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Why We Should Switch To A Base-12 Counting System (http://io9.com/5977095/why-we-should-switch-to-a-base-12-counting-system)

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152,357 9George Dvorsky (http://georgedvorsky.kinja.com)Filed to: DAILY EXPLAINER (/TAG/DAILY-EXPLAINER) 1/18/13 11:36am (http://io9.com/5977095/why-we-should-switch-to-a-base-12-counting-system)6

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?

Humans,forthemostpart,countinchunksof10that'sthefoundationofthedecimalsystem.Despiteitsnearuniversaladoption,however,it'sacompletelyarbitrarynumberingsystemthatemergedforoneverysimplereason:Wehavefivefingersoneachhand.Butasmanymathematiciansliketopointout,base10isnotwithoutitsproblems.Thenumber12,theyargue,iswhereit'sreallyat.Here'swhyweshouldhaveadoptedabase12countingsystemandhowwecouldstillmakeitwork.

Indeed,it'sregrettablethatwefailedtoevolveanidealsetoffingerstohelpuscomeupwithnumberingsystemsuitableforcountingandcalculating.Instead,withour10fingers,wearestuckwiththeclunkydecimalsystem.

Takingacloserlookatbase10,wecanseehowfrustratinglylimiteditreallyis.Tenhasapaltrytwofactors(adivisorthatproduceswholenumbers),namely5and2.Moreover,thesenumbersarenotveryusefulinandofthemselves5isaprimenumberthatcannotdivideanyfurther,and2isafrustratinglysmallintegertoworkwith.

Defendersofbase10highlightitsabilitytoallowforthemovingoffractionpointsaftermultiplicationordivisionbutthat'snotatraitexclusivetobase10.It'snottennessthatallowsforthisproperty.Moreaccurately,it'sacharacteristicthatbelongstoallbasesapropertyoftheplacevaluenotationweuseforexpressingnumbers,alongwithasymbolforzero.

Interestingly,base10isnotuniversalacrosshumansocieties.TheMayanswereknowntouseabase20system(http://wwwhistory.mcs.stand.ac.uk/HistTopics/Mayan_mathematics.html),andtheBabyloniansdevelopedasystemusingsetsof60(http://wwwhistory.mcs.stand.ac.uk/HistTopics/Babylonian_numerals.html).Base8andbase16(thehexadecimalsystem)havealsobeenused,mostlyforcomputationalreasons(quartersandeighthsaresimplified).

Butthesealternativesetsarestillnotidealfordaytoday,humanapplications.Base20isnotgreatforfingercountingmanyofuswearshoeswhenwe'redoingmath,norcanwemoveourtoeswithanykindofdexterity.Base8issimplytoosmall,andbase16andbase60aretoounwieldy.

Luckily,there'sabasethatsitsinbetweentheseanumberingsystemthathasaplethoraofcharacteristicsthatsimplymakeitthebestchoiceforcountingandcalculating.

Introducing the Dozenal System

Alsocalledtheduodecimalsystem,the"dozenal"systemwasinitiallypopularizedinthe17thcenturywhenmathematiciansbegantorecognizethelimitationsofbase10.

Later,duringthe1930s,F.EmersonAndrewspublishedabook,NewNumbers:HowAcceptanceofaDuodecimalBaseWouldSimplifyMathematics(http://books.google.ca/books/about/New_numbers.html?id=zfXuAAAAMAAJ&redir_esc=y),inwhichhecogentlyarguedforthechange.Henoticedthat,duetothemyriadoccurrencesof12inmanytraditionalunitsofweightandmeasures,manyoftheadvantagesclaimedforthemetricsystemcouldalsobeadoptedbythedozenalsystem.

Indeed,examplesofbase12systemsabound.Acarpenter'srulerhas12subdivisions,grocersdealindozensandgrosses(12dozenequalsagross),pharmacistsandjewelersusethe12ouncepound,andmintersdivideshillingsinto12pence.Evenourtiminganddatingsystemdependsonitthereare12monthsintheyear,andourdayismeasuredin2setsof12.Additionally,ingeometry,acircleisrepletewithsubsetsandsupersetsof12what'smeasuredindegrees(a360degreecircleconsistsof30setsof12).

It'salsoobviousthatsomeoneinourhistorywasthinkingalongtheselines.It'sthelargestnumberwithasinglemorphemenameinEnglish(i.e.theword"twelve").Afterthat,wehitthirteen,fourteen,fifteen,andsoonderivativesofthree,fourandfive.Clearly,itwasnaturaltothinkintermsofdozens.

ThreedecadesafterAndrews'sbook,thebrilliantmathematicianA.C.Aitken(http://wwwhistory.mcs.stand.ac.uk/Mathematicians/Aitken.html)madeasimilarcase.WritinginTheListenin1962,henoted:

Theduodecimaltablesareeasytomaster,easierthanthedecimalonesandinelementaryteachingtheywouldbesomuchmoreinteresting,sinceyoungchildrenwouldfindmorefascinatingthingstodowithtwelverodsorblocksthanwithten.Anyonehavingthesetablesatcommandwilldothesecalculationsmorethanoneandahalftimesasfastintheduodecimalscaleasinthedecimal.ThisismyexperienceIamcertainthatevenmoresoitwouldbetheexperienceofothers.

SincethetimeofAndrewsandAitken,thedozenalmovementhasgarneredanumberofenthusiasticsupporters,includingtheadventoftheDozenalSocietyofAmerica(http://www.dozenal.org)andtheDozenalSocietyofGreatBritain(http://www.dozenalsociety.org.uk/).

Thebasicargumentfromthesesocalleddozenalistsisthatitmakesmathematicseasiertoconceptualizeandunderstand,especiallyforchildrenandstudents.Here'swhythey'reright.

It's All About the Factors

Firstandforemost,12isahighlycompositenumberthesmallestnumberwithexactlyfourdivisors:2,3,4,and6(sixifyoucount1and12).Asnoted,10hasonlytwo.Consequently,12ismuchmorepracticalwhenusingfractionsit'seasiertodivideunitsofweightsandmeasuresinto12parts,namelyhalves,thirds,andquarters.

Moreover,withbase12,wecanusethesethreemostcommonfractionswithouthavingtoemployfractionalnotations.Thenumbers6,4,and3areallwholenumbers.Ontheotherhand,withbase10,wehavetodealwithunwieldydecimals,=0.5,=0.25,andworstofall,thehighlyproblematic=0.333333333333333333333.

Andsimilartothebase16hexadecimalsystem,thedozenalsystemisexceptionallyfriendlytocomputerscience.Thenumber12hastwofactorsthatareprimenumbers,2and3.Thismeansthatthereciprocalsofallsmoothnumbers(anumberwhichfactorscompletelyintosmallprimenumbers),suchas2,3,4,6,7,8,haveaterminatingrepresentationinduodecimal(we'llgettocountinginduodecimalinjustabit).Twelvejusthappenstobethesmallestnumberwiththisfeature,thusmakingitanextremelyefficientnumberforencryptionpurposesandforcomputingfractionsandthisincludesthedecimal,vigesimal,binary,octal,andhexadecimalsystems.

Interestingly,thedozenalsystemwouldalsomakeiteasiertotelltime.Fiveminutesisa12thofanhour,soinsteadofsaying"fivepastone,"wecouldsay"oneandatwelfth"hours.Tenpastonewouldbe12,aquarterpastone13,andsoon(thesymbol""isusedasthefractionalpoint).

Butthiswouldrequireanewclock.Forittowork,boththehourhandandtheminutehandwouldpointtotheprecisetime.Intheconventionaldecimalclock,theminutehandawkwardlypointstoanumberthathastobemultipliedbyfive.

Notation and Pronunciation

Asyoulookatthegraphicoftheclocktoyouraboveleft,you'reprobablywonderingwhatthosefunnysymbolsandwordsare.That'sbecause,forabase12towork,weneedtoaddtwonewsymbolsfor11and12(remember,thesearerepresentationsofnumbers,andarenotalphabeticthenumber12isderivedfromhavingonecompletesetof10(hencethe1inthefirstcolumn),andanadditionalnumber2inthesecondcolumntodenotetwoadditionalincrements).

Recognizingtheadvantagesofabase12system,Andrewsdesignedanewnotationtoaccountfortwonewnumbers.Insteadofusing"A"and"B"for10and11(asperthehexadecimalsystem),AndrewssuggestedascriptX(U+1D4B3)andE(U+2130),with10duodecimalrepresenting12decimal.Sothefirst12numberswouldlooklike1,2,3,4,5,6,7,8,9,X,E,10.

Othershavesuggestedthat10couldbewrittenas"T"andthenumbereleven"E."MathematicianIsaacPitmanwantedtousearotated"2"fortenandareversed"3"foreleven(aspertheclockabove).Otherschemasuse"*"for10and"#"for11(whichisphoneandcomputerkeyboardfriendly).

Forfractions,thedecimal0.5wouldbewritteninduodecimalas06(remember,ahalfof10isdifferentthanahalfof12).

Ifthisisconfusing,youcanalwaysusethedozenal/decimalcalculator(http://flud.org/dozenalcalc.html).

Fornumbersthatgobeyond12,wewouldaddaprefixtothevaluedenotingthenumberofsets.So,forthenumbers13,14,and15,we'dwrite11,12,and13.Andforthenumbers22,23,and24,we'dwrite1X,1E,and20.

Intermsofpronunciation,DonaldP.Goodman,presidentoftheDozenalSocietyofAmerica,saysthatXshouldbecalled"ten",Ecalled"elv"and10pronounced"unqua."So,whencounting,we'dsay,"...eight,nine,tenelv,unqua."

Interestingly,inthe1973episode"LittleTwelvetoes"oftheSchoolhouseRock!televisionseries,analienchildusesabase12systemandpronouncesthelastthreenumbers"dek,""el"and"doh.""Dek"wasderivedfromtheprefix"deca",while"el"wasshortfor"eleven,"and"doh"ashorteningof"dozen."Manydozenalistshaveadoptedthisparticularpronunciationsystem.

Now,topronouncenumbersgreaterthan12,likeduodecimal15,wewouldsaydohfive,whichisacompoundofdoh,whichistwelve,andfive.Wecanextendthisforothernumberssuchasduodecimal64,whichwouldbepronouncedassixdohfour.IfweweretoreachandsurpassthenumberEE,(eldohel),weneedanewwordforthedigitsinthethirdcolumnover.

Thewordfor144decimal,or100dozenal,iscalled"gros"(thes'issilent)So,athreedigitdozenalnumber,suchas25X,wouldbepronouncedas"twogrosfivedohdek."Indecimal,thisnumberis358.

Counting Fingers

Criticsofthedozenalsystemsaythatitwouldunderminethebenefitsoffingercounting.

Butasdozenalistsarehappytopointout,eachfingerconsistsofthreeparts.So,startingwiththeindexfinger,andusingthethumbasapointer,wecanimmediatelydenotethefirstthreedigits(workingourwayfrombottomtothetopofthefinger).Then,themiddlefingercandenote4,5,6,themiddlefinger,7,8,9,andsoon.Usingthissystem,ourtwohandsgivesusatotalof24numberstoworkwith.Somefingercountersworktheirwayfromlefttoright,designatingthetipsoftheirfingers1,2,3,4.

Evenbetter,wecanuseoursecondhandtodisplaythenumberofcompletedbase12's.Consequently,wecanuseourfingerstogoupto144(12x12).

Forexample,ifyoutakethethumbofyourlefthandandplaceitonthemiddlejointofyourmiddlefinger(whichisthe5thbase12,equalling60decimal),andyoudothesameonyourrighthand(whichsignifiesthe5thincrement),wegetthenumber65decimal.

Could We Ever Switch Over?

9 446 Reply

Unfortunately,convertingtothedozenalsystematthispointwouldbeexceptionallydifficult,andoverthetopexpensive.Whilethelongtermbenefitsareobvious,it'sprobablynotworththeshorttermpain.Butthatsaid,livingwithasuboptimalcountingsystemfromheretoeternityseemssad.

Thatsaid,dozenalistslikeDonaldGoodmansayit'snotcompletelyimpossible.Hearguesthatconvertingthecurrencywouldbethefirstandmostcrucialstep,followedbyanorganizededucationcampaignonthematterintheschools(Asanaside,andinregardstothislaststep,thisisexactlyhowthemetricsystemwaspopularizedandtaughtinCanadaIvividlyrememberthedaywhen,asachild,ourteachercameinandsaid,"Kids,fromhereonin,it'sthemetricsystemnoexceptions").

Goodmanisskeptical,however,thatanyoneprocedurecouldworkeverywhere,suggestingthatitwouldhavetobetailoredtolocalcircumstances.

"Mostdozenalistsbelievethatweshouldletdozenalsspeakforthemselves,"hetoldtheGuardian(http://www.guardian.co.uk/science/alexsadventuresinnumberland/2012/dec/12/dozenalistsworldunitetyrannyten)."Astimegoeson,andasmorepeoplelearnaboutdozenals,morepeoplewillusethemafterawhile,peoplewon'twanttousedecimalsanymore."Noofficial,topdownchangeisreallyneeded,heargues,exceptforthingslikemoneyandlegalrecognitionfordozenalmeasurementsystems.

So,whatdoyouthink?Hasthetimecomeforthedozenalsystem?

SpecialthankstoCalvinDvorskyforhelpingmewritethisarticle!

Sources:DozenalSocietyofGreatBritain(http://www.dozenalsociety.org.uk/),DozenalSocietyofAmerica(http://www.dozenal.org/articles/articles.html),TheGuardian(http://www.guardian.co.uk/science/alexsadventuresinnumberland/2012/dec/12/dozenalistsworldunitetyrannyten).

Images:Shutterstock/ArtisticPhoto,Guardian,gorpub(http://gorpub.freeshell.org/dozenal/blosxom.cgi/compbase.html).

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George Dvorskys Discussions (http://io9.com/5977095/why-we-should-switch-to-a-base-12-counting-system)

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George Dvorsky (http://georgedvorsky.kinja.com)1/18/13 11:42am (http://io9.com/gonna-date-myself-here-and-mention-that-i-actually-reme-263239238)

(http://georgedvorsky.kinja.com)

GonnadatemyselfhereandmentionthatIactuallyrememberwatchingthatSchoolhouseRock!episode.ThoughIstruggledtounderstandit.

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Jakob Pfender (http://norknork.kinja.com)1/18/13 11:59am (http://io9.com/mr-dvorsky-i-do-enjoy-your-articles-however-someth-263239669)

(http://norknork.kinja.com)

Mr.Dvorsky,Idoenjoyyourarticleshowever,somethingkeepsbuggingme.Yourconstantuseofdashesit'sfranklyirritating.Youkeeprippingyoursentencesapartthat'snotverynice.Inoticedthisawhilebackhaven'tbeenabletounnoticeitsince.ThisisnotmeantasaseriouscriticismjusttoannoythehelloutofotherOCD'swhowillnowhavepainsreadingyourarticles.Keepupyourgoodworkit'sappreciated.

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George Dvorsky (http://georgedvorsky.kinja.com)1/18/13 12:20pm (http://io9.com/point-taken-i-do-loves-me-my-dashes-but-it-often-se-263240310)

(http://georgedvorsky.kinja.com)

Pointtaken.Idolovesmemydashesbutitoftenservestoaccentuateadramaticpoint.I'lltrytoscaleback:)

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tech2120 (http://tech2120.kinja.com)1/18/13 11:55am (http://io9.com/first-and-foremost-12-is-a-highly-composite-number-263239547)

(http://tech2120.kinja.com)

"Firstandforemost,12isahighlycompositenumberthesmallestnumberwithexactlysixdivisors:2,3,4,and6.Asnoted,10hasonlytwo."

All replies (http://io9.com/5977095/why-we-should-switch-to-a-base-12-counting-system/all)

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Wouldn't2,3,4,and6makefourdivisors?It'sonlysixifyoucount1and12,butthen10wouldhavefourdivisorsinsteadoftwo.#corrections

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George Dvorsky (http://georgedvorsky.kinja.com)1/18/13 12:25pm (http://io9.com/fixed-but-i-made-a-note-that-technically-it-is-six-i-263240549)

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Fixed,butImadeanotethattechnicallyit*is*sixifyoucount1anditself.

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KenCole (http://kencole.kinja.com)1/18/13 11:44am (http://io9.com/so-the-widely-used-in-computer-science-base-16-is-sum-263239247)

(http://kencole.kinja.com)

Sothewidelyused(incomputerscience)base16issummarilydismissedwithtwowords:"toounwieldy".Whatdoesthatevenmean?Igetthatthereareadvantagestobase12thatbase16doesn'tshare,butthereverseisalsotrue.Surelybase16deservesmorethanatwoworddismissal?

/No,IwillnotstopcallingyouShirley

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George Dvorsky (http://georgedvorsky.kinja.com)1/18/13 12:27pm (http://io9.com/its-not-great-for-human-applications-like-counting-and-263240581)

(http://georgedvorsky.kinja.com)

It'snotgreatforhumanapplications,likecountingandcalculatinginourheads.

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