKobon,for1–11)areusedinreverseorderfortheothersideofthebody(Kobon13–23).Arithmeticproperties(withoutanyclaimstopsychologicalreality):a)Onthefirstpassacrossthebody,anygivennumbertermnalsodenotes24–n,andsimilarly,mutatismutandis,forsubsequentpassesacrossthebody.b)Thesystemhasabase23.

(20) agɨlɨŋ littlefinger 1 19 agɨlɨŋrolyöbö ringfinger 2 20 wölömɨl middlefinger 3 21 köñönɨgɨb indexfinger 4 22 mömɨd thumb 5 23 urapcɨgɨb wrist 6 18 24 36 mɨj forearm 7 17 25 35 amɨñaxɨb insideofelbow 8 16 26 34 mac biceps 9 15 27 33 möib shoulder 10 14 28 32 katlöi collarbone 11 13 29 31 mɨgan holeabovebreastbone 12 30 [Haruai;Comrie:Ownfieldwork] [Somenumerals,likethecorrespondingbodyparts,havealternativeforms;urap

cɨgɨbandmacareproperlybraceletswornatthegivenbodypart.]

Notes:a)Thesecondhalfofthebodyoneachpassacrossthebodyisdistinguishedbyadökwebö‘ofthatside’.b)Thesecondpassacrossthebodyisdistinguishedbyhöuöilɨb‘turning’orhöbkalɨb‘returning’.c)Theendofthefirstpassacrossthebody(i.e.23)canbeidentifiedbysayingpadökwopaŋyöbödua‘housepostthereonewent’,subsequentpassesbyreplacingpaŋyöbö‘one’withahighernumeralandusingthepluralverbformdumä.d)Thesystemis“asymmetrical”,i.e.thebodypartsusedononesideofthebody(inHaruai,for1–11)arenotallusedinreverseorderfortheothersideofthebody(cf.inparticularthefingers).

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(20') agɨlɨŋ littlefinger 1 19 37 agɨlɨŋrolyöbö ringfinger 2 20 38 wölömɨl middlefinger 3 21 39 köñönɨgɨb indexfinger 4 22 40 mömɨd thumb 5 23 41 urapcɨgɨb wrist 6 18 24 36 mɨj forearm 7 17 25 35 amɨñaxɨb insideofelbow 8 16 26 34 mac biceps 9 15 27 33 möib shoulder 10 14 28 32 katlöi collarbone 11 13 29 31 mɨgan holeabovebreastbone 12 30

Arithmeticproperties:a)Thereisnoconsistentrelationbetweenthenumericalvaluesofthesamebodypartondifferentsidesofthebody.For6–18,thetwoaddupto24;for1–5and19–23,thetotalvariesfrom20to28.b)Thebaseshiftsfrom23onthefirstpassacrossthebodyto18onsubsequentpasses,i.e.passesendat23,41,59,etc.Seealso(onOksapmin):Saxe,GeoffreyB.,withIndigoEsmonde,Culturaldevelopmentofmathematicalideas:PapuaNewGuineastudies,NewYork,2012:CambridgeUniversityPress.

(21) Hypothesis:Arithmeticbasesofnumeralsystemshaveeitherasomaticora

commercial(transactional)origin;lowerbasesaretypicallysomatic,higherbasescommercial,butNewGuineaHighlandbody-partcountingsystemshaverelativelyhighsomatic-originbases.

10 fingers 20 fingersandtoes;eachfingertwice(twophalanges/knuckles) 8 spacesbetweenfingers(attestedforsomeCalifornialanguages) 12 phalangesorknucklesoffingers(excludingthumbs) Forhigherbaseswithacommercialorigin,cf.Englishscore‘20’,whichinsome

varietieshasmadeitintothenumeralsystem.(22) Ifthelowestmultiplicativebaseishigherthanabout12,andthesystemisnotan

extendedbody-partcountingsystem,thenthehighernumbersbelowthelowestmultiplebasemakeuseofanadditivebase,e.g.Georgian(lowestmultiplicativebase:20);somelanguagesusethisevenforasmallerlowestmultiplicativebase,e.g.Nauete(lowestmultiplicativebase:10);seealsoChukchiexample(12)above.

(23) 1 ert-i -inominativecase(NOM) 2 or-i

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5 xut-i 6 ekvs-i 9 cxra NOMdropsaftera 10 at-i 11 t-ert-met’-i Ø-one-teen-NOM 15 t-xut-met’-i 16 t-ekvs-met’-i 19 [t-]cxra-met’-i (t+c>c) 20 oc-i 21 oc-da-ert-i twenty-and-one-NOM 30 oc-da-at-i twenty-and-ten-NOM 31 oc-da-t-ert-met’-i twenty-and-Ø-three-teen-NOM 40 or-m-oc-i two-Ø-twenty-NOM [Georgian;Hewitt1995:51-52](24) 2 kai-rua CL-two 3 kai-tolu CL-three 5 kai-lima CL-five 7 kai-limaresikai-rua CL-fiveandCL-two 8 kai-limaresikai-tolu CL-fiveandCL-three 10 weli-see ten-one 12 weli-seeresikai-rua ten-oneandCL-two 17 weli-seeresikai-limaresikai-rua ten-oneandCL-fiveandCL-two 20 weli-kai-rua ten-CL-two [Nauete;https://mpi-lingweb.shh.mpg.de/numeral/Naute.htm,withdata

providedbyGeoffreyHullin1996;seealsoSchapper&Hammarström2013:429,andhttps://abvd.shh.mpg.de/austronesian/language.php?id=1365,thelatterwithdataprovidedbyAlexandreVelosoin2016.Therearedifferencesofdetailamongthesources,notaffectingthemainpoint.]

Note:Nauetenumerals2-5requireanumeralclassifierprefix,withkai-

beingthedefaultusedincounting.(25) Distributionofdifferentbasesacrossasampleoflanguagesoftheworld [See:Comrie(2013)]D.IdiosyncrasiesrelatingtobasesPortmanteauforms(26) sorok forty

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40[expected4x10] [Russian](27) eleven 11[expected10+1] [English](28) 25 se-lae ‘onethread(ofChinesecoins)’ 45 se-timan ‘oneopiumpacket(costing45Chinesecoins)’ 50 se-ket ‘onetie(i.e.twothreadsof25Chinesecoins)’ 75 telungbenang ‘threethreads(ofChinesecoins)’ 200 s-atak ‘onebundleof200Chinesecoins’ 400 s-aman ‘onegold(coinworth400Chinesecoins)’ 900 sanga [etymologyunclear] [Balinese;Eiseman1990:162-168](29) Comparelessspectacularirregularities fif-teen(*five-teen) five-ten 15 twenty;twelve [English](30) InHindi,arguablyallthenumerals1–100areirregular 0 1 2 3 4 5 6 7 8 9- ek do tīn cār pāṁc chah sāt āṭh nau10 das gyārah bārah terah caudah pandrah solah satrah aṭhārah unnīs20 bīs ikkīs bāīs teīs caubīs paccīs chabbīs sattāīs aṭṭāīs untīs30 tīs ikattīs battīs taiṁtīs cauṁtīs paiṁtīs chattīs saiṁtīs aṛtīs untālīs40 cālīs iktālīs bayālīs taiṁtālīs cavālīs paiṁtālīs chiyālīs saiṁtālīs aṛtālīs uncās50 pacās ikyāvan bāvan tirpan cauvan pacpan chappan sattāvan aṭṭhāvan unsaṭh60 sāṭh iksaṭh bāsaṭh tirsaṭh cauṁsaṭh paiṁsaṭh chiyāsaṭh sarsaṭh aṛsaṭh unhattar70 sattar ikhattar bahattar tihattar cauhattar pachattar chihattar sathattar aṭhhattar unyāsī80 assī ikyāsī bayāsī tirāsī caurāsī pacāsī chiyāsī sattāsi aṭṭhāsī navāsī90 nave ikyānve bānve tirānve caurānve pacānve chiyānve sattānve aṭṭhānve ninyānve

[Hindi;McGregor1972:61-62;notethatsomenumeralshaveminorvariants]Isolated“bases”(31) quatre-vingt-douze

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four-twenty-twelve 92[(4x20)+12] [French](32) deu-naw two-nine 18[2x9] [Welsh;King1993:113]Overrunning(33) disąt-nocti 10-teen 20[10+10] Note:21isjanüdisąt-nocti,i.e.[1+20],not[11-teen] [Polabian;Polański&Sehnert1972:52,73](34) soixante-dix sixty-ten 70[60+10] (35) soixante-douze sixty-twelve 72[60+12] (36) soixante-dix-sept sixty-ten-seven 77[60+(10+7),ratherthan[(60+10)+7] [French]E.Exponentiationandotherhigherbases(37) 10¹ 10² 10³ 10⁶ ten hundred thousand million [English]Absenceofexponentiation(38) qliq-qlikkin twenty-twenty

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400(20x20)–highestnumeralintraditionalsystem [Chukchi;Skorik1961:388,391]Effectivelymonomorphemicseriesofbases(39) 10 dáśa 100 śatá 1,000 sahásra 10,000 ayúta 100,000 lakṣá 1,000,000 prayúta 10⁷ kóṭi 10⁸ arbudá 10⁹ mahārbuda 10¹⁰ kharvá 10¹¹ nikharva [Sanskrit;Whitney1889:177-178,whonotesfurther“Theseriesofdecimalnumbersmaybecarriedstillfurther;buttherearegreatdifferencesamongthedifferentauthoritieswithregardtotheirnames;andthereismoreorlessofdiscordanceevenfromayútaon.”] Note:IndianEnglishcurrentlyuses:lakh10⁵;crore10⁷;inIndia,133,435,360wouldbewrittenas13,34,35,360(13crore,34lakh,35thousand,360).(40) wàn 万 萬 10⁴

yì 亿 億 10⁸

zhào 兆 10¹²

jīng 京 10¹⁶ etc.

[MandarinChinese.ThissystemisusedinChinese,Japanese,Korean;butseealso(92)](41) cem-pōhual-li one-twenty-ABS 20 cen-tzon-tli one-four.hundred-ABS 400(20²) cen-xiquipil-li one-eight.thousand-ABS 8000(20³) [ClassicalNahuatl;Andrews1975:397-398,464,482,484] Note:pōhua-isalsoaverb‘tocount’;tzon-alsomeans‘hair’;xiquipil-alsomeans

‘bag,sack’.(Semi-)productivesystems

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(42) longscale shortscale million first 10⁶ 10⁶ billion second 10¹² 10⁹ trillion third 10¹⁸ 10¹² quadrillion fourth 10²⁴ 10¹⁵ [generalpattern] nth 10⁶ⁿ 10³⁽ⁿ⁺¹⁾ [English] Seehttp://www.isthe.com/chongo/tech/math/number/howhigh.htmlfora

proposalonhowtocountindefinitelyhighusing(pseudo-)Latinprefixes.(43) Contrastinnovative/metric kilo- 10³ mega- 10⁶ giga- 10⁹ tera- 10¹² peta- 10¹⁵ exa- 10¹⁸ zetta- 10²¹ yotta- 10²⁴ [English/International]Sequenceofbasesthatarenot(all)powersofasinglebase(44) cxr-as otxm-oc-da-cxra-met’i nine-hundred four-twenty-and-nine-teen 999[(9x100)+(4x20)+(10+9)] Bases:(10,)20,100 [Georgian;Hewitt1995:54](45) kàmpwòò ná ŋkwuu sicyɛɛré 'ná béé-tàànrè ná kɛ 'ná báár-ìcyɛɛrè fourhundred and eighty four and twenty-three and ten and five-four 799[i.e.400+(4x80)+(3x20)+{10+(5+4)}]’ Bases:(5,10,)20,80,400 [SupyireSenoufo;Carlson1994:169](46) Notethatthenexthigherbase(nhb)isnearlyalwaysequaltoorlessthanthe

squareofthecurrentbase(cb): nhb≤cb²

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ButnotetheolderGermanic“longhundred”:(47) 10 tíu 30 þrírtigir 100 tíutigir 110 ellifutigir 120 hundrað (“longhundred”) 240 tvauhundrað 1200 þúsund (“longthousand”) [OldNorse;Gordon1957:292-293](48) Ahighermultiplicativebaseisnearlyalwaysaproductofthelowestmultiplicative

base.AnexceptionisprovidedbyCoahuiltecan,withmultiplicativebases3(fornumerals12-19)and20;surprisingly,thelowerbase3isitselfcomplex(2+1).

(49) 12 4x3 13 4x3+1 14 4x3+2 15 5x3 18 6x3(?) 20 20 30 20+10 40 2x20(50) puwāntz’an axti-k-pil’ ko pil’ four two-and-one and one 13[i.e.{4x(2+1)}+1] [Coahuiltecan;Swanton1940:48]Alternatingbases(51) 10 desat 10 20 dwísti 2x10 30 trəsti 3x10 40 dwákratdwísti 2x20 50 patardú 5x10 60 trıkratdwísti 3x20 [ResiaSlovenian;Steenwijk1992:125]F.OtherarithmeticprocessesSubtraction

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(52) un-de-viginti one-from-twenty 19[20–1] [Latin]Division(actually:multiplicationbyfraction)(53) hanner cant half hundred 50[½x100] [Welsh;King1993:113]Subtractionandaddition(54) éks bənsaŋ kiʔ twenty without hundred 80[100–20] (55) ynam əkam éks bənsaŋ kiʔ two left.over twenty without hundred 82[(100–20)+2;NB:not100–(20+2)] [Ket;Georg2007:179-181](56) Successiveapproximation,cf.timeexpressionsinsomelanguages drie (uur) three hour 03:00 (57) half drie half three 02:30 (58) vijf over half drie five after half three 02:35 [Dutch]Overcounting(59) halv-tred-sinds-tyve

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half-third-times-twenty 50[halfofthethird,timestwenty] [Danish;Allanetal.1995:127;nowusually:halvtreds](60) paüne tini šata less.quarter three hundred 275[lessaquarterofthethirdhundred] [Oriya;Karpuškin1964:38]Pairing(61) 1 seénu,wepul 2 woói 3 báhi 4 naíki 5 mámni 6 búsani 7 wóo-búsani two-six(i.e.‘secondsix’?) 8 wóh-naíki two-four(i.e.2x4) ← 9 bátani 10 wóh-mámni two-five(i.e.2x5) ← [Yaqui;Dedrick&Casad1999:229](62) 1 hito 2 huta 3 mi 6 mu 4 yo 8 ya [Japanese]Non-arithmeticstructures(63) 10⁸ arbudá- 10⁹ mahārbuda- (maha-‘big’) [Sanskrit;Whitney1889:177](64) 10³ mille(PLmila) 10⁶ milione (-oneAUGMENTATIVE) [Italian] G.Orderingofconstituents

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Fromlargertosmaller(65) sān-bǎi wǔ-shí sì three-hundred five-ten four 354[i.e.300+50+4] [MandarinChinese]Fromsmallertolarger(66) efatra amby dima-mpolo sy telo-njato four plus five-ten and three-hundred 354[i.e.4+50+300] [Malagasy(Standard);Rajaonarimanana2001:67]Fromsmallertolargerforsmallercombinations,fromlargertosmallerforlargercombinations(67) drei-hundert-vier-und-fünf-zig three-hundred-four-and-five-ten 354[i.e.300+4+50] (68) zwei-hundert-sechs-und-fünf-zig-tausend-drei-hundert-vier-und-sieb-zig two-hundred-six-and-five-ten-thousand-three-hundred-four-and-seven-ty 256374[i.e.(200+6+50)x1000+(300+4+70)] [German](69) ordersmaller-larger(contrastingwithhigherlarger-smaller) through12(ðó-ðeka,cf.13ðeka-trís) ModernGreek through15(quin-ce,cf.16diec-i-séis) Spanish through16(se-dici,cf.17dici-as-sette) Italian through19(nine-teen) English through99(neun-und-neun-zig) German [ForModernGreek:Holtonetal.1997:103-104]Fromlargertosmallerforsmallercombinations,fromsmallertolargerforlargercombinations(70) limam-polo roe amby, amby telon-jato five-ten two plus plus three-hundred ‘352[i.e.50+2+300]’ [Malagasy(NosyBe);Dahl1968:14]

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(71) Hypothesis:Theorderfromlargertosmallerispreferredbecauseitgivesearlier

recognitionoftheapproximatequantityinvolved,i.e.in354the300ismoresignificantthanthe4.Localinversionoflowerpositions(e.g.tensandunits)isminimallydisruptive.Soweexpectprevalenceoftheorderfromlargertosmaller,withpossiblelocalinversionofthelowerpositions.

H.Finitenessandgaps(Comrie2020)(72) “Everylanguagehasanumeralsystemoffinitescope.” (Generalization1,Greenberg1978)(73) Shortscale Longscale withoutintermediaries withintermediaries 10⁶ million million million 10⁹ billion milliard 10¹² trillion billion billion 10¹⁵ quadrillion billiard 10¹⁸ quintillion trillion trillion(74) a. anewsingle-wordtermisavailableeachtimeonemultipliesthepreviously

highestsingle-wordtermby1,000,000,uptotrillion,i.e.therearenoterms*quadrillionandbeyond;

b. therearenointermediaries,e.g.*milliard,so10⁹canonlybeexpressedasthousandmillion;

c. numeralsfrommillionupcanbecombinedtoexpressmultiplication,e.g.milliontrillion([10⁶x10¹⁸]=10²⁴);

d. thereisastrongpreferencetoimplementthe“PackingStrategy”(Hurford1975:67),i.e.,informallyspeaking,toavoidcombiningtermswithlowernumericalvaluewhenthereisanavailableappropriatetermwithhighernumericalvalue;thus,althoughmillionmillionispossible,billionispreferred;

e. itisnotpossibleforahigher-valuetermtoprecedealower-valueterm,e.g.for10³⁰onecansaybilliontrillionbutnot*trillionbillion.

(75) a. 10⁶ million b. 10⁹ thousandmillion c. 10¹² billion d. 10¹⁵ thousandbillion e. 10¹⁸ trillion f. 10²⁴ milliontrillion g. 10³⁰ billiontrillion h. 10³⁶ trilliontrillion i. 10⁵⁴ trilliontrilliontrillion j. 8x10⁶⁷ eightybilliontrilliontrilliontrillion TraditionalBritishlong-scalesystem(myversion)

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(76) Everynumbern(0<n<L)canbeexpressedaspartofthenumericalsystemin

anylanguage. [Generalization2,Greenberg1978](77) decillion,vigintillion,centillion Whatisthe23rdtermintheseries? tresvigintillion[https://en.wikipedia.org/wiki/Names_of_large_numbers]I.AmbiguityParsingambiguities(78) amillionandahalf(apples) (i)1½million,i.e.1,500,000 (ii)1,000,000½ [English](79) un fil a thri ugain o asynod one thousand and three twenty of asses (i)1060asses(possibleinterpretation) (ii)61,000asses(Numbers31.34–intendedinterpretation)(80) deuddeg a thri ugain mil o eidionau twelve and three twenty thousand of cattle 72,000cattle(Numbers31.33)(81) saith mil a phedwar ugain mil seven thousand and four twenty thousand 87,000(IChronicles7.5) [Welsh(Biblical);Hurford1975:192,184,176]Abbreviation(82) (nyŋ) phan sɔɔŋ one thousand two (i)1002 (ii)1200 (iii)*1020 (83) (nyŋ) phan sɔɔŋ rɔɔy one thousand two hundred 1200

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(84) (nyŋ) phan kàp sɔɔŋ one thousand with two 1002 [Thai;Smyth2002:173](85) (a) sān-bǎi liù-shí three-hundred six-ten (b) sān-bǎi liù three-hundred six 360(86) sān-bǎi líng liù three-hundred zero six 306 Note:Obligatoryuseoflíng,creatingapartialplace-valuesystem,avoids

ambiguity.(87) sān-qiān líng liù three-thousand zero six 3006 (88) sān-qiān líng liù-shí three-thousand zero six-ten 3060 [MandarinChinese] Note:Anolderusagerequiredlíngtoberepeatedasmanytimesastherearezeroes

inthe“Arabic”numeralrepresentation,so(87)wouldbesān-qiānlínglíngliù(Chao1968:575).

Diachronicmerger(89) thirty,thirteen [English](90) 7 sāt 100 sāt>janiɣsāt(janiɣ‘big’) [NorthernMansi;Honti1993:125]Semanticchange

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(91) billion (i)older,“longscale” 10¹² (ii)newer,“shortscale”=USEnglish 10⁹ (longscale:10⁶ⁿ;shortscale:10³⁽ⁿ⁺¹⁾) [BritishEnglish](92) 万/萬 亿/億 兆 京 wàn yì zhào jīng (i) 10⁴ 10⁵ 10⁶ 10⁷ (ii) 10⁴ 10⁸ 10¹² 10¹⁶ (iii) 10⁴ 10⁸ 10¹⁶ 10²⁴ (iv) 10⁴ 10⁸ 10¹⁶ 10³² [EastAsian] [Row(ii)istheoneincurrentuseforChinese,Japanese,andKorean,thoughwith

occasionalvariation.Rows(i),(iii),(iv)arepresentedbyMartzloff(1997:99),basedontheChinesemathematicalclassicShushujiyi.]

Specializeduse(93) bak usually400 but360days(longcalendar) [Mayan] [Tozzer1921:97,andmoregenerally97-103.TheformscitedbyTozzeraremore

specificallyYucatecMaya;notethatTozzerusesthespellingbaq.TheClassicalMayaninscriptionsandcodicesdonotclearlyincludelargenumbersotherthanincalendricalaccounts(Chrisomalis2010:292-294).]

(94) kilo- 1000 but:kilobyte‘1024bytes’ [English/International]Bodypartsystems(95) siduŋ‘shoulder’=10,14,33,37,56,60 [Kobon;Davies1981:206-208]

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J.Internalstructureandpsychologicalreality(96) Issuesatinterfacewithpsychomathematics(psychoarithmetic) a) Towhatextentareproblemscausedbyhavingalinguisticrepresentationthat

doesnotcorrespondtothearithmeticnotation,e.g.,assumingadecimalnotationindescendingorder

(i)ifaformationisnon-decimal,e.g.Frenchquatre-vingt-onze,literally‘fourtwentiesandeleven’,for91?

(ii)iftheorderis(partially)inverted,e.g.Germansechsundfünfzig,literally‘sixandfifty’,for56?

b) Whenpeopledoarithmetic,dotheyoperatewithlinguisticrepresentationsof

numerals(inwhichcaseproblems(a)mightarise)ordotheyoperatewithabstractquantitiesorarithmeticnotations(inwhichcasetheyshouldnot)?

c) Butnotethatsomearithmeticcalculationsmightbeeasierinanon-decimal

base,e.g.68+7mightbeeasierinavigesimalbase,whereonedoesnothaveto“carryover”anumber;indecimalnotation,68+7=75;invigesimalnotation38+7=3F(whereF=15)

(97) Writing‘sixandfifty’for56 Dutch zesenvijftig German sechsundfünfzig Dutchspeakerstypicallywritethe6andthengobackto“fillin”the5. Germanspeakerstypicallywritethe5thenthe6,andareexplicitlyinstructedto

dothisatschool.(98) Butnotethatinsomeinstancestheoverallstructureofthenumeralsystemmeans

thattheformalstructureofindividualnumeralscanbeoverridden.(99) Cardinal Ordinal 10 ti tiende 20 tyve tyvende 30 tred(i)ve tred(i)vte 40 fyrre fyrretyvende 50 halvtreds halvtredsindstyvende cf.tredje3rd 60 tres tresindstyvende cf.tre3 70 halvfjerds halvfjerdsindstyvende cf.fjerde4th 80 firs firsindstyvende cf.fire4 90 halvfems halvfemsindstyvende cf.femte5th 100 (et)hundred(e) hundrede [Danish;Allanetal.1995:122]

21

(100) Therecanbeotherdiscrepanciesbetweenformalstructure(oftenhistorical/etymological)andcurrentanalysisfornativespeakers,cf.

(101) 10 at-i 20 oc-i 100 as-i 1000 at-as-i ten-hundred-NOM 2000 or-iat-as-i(not*oc-as-i) two-nom-ten-hundred-NOM (*twenty-hundred-NOM) [Georgian;Hewitt1995:51-54](102) 10⁶ énaekato-míri-o onehundred-ten.thousand-SG 10⁷ ðékaekato-míri-a tenhundred-ten.thousand-PL (not*thousand-ten.thousand) [ModernGreek;LKN1998:427](103) Possiblecompetingpedagogicaladvantages a) Havinganumeralsystemthatcorrespondstothearithmeticnotation

facilitateslearningarithmetic. b) Havinganumeralsystemthatdoesnotcorrespondtothearithmeticnotation

providesextensivepracticeinarithmeticandleadstoahigherlevelofarithmeticskill.

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Osaka:NationalMuseumofEthnology.IndexofLanguagesTheclassificationfollowsGlottolog4.3<http://glottolog.org>.Thethree-letterabbreviationsareISO639-3codes;parenthesesindicateapartialmatch.Adzera ADZ Oceanic,Austronesian MorobePr.,PapuaNew

GuineaBalinese BAN Malayo-Sumbawan,

AustronesianBali,Indonesia

Berom BOM Benue-Congo,Atlantic-Congo

PlateauState,Nigeria

Chukchi CKT Chukotko-Kamchatkan Chukotka,RussiaClassicalNahuatl NCI Uto-Aztecan Mexico[extinct]Coahuilteco XCW isolate Texas,USA;Mexico

[extinct]Danish DAN Germanic,Indo-European DenmarkDutch NLD Germanic,Indo-European Netherlands;etc.Ekari EKG PaniaiLakes,Nuclear

TransNewGuineaPapua,Indonesia

English ENG Germanic,Indo-European England;USA;etc.French FRA Italic,Indo-European France;etc.Georgian KAT Kartvelian Rep.ofGeorgiaGerman DEU Germanic,Indo-European Germany;etc.Haruai TMD Piawi MadangPr.,PapuaNew

GuineaHindi HIN Indo-Aryan,Indo-

EuropeanNorth-CentralIndia

Hixkaryána HIX Cariban Amazonas,BrazilItalian ITA Italic,Indo-European Italy

25

Japanese JPN Japonic JapanKet KET Yeniseian W.Siberia,RussiaKobon KPW Madang,NuclearTrans

NewGuineaMadangPr.,PapuaNewGuinea

Komnzo (TCI) Morehead-Maro,Yam WesternPr.,PapuaNewGuinea

Latin LAT Italic,Indo-European Rome[extinct]Malagasy,NosyBe(varietyofSakalavaMalagasy

(SKG) Basap-GreaterBarito,Austronesian

NWMadagascar

Malagasy,Standard(=PlateauMalagasy)

PLT Basap-GreaterBarito,Austronesian

Madagascar

MandarinChinese CMN Sinitic,Sino-Tibetan ChinaMangarrayi MPC Mangarrayi-Maran NorthernTerr.,

AustraliaMayan [languagefamily] MesoamericaModernGreek ELL Greek,Indo-European Greece,CyprusMundurukú MYU Tupian Amazonas/Mato

Grosso/Pará,BrazilNauete NXA CentralMalayo-

Polynesian,AustronesianEastTimor

Ngiti NIY CentralSudanic OrientalePr.,DRCongoNorthernMansi (MNS) Mansic,Uralic W.Siberia,RussiaNorthernPame PMQ Pamean,Otomanguean SanLuisPotosí,MexicoOksapmin OPM Asmat-Awyu-Ok,Nuclear

TransNewGuineaSandaunPr.,PapuaNewGuinea

OldNorse NON Germanic,Indo-European Scandinavia;Iceland[extinct]

Oriya ORI Indo-Aryan,Indo-European

Orissa,India

Pirahã MYP isolate Amazonas,BrazilPolabian POX Slavic,Indo-European Germany[extinct]ResiaSlovenian (SLV) Slavic,Indo-European UdinePr.,ItalyRussian RUS Slavic,Indo-European RussiaSanskrit SAN Indo-Aryan,Indo-

EuropeanIndia[extinct]

Spanish SPA Italic,Indo-European Spain;LatinAmericaSupyireSenoufo SPP Senufo,Atlantic-Congo MaliThai THA Tai-Kadai ThailandWelsh CYM Celtic,Indo-European Wales,UnitedKingdomYaqui YAQ Uto-Aztecan MexicoYidiny YII Pama-Nyungan Queensland,AustraliaYucatecMaya YUA Mayan Yucatán,Mexico