Elementary arithmetic From Wikipedia, the free encyclopedia Chapter 1 0 (number) “Zero”redirectshere. Forotheruses,seeZero(disambiguation). 0(zero;BrE:/ˈzɪərəʊ/orAmE:/ˈziːroʊ/)isbothanumber[1]andthenumericaldigitusedtorepresentthatnumberin numerals. Itfulfillsacentralroleinmathematicsastheadditiveidentityoftheintegers,realnumbers,andmanyother algebraicstructures. Asadigit,0isusedasaplaceholderinplacevaluesystems. Namesforthenumber0inEnglish includezero,noughtor(US)naught(/ˈnɔːt/),nil,or—incontextswhereatleastoneadjacentdigitdistinguishes itfromtheletter“O”—ohoro(/ˈoʊ/). Informalorslangtermsforzeroincludezilchandzip.[2] Ought andaught (/ˈɔːt/),aswellascipher,havealsobeenusedhistorically.[3] 1.1 Etymology Mainarticles: Namesforthenumber0andNamesforthenumber0inEnglish ThewordzerocameintotheEnglishlanguageviaFrenchzérofromItalianzero,ItaliancontractionofVenetianzevero formof'Italianzefiroviaṣafiraorṣifr.[4]Inpre-Islamictimethewordṣifr(Arabicرفص)hadthemeaning'empty'.[5] Sifrevolvedtomeanzerowhenitwasusedtotranslateśūnya(Sanskrit: शून्य)fromIndia.[5]ThefirstknownEnglish useofzerowasin1598.[6] TheItalianmathematicianFibonacci(c.1170–1250),whogrewupinNorthAfricaandiscreditedwithintroducing thedecimalsystemtoEurope,usedthetermzephyrum. ThisbecamezefiroinItalian,andwasthencontractedtozero inVenetian. TheItalianwordzefirowasalreadyinexistence(meaning“westwind”fromLatinandGreekzephyrus) andmayhaveinfluencedthespellingwhentranscribingArabicṣifr.[7] Modernusage Therearedifferentwordsusedforthenumberorconceptofzerodependingonthecontext. Forthesimplenotion oflacking,thewordsnothingandnoneareoftenused. Sometimesthewordsnought,naught andaught[8] areused. Several sports have specific words for zero, such as nil in football, love in tennis and a duck in cricket. It is often calledohinthecontextoftelephonenumbers. Slangwordsforzeroincludezip,zilch,nada,andscratch. Duckegg andgooseeggarealsoslangforzero.[9] 1.2 History 1.2.1 AncientNearEast Ancient Egyptian numerals were base 10. They used hieroglyphs for the digits and were not positional. By 1740 BC,theEgyptianshadasymbolforzeroinaccountingtexts. Thesymbolnfr,meaningbeautiful,wasalsousedto indicatethebaselevelindrawingsoftombsandpyramidsanddistancesweremeasuredrelativetothebaselineas beingaboveorbelowthisline.[10] 2 1.2. HISTORY 3 By the middle of the 2nd millennium BC, the Babylonian mathematics had a sophisticated sexagesimal positional numeralsystem. Thelackofapositionalvalue(orzero)wasindicatedbyaspacebetweensexagesimalnumerals. By 300BC,apunctuationsymbol(twoslantedwedges)wasco-optedasaplaceholderinthesameBabyloniansystem. InatabletunearthedatKish(datingfromabout700BC),thescribeBêl-bân-apluwrotehiszeroswiththreehooks, ratherthantwoslantedwedges.[11] The Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the end of a number. Thusnumberslike2and120(2×60),3and180(3×60),4and240(4×60),lookedthesamebecausethe largernumberslackedafinalsexagesimalplaceholder. Onlycontextcoulddifferentiatethem. 1.2.2 ClassicalAntiquity RecordsshowthattheancientGreeksseemedunsureaboutthestatusofzeroasanumber. Theyaskedthemselves, “Howcannothingbesomething?",leadingtophilosophicaland,bytheMedievalperiod,religiousargumentsabout thenatureandexistenceofzeroandthevacuum. TheparadoxesofZenoofEleadependinlargepartontheuncertain interpretationofzero. ExampleoftheearlyGreeksymbolforzero(lowerrightcorner)froma2nd-centurypapyrus By 130 AD, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for zero (a small circle with a long overbar) within a sexagesimal numeral system otherwise using alphabetic Greek numerals. Because it wasusedalone,notjustasaplaceholder,thisHellenisticzerowasperhapsthefirstdocumenteduseofanumberzero in the Old World. However, the positions were usually limited to the fractional part of a number (called minutes, seconds,thirds,fourths,etc.)—theywerenotusedfortheintegralpartofanumber. InlaterByzantinemanuscripts ofPtolemy’sSyntaxisMathematica(alsoknownastheAlmagest),theHellenisticzerohadmorphedintotheGreek letteromicron(otherwisemeaning70). AnotherzerowasusedintablesalongsideRomannumeralsby525(firstknownusebyDionysiusExiguus),butasa word,nullameaning“nothing”,notasasymbol. Whendivisionproducedzeroasaremainder,nihil,alsomeaning “nothing”,wasused. Thesemedievalzeroswereusedbyallfuturemedievalcomputists(calculatorsofEaster). The initial“N”wasusedasazerosymbolinatableofRomannumeralsbyBedeorhiscolleaguearound725. 1.2.3 IndiaandSoutheastAsia The concept of zero as a digit in the decimal place value notation was developed in India, presumably as early as duringtheGuptaperiod(c. 5thcentury),withtheoldestunambiguousevidencedatingtothe7thcentury.[12] TheIndianscholarPingala(c. 200BC)usedabinarynumbersintheformofshortandlongsyllables(thelatterequal 4 CHAPTER1. 0(NUMBER) inlengthtotwoshortsyllables),anotationsimilartoMorsecode.[13]PingalausedtheSanskritwordśūnyaexplicitly torefertozero.[14] The earliest text to use a decimal place-value system, including a zero, the Lokavibhāga, a Jain text surviving in a medievalSanskrittranslationofthePrakritoriginal,whichisinternallydatedtoAD458(Sakaera380). Inthistext, śūnya(“void,empty”)isalsousedtorefertozero.[15] Theoriginofthemoderndecimal-basedplacevaluenotationcanbetracedtotheAryabhatiya(c. 500),whichstates sthānātsthānaṁdaśaguṇaṁsyāt“fromplacetoplaceeachistentimesthepreceding”,[16][16][17][18] TherulesgoverningtheuseofzeroappearedforthefirsttimeintheBrahmasputhaSiddhanta(7thcentury). This workconsidersnotonlyzero,butnegativenumbers,andthealgebraicrulesfortheelementaryoperationsofarithmetic with such numbers. In some instances, his rules differ from the modern standard, specifically the definition of the valueofzerodividedbyzeroaszero.[19] Epigraphy Thenumber605inKhmernumerals, fromtheSamborinscription(Sakaera605correspondstoAD683). Theearliestknown materialuseofzeroasadecimalfigure. Therearenumerouscopperplateinscriptions,withthesamesmallointhem,someofthempossiblydatedtothe6th century,buttheirdateorauthenticitymaybeopentodoubt.[11] AstonetabletfoundintheruinsofatemplenearSamborontheMekong,KratiéProvince,Cambodia,includesthe inscriptionof“605”inKhmernumerals(asetofnumeralglyphsoftheHindunumeralsfamily). Thenumberisthe yearoftheinscriptionintheSakaera,correspondingtoadateofAD683.[20] Thefirstknownuseofspecialglyphsforthedecimaldigitsthatincludestheindubitableappearanceofasymbolfor the digit zero, a small circle, appears on a stone inscription found at the Chaturbhuja Temple at Gwalior in India, dated876.[21][22] 1.2.4 China TheSunziSuanjing,ofunknowndatebutestimatedtobedatedfromthe1stto5thcenturies,andJapaneserecords datedfromtheeighteenthcentury,describehowcountingrodswereusedforcalculations. AccordingtoAHistory 1.2. HISTORY 5 ThisisadepictionofzeroexpressedinChinesecountingrods,basedontheexampleprovidedbyAHistoryofMathematics. An emptyspaceisusedtorepresentzero.[23] ofMathematics,therods“gavethedecimalrepresentationofanumber,withanemptyspacedenotingzero.”[23]The countingrodsystemisconsideredapositionalnotationsystem.[24] Zero was not treated as a number at that time, but as a “vacant position”, unlike the Indian mathematicians who developedthenumericalzero.[25]Ch'inChiu-shao's1247MathematicalTreatiseinNineSectionsistheoldestsurviving Chinese mathematical text using a round symbol for zero.[26] Chinese authors had been familiar with the idea of negative numbers by the Han Dynasty (2nd century AD), as seen in the The Nine Chapters on the Mathematical Art,[27]muchearlierthanthefifteenthcenturywhentheybecamewellestablishedinEurope.[26] 1.2.5 MiddleAges TransmissiontoIslamicculture Seealso: HistoryoftheHindu-Arabicnumeralsystem Positionalnotationwithouttheuseofzero(usinganemptyspaceintabulararrangements,orthewordkha“empti- ness”)isknowntohavebeeninuseinIndiafromaboutthe6thcentury. Theglyphforthezerodigitwaswrittenin theshapeofadot,andconsequentlycalledbindu(“dot”). AdothadalsobeenusedinGreeceduringearlierciphered numeralperiods. TheArabic-languageinheritanceof sciencewaslargelyGreek,[28] followedbyHinduinfluences.[29] In 773, at Al- Mansur'sbehest,translationsweremadeofmanyancienttreatisesincludingGreek,Latin,Indian,andothers. In AD 813, astronomical tables were prepared by a Persian mathematician, Muḥammad ibn Mūsā al-Khwārizmī, using Hindu numerals;[29] and about 825, he published a book synthesizing Greek and Hindu knowledge and also containedhisowncontributiontomathematicsincludinganexplanationoftheuseofzero.[30] Thisbookwaslater translatedintoLatininthe12thcenturyunderthetitleAlgoritmidenumeroIndorum. Thistitlemeans“al-Khwarizmi ontheNumeralsoftheIndians”. Theword“Algoritmi”wasthetranslator’sLatinizationofAl-Khwarizmi’sname, andtheword“Algorithm”or“Algorism”startedmeaninganyarithmeticbasedondecimals.[29] MuhammadibnAhmadal-Khwarizmi,in976,statedthatifnonumberappearsintheplaceoftensinacalculation, alittlecircleshouldbeused“tokeeptherows”. Thiscirclewascalledṣifr.[31] TransmissiontoEurope TheHindu-Arabicnumeralsystem(base10)reachedEuropeinthe11thcentury,viatheIberianPeninsulathrough SpanishMuslims,theMoors,togetherwithknowledgeofastronomyandinstrumentsliketheastrolabe,firstimported byGerbertofAurillac. Forthisreason,thenumeralscametobeknowninEuropeas“Arabicnumerals”. TheItalian mathematicianFibonacciorLeonardoofPisawasinstrumentalinbringingthesystemintoEuropeanmathematics in1202,stating: 6 CHAPTER1. 0(NUMBER) Aftermyfather’sappointmentbyhishomelandasstateofficialinthecustomshouseofBugiaforthe Pisanmerchantswhothrongedtoit,hetookcharge;andinviewofitsfutureusefulnessandconvenience, hadmeinmyboyhoodcometohimandtherewantedmetodevotemyselftoandbeinstructedinthe studyofcalculationforsomedays. There, followingmyintroduction, asaconsequenceofmarvelous instructionintheart,totheninedigitsoftheHindus,theknowledgeoftheartverymuchappealedtome beforeallothers,andforitIrealizedthatallitsaspectswerestudiedinEgypt,Syria,Greece,Sicily,and Provence,withtheirvaryingmethods; andattheseplacesthereafter,whileonbusiness. Ipursuedmy studyindepthandlearnedthegive-and-takeofdisputation. Butallthiseven,andthealgorism,aswell astheartofPythagoras,IconsideredasalmostamistakeinrespecttothemethodoftheHindus(Modus Indorum). Therefore,embracingmorestringentlythatmethodoftheHindus,andtakingstricterpainsin itsstudy,whileaddingcertainthingsfrommyownunderstandingandinsertingalsocertainthingsfrom thenicetiesofEuclid’sgeometricart. Ihavestriventocomposethisbookinitsentiretyasunderstandably asIcould,dividingitintofifteenchapters. AlmosteverythingwhichIhaveintroducedIhavedisplayed withexactproof,inorderthatthosefurtherseekingthisknowledge,withitspre-eminentmethod,might be instructed, and further, in order that the Latin people might not be discovered to be without it, as theyhavebeenuptonow. IfIhaveperchanceomittedanythingmoreorlessproperornecessary,Ibeg indulgence,sincethereisnoonewhoisblamelessandutterlyprovidentinallthings. ThenineIndian figures are: 9 8 7 6 5 4 3 2 1. With these nine figures, and with the sign 0 ... any number may be written.[32][33] HereLeonardoofPisausesthephrase“sign0”,indicatingitislikeasigntodooperationslikeadditionormultipli- cation. Fromthe13thcentury,manualsoncalculation(adding,multiplying,extractingroots,etc.) becamecommon inEuropewheretheywerecalledalgorismusafterthePersianmathematicianal-Khwārizmī. Themostpopularwas writtenbyJohannesdeSacrobosco,about1235andwasoneoftheearliestscientificbookstobeprintedin1488. Until thelate15thcentury,Hindu-Arabicnumeralsseemtohavepredominatedamongmathematicians,whilemerchants preferredtousetheRomannumerals. Inthe16thcentury,theybecamecommonlyusedinEurope. 1.2.6 Pre-ColumbianAmericas TheMesoamericanLongCountcalendardevelopedinsouth-centralMexicoandCentralAmericarequiredtheuse ofzeroasaplace-holderwithinitsvigesimal(base-20)positionalnumeralsystem. Manydifferentglyphs,including thispartialquatrefoil— —wereusedasazerosymbolfortheseLongCountdates,theearliestofwhich(onStela 2atChiapadeCorzo,Chiapas)hasadateof36BC.[34] SincetheeightearliestLongCountdatesappearoutsidetheMayahomeland,[35]itisassumedthattheuseofzeroin theAmericaspredatedtheMayaandwaspossiblytheinventionoftheOlmecs.[36] ManyoftheearliestLongCount dateswerefoundwithintheOlmecheartland,althoughtheOlmeccivilizationendedbythe4thcenturyBC,several centuriesbeforetheearliestknownLongCountdates. AlthoughzerobecameanintegralpartofMayanumerals,withadifferent,emptytortoise-like“shellshape”usedfor manydepictionsofthe“zero”numeral,itdidnotinfluenceOldWorldnumeralsystems. Quipu,aknottedcorddevice,usedintheIncaEmpireanditspredecessorsocietiesintheAndeanregiontorecord accountingandotherdigitaldata,isencodedinabasetenpositionalsystem. Zeroisrepresentedbytheabsenceofa knotintheappropriateposition. 1.3 Mathematics Seealso: parityofzero 0istheintegerimmediatelypreceding1. Zeroisanevennumber,[37]becauseitisdivisibleby2. 0isneitherpositive nornegative. Bymostdefinitions[38]0isanaturalnumber,andthentheonlynaturalnumbernottobepositive. Zero isanumberwhichquantifiesacountoranamountofnullsize. Inmostcultures,0wasidentifiedbeforetheideaof negativethings,orquantitieslessthanzero,wasaccepted. The value, or number, zero is not the same as the digit zero, used in numeral systems using positional notation. Successivepositionsofdigitshavehigherweights,soinsideanumeralthedigitzeroisusedtoskipapositionandgive 1.3. MATHEMATICS 7 ThebackofOlmecstelaCfromTresZapotes,thesecondoldestLongCountdatediscovered. Thenumerals7.16.6.16.18translate toSeptember,32BC(Julian). TheglyphssurroundingthedatearethoughttobeoneofthefewsurvivingexamplesofEpi-Olmec script. appropriateweightstotheprecedingandfollowingdigits. Azerodigitisnotalwaysnecessaryinapositionalnumber system,forexample,inthenumber02. Insomeinstances,aleadingzeromaybeusedtodistinguishanumber. 8 CHAPTER1. 0(NUMBER) 1.3.1 Elementaryalgebra The number 0 is the smallest non-negative integer. The natural number following 0 is 1 and no natural number precedes0. Thenumber0mayormaynotbeconsideredanaturalnumber, butitisawholenumberandhencea rationalnumberandarealnumber(aswellasanalgebraicnumberandacomplexnumber). Thenumber0isneitherpositivenornegativeandappearsinthemiddleofanumberline. Itisneitheraprimenumber noracompositenumber. Itcannotbeprimebecauseithasaninfinitenumberoffactorsandcannotbecomposite because it cannot be expressed by multiplying prime numbers (0 must always be one of the factors).[39] Zero is, however,even. The following are some basic (elementary) rules for dealing with the number 0. These rules apply for any real or complexnumberx,unlessotherwisestated. (cid:15) Addition: x+0=0+x=x. Thatis,0isanidentityelement(orneutralelement)withrespecttoaddition. (cid:15) Subtraction: x−0=xand0−x=−x. (cid:15) Multiplication: x·0=0·x=0. (cid:15) Division: 0⁄x=0,fornonzerox. Butx⁄ isundefined,because0hasnomultiplicativeinverse(norealnumber 0 multipliedby0produces1),aconsequenceofthepreviousrule. (cid:15) Exponentiation: x0=x/x=1,exceptthatthecasex=0maybeleftundefinedinsomecontexts. Forallpositive realx,0x =0. Theexpression0⁄ ,whichmaybeobtainedinanattempttodeterminethelimitofanexpressionoftheformf(x)⁄g₍x₎ 0 asaresultofapplyingthelimoperatorindependentlytobothoperandsofthefraction,isaso-called"indeterminate form". That does not simply mean that the limit sought is necessarily undefined; rather, it means that the limit of f(x)⁄g₍x₎,ifitexists,mustbefoundbyanothermethod,suchasl'Hôpital’srule. Thesumof0numbersis0,andtheproductof0numbersis1. Thefactorial0! evaluatesto1. 1.3.2 Otherbranchesofmathematics (cid:15) In set theory, 0 is the cardinality of the empty set: if one does not have any apples, then one has 0 apples. Infact, incertainaxiomaticdevelopmentsofmathematicsfromsettheory, 0isdefinedtobetheemptyset. Whenthisisdone,theemptysetistheVonNeumanncardinalassignmentforasetwithnoelements,which istheemptyset. Thecardinalityfunction,appliedtotheemptyset,returnstheemptysetasavalue,thereby assigningit0elements. (cid:15) Alsoinsettheory,0isthelowestordinalnumber,correspondingtotheemptysetviewedasawell-orderedset. (cid:15) Inpropositionallogic,0maybeusedtodenotethetruthvaluefalse. (cid:15) Inabstractalgebra,0iscommonlyusedtodenoteazeroelement,whichisaneutralelementforaddition(if definedonthestructureunderconsideration)andanabsorbingelementformultiplication(ifdefined). (cid:15) Inlatticetheory,0maydenotethebottomelementofaboundedlattice. (cid:15) Incategorytheory,0issometimesusedtodenoteaninitialobjectofacategory. (cid:15) Inrecursiontheory,0canbeusedtodenotetheTuringdegreeofthepartialcomputablefunctions. 1.3.3 Relatedmathematicalterms (cid:15) Azeroofafunctionf isapointx inthedomainofthefunctionsuchthatf(x)=0. Whentherearefinitely manyzerosthesearecalledtherootsofthefunction. Thisisrelatedtozerosofaholomorphicfunction. (cid:15) Thezerofunction(orzeromap)onadomainDistheconstantfunctionwith0asitsonlypossibleoutputvalue, i.e.,thefunctionf definedbyf(x)=0forallxinD.Thezerofunctionistheonlyfunctionthatisbotheven andodd. Aparticularzerofunctionisazeromorphismincategorytheory;e.g.,azeromapistheidentityin theadditivegroupoffunctions. Thedeterminantonnon-invertiblesquarematricesisazeromap. 1.4. PHYSICS 9 (cid:15) Several branches of mathematics have zero elements, which generalise either the property 0 + x = x, or the property0×x=0,orboth. 1.4 Physics The value zero plays a special role for many physical quantities. For some quantities, the zero level is naturally distinguished from all other levels, whereas for others it is more or less arbitrarily chosen. For example, for an absolutetemperature(asmeasuredinkelvins)zeroisthelowestpossiblevalue(negativetemperaturesaredefined, but negative-temperature systems are not actually colder). This is in contrast to for example temperatures on the Celsius scale, where zero is arbitrarily defined to be at the freezing point of water. Measuring sound intensity in decibelsorphons, thezerolevelisarbitrarilysetatareferencevalue—forexample, atavalueforthethresholdof hearing. Inphysics,thezero-pointenergyisthelowestpossibleenergythataquantummechanicalphysicalsystem maypossessandistheenergyofthegroundstateofthesystem. 1.5 Chemistry Zerohasbeenproposedastheatomicnumberofthetheoreticalelementtetraneutron. Ithasbeenshownthatacluster offourneutronsmaybestableenoughtobeconsideredanatominitsownright. Thiswouldcreateanelementwith noprotonsandnochargeonitsnucleus. As early as 1926, Professor Andreas von Antropoff coined the term neutronium for a conjectured form of matter madeupofneutronswithnoprotons,whichheplacedasthechemicalelementofatomicnumberzeroatthehead of his new version of the periodic table. It was subsequently placed as a noble gas in the middle of several spiral representationsoftheperiodicsystemforclassifyingthechemicalelements. 1.5.1 Astronomy (cid:15) TheSarosnumberofthesolareclipseserieswhichbeganonMay23,2955BCandendedonJune29,1675 BC.ThedurationofSarosseries0was1280.14years,anditcontained72solareclipses. (cid:15) TheSarosnumberofthelunareclipseserieswhichbeganonMarch1,2653BCandendedonApril30,1337 BC.ThedurationofSarosseries0was1316.2years,anditcontained74lunareclipses. 1.6 Computer science Themostcommonpracticethroughouthumanhistoryhasbeentostartcountingatone, andthisisthepracticein earlyclassiccomputerscienceprogramminglanguagessuchasFortranandCOBOL.However,inthelate1950sLISP introducedzero-basednumberingforarrayswhileAlgol58introducedcompletelyflexiblebasingforarraysubscripts (allowing any positive, negative, or zero integer as base for array subscripts), and most subsequent programming languagesadoptedoneorotherofthesepositions. Forexample,theelementsofanarrayarenumberedstartingfrom 0inC,sothatforanarrayofnitemsthesequenceofarrayindicesrunsfrom0ton−1. Thispermitsanarrayelement’s locationtobecalculatedbyaddingtheindexdirectlytoaddressofthearray,whereas1basedlanguagesprecalculate thearray’sbaseaddresstobethepositiononeelementbeforethefirst. Therecanbeconfusionbetween0and1basedindexing,forexampleJava’sJDBCindexesparametersfrom1although Javaitselfuses0-basedindexing. Indatabases,itispossibleforafieldnottohaveavalue. Itisthensaidtohaveanullvalue. Fornumericfieldsitisnot thevaluezero. Fortextfieldsthisisnotblanknortheemptystring. Thepresenceofnullvaluesleadstothree-valued logic. No longer is a condition either true or false, but it can be undetermined. Any computation including a null valuedeliversanullresult. Askingforallrecordswithvalue0orvaluenotequal0willnotyieldallrecords,since therecordswithvaluenullareexcluded. A null pointer is a pointer in a computer program that does not point to any object or function. In C, the integer constant 0 is converted into the null pointer at compile time when it appears in a pointer context, and so 0 is a 10 CHAPTER1. 0(NUMBER) standardwaytorefertothenullpointerincode. However,theinternalrepresentationofthenullpointermaybeany bitpattern(possiblydifferentvaluesfordifferentdatatypes). In mathematics −0 = +0 = 0, both −0 and +0 represent exactly the same number, i.e., there is no “negative zero” distinct from zero. In some signed number representations (but not the two’s complement representation used to represent integers in most computers today) and most floating point number representations, zero has two distinct representations, one grouping it with the positive numbers and one with the negatives; this latter representation is knownasnegativezero. 1.7 Other fields (cid:15) Intelephony,pressing0isoftenusedfordiallingoutofacompanynetworkortoadifferentcityorregion,and 00isusedfordiallingabroad. Insomecountries,dialling0placesacallforoperatorassistance. (cid:15) DVDsthatcanbeplayedinanyregionaresometimesreferredtoasbeing"region0" (cid:15) Roulettewheelsusuallyfeaturea“0”space(andsometimesalsoa“00”space),whosepresenceisignoredwhen calculatingpayoffs(therebyallowingthehousetowininthelongrun). (cid:15) InFormulaOne, ifthereigningWorldChampionnolongercompetesinFormulaOneintheyearfollowing theirvictoryinthetitlerace,0isgiventooneofthedriversoftheteamthatthereigningchampionwonthetitle with. Thishappenedin1993and1994,withDamonHilldrivingcar0,duetothereigningWorldChampion (NigelMansellandAlainProstrespectively)notcompetinginthechampionship. 1.8 Symbols and representations Mainarticle: Symbolsforzero Themodernnumericaldigit0isusuallywrittenasacircleorellipse. Traditionally,manyprinttypefacesmadethe capitalletterOmoreroundedthanthenarrower,ellipticaldigit0.[40] Typewritersoriginallymadenodistinctionin shape between O and 0; some models did not even have a separate key for the digit 0. The distinction came into prominenceonmoderncharacterdisplays.[40] Aslashedzerocanbeusedtodistinguishthenumberfromtheletter. Thedigit0withadotinthecenterseemsto haveoriginatedasanoptiononIBM3270displaysandhascontinuedwithsomemoderncomputertypefacessuch asAndaléMono,andinsomeairlinereservationsystems. Onevariationusesashortverticalbarinsteadofthedot. Somefontsdesignedforusewithcomputersmadeoneofthecapital-O–digit-0pairmoreroundedandtheothermore angular(closertoarectangle). Afurtherdistinctionismadeinfalsification-hinderingtypefaceasusedonGerman