Addition - Wikipedia

Addition is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole ... Addition FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Arithmeticoperation Forotheruses,seeAddition(disambiguation). "Add"redirectshere.Forotheruses,seeADD(disambiguation). 3+2=5withapples,apopularchoiceintextbooks[1] Addition(usuallysignifiedbytheplussymbol+)isoneofthefourbasicoperationsofarithmetic,theotherthreebeingsubtraction,multiplicationanddivision.Theadditionoftwowholenumbersresultsinthetotalamountorsumofthosevaluescombined.Theexampleintheadjacentimageshowsacombinationofthreeapplesandtwoapples,makingatotaloffiveapples.Thisobservationisequivalenttothemathematicalexpression"3+2=5"(thatis,"3plus2isequalto5"). Besidescountingitems,additioncanalsobedefinedandexecutedwithoutreferringtoconcreteobjects,usingabstractionscallednumbersinstead,suchasintegers,realnumbersandcomplexnumbers.Additionbelongstoarithmetic,abranchofmathematics.Inalgebra,anotherareaofmathematics,additioncanalsobeperformedonabstractobjectssuchasvectors,matrices,subspacesandsubgroups. Additionhasseveralimportantproperties.Itiscommutative,meaningthatorderdoesnotmatter,anditisassociative,meaningthatwhenoneaddsmorethantwonumbers,theorderinwhichadditionisperformeddoesnotmatter(seeSummation).Repeatedadditionof1isthesameascounting.Additionof0doesnotchangeanumber.Additionalsoobeyspredictablerulesconcerningrelatedoperationssuchassubtractionandmultiplication. Performingadditionisoneofthesimplestnumericaltasks.Additionofverysmallnumbersisaccessibletotoddlers;themostbasictask,1+1,canbeperformedbyinfantsasyoungasfivemonths,andevensomemembersofotheranimalspecies.Inprimaryeducation,studentsaretaughttoaddnumbersinthedecimalsystem,startingwithsingledigitsandprogressivelytacklingmoredifficultproblems.Mechanicalaidsrangefromtheancientabacustothemoderncomputer,whereresearchonthemostefficientimplementationsofadditioncontinuestothisday. Arithmeticoperationsvte Addition(+) term + term summand + summand addend + addend augend + addend } = {\displaystyle\scriptstyle\left.{\begin{matrix}\scriptstyle{\text{term}}\,+\,{\text{term}}\\\scriptstyle{\text{summand}}\,+\,{\text{summand}}\\\scriptstyle{\text{addend}}\,+\,{\text{addend}}\\\scriptstyle{\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,} sum {\displaystyle\scriptstyle{\text{sum}}} Subtraction(−) term − term minuend − subtrahend } = {\displaystyle\scriptstyle\left.{\begin{matrix}\scriptstyle{\text{term}}\,-\,{\text{term}}\\\scriptstyle{\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,} difference {\displaystyle\scriptstyle{\text{difference}}} Multiplication(×) factor × factor multiplier × multiplicand } = {\displaystyle\scriptstyle\left.{\begin{matrix}\scriptstyle{\text{factor}}\,\times\,{\text{factor}}\\\scriptstyle{\text{multiplier}}\,\times\,{\text{multiplicand}}\end{matrix}}\right\}\,=\,} product {\displaystyle\scriptstyle{\text{product}}} Division(÷) dividend divisor   numerator denominator } = {\displaystyle\scriptstyle\left.{\begin{matrix}\scriptstyle{\frac{\scriptstyle{\text{dividend}}}{\scriptstyle{\text{divisor}}}}\\\scriptstyle{\text{}}\\\scriptstyle{\frac{\scriptstyle{\text{numerator}}}{\scriptstyle{\text{denominator}}}}\end{matrix}}\right\}\,=\,} fraction quotient ratio {\displaystyle{\begin{matrix}\scriptstyle{\text{fraction}}\\\scriptstyle{\text{quotient}}\\\scriptstyle{\text{ratio}}\end{matrix}}} Exponentiation base exponent = {\displaystyle\scriptstyle{\text{base}}^{\text{exponent}}\,=\,} power {\displaystyle\scriptstyle{\text{power}}} nthroot(√) radicand degree = {\displaystyle\scriptstyle{\sqrt[{\text{degree}}]{\scriptstyle{\text{radicand}}}}\,=\,} root {\displaystyle\scriptstyle{\text{root}}} Logarithm(log) log base ⁡ ( anti-logarithm ) = {\displaystyle\scriptstyle\log_{\text{base}}({\text{anti-logarithm}})\,=\,} logarithm {\displaystyle\scriptstyle{\text{logarithm}}} Contents 1Notationandterminology 2Interpretations 2.1Combiningsets 2.2Extendingalength 3Properties 3.1Commutativity 3.2Associativity 3.3Identityelement 3.4Successor 3.5Units 4Performingaddition 4.1Innateability 4.2Childhoodlearning 4.2.1Table 4.3Decimalsystem 4.3.1Carry 4.3.2Decimalfractions 4.3.3Scientificnotation 4.4Non-decimal 4.5Computers 5Additionofnumbers 5.1Naturalnumbers 5.2Integers 5.3Rationalnumbers(fractions) 5.4Realnumbers 5.5Complexnumbers 6Generalizations 6.1Abstractalgebra 6.1.1Vectors 6.1.2Matrices 6.1.3Modulararithmetic 6.1.4Generaltheory 6.2Settheoryandcategorytheory 7Relatedoperations 7.1Arithmetic 7.2Ordering 7.3Otherwaystoadd 8Seealso 9Notes 10Footnotes 11References 12Furtherreading Notationandterminology[edit] Theplussign Additioniswrittenusingtheplussign"+"betweentheterms;[2]thatis,ininfixnotation.Theresultisexpressedwithanequalssign.Forexample, 1 + 1 = 2 {\displaystyle1+1=2} ("oneplusoneequalstwo") 2 + 2 = 4 {\displaystyle2+2=4} ("twoplustwoequalsfour") 1 + 2 = 3 {\displaystyle1+2=3} ("oneplustwoequalsthree") 5 + 4 + 2 = 11 {\displaystyle5+4+2=11} (see"associativity"below) 3 + 3 + 3 + 3 = 12 {\displaystyle3+3+3+3=12} (see"multiplication"below) Columnaraddition–thenumbersinthecolumnaretobeadded,withthesumwrittenbelowtheunderlinednumber. Therearealsosituationswhereadditionis"understood",eventhoughnosymbolappears: Awholenumberfollowedimmediatelybyafractionindicatesthesumofthetwo,calledamixednumber.[3]Forexample, 3 1 2 = 3 + 1 2 = 3.5. {\displaystyle3{\frac{1}{2}}=3+{\frac{1}{2}}=3.5.} Thisnotationcancauseconfusion,sinceinmostothercontexts,juxtapositiondenotesmultiplicationinstead.[4] Thesumofaseriesofrelatednumberscanbeexpressedthroughcapitalsigmanotation,whichcompactlydenotesiteration.Forexample, ∑ k = 1 5 k 2 = 1 2 + 2 2 + 3 2 + 4 2 + 5 2 = 55. {\displaystyle\sum_{k=1}^{5}k^{2}=1^{2}+2^{2}+3^{2}+4^{2}+5^{2}=55.} Thenumbersortheobjectstobeaddedingeneraladditionarecollectivelyreferredtoastheterms,[5]theaddends[6][7][8]orthesummands;[9] thisterminologycarriesovertothesummationofmultipleterms. Thisistobedistinguishedfromfactors,whicharemultiplied. Someauthorscallthefirstaddendtheaugend.[6][7][8]Infact,duringtheRenaissance,manyauthorsdidnotconsiderthefirstaddendan"addend"atall.Today,duetothecommutativepropertyofaddition,"augend"israrelyused,andbothtermsaregenerallycalledaddends.[10] AlloftheaboveterminologyderivesfromLatin."Addition"and"add"areEnglishwordsderivedfromtheLatinverbaddere,whichisinturnacompoundofad"to"anddare"togive",fromtheProto-Indo-Europeanroot*deh₃-"togive";thustoaddistogiveto.[10]Usingthegerundivesuffix-ndresultsin"addend","thingtobeadded".[a]Likewisefromaugere"toincrease",onegets"augend","thingtobeincreased". RedrawnillustrationfromTheArtofNombryng,oneofthefirstEnglisharithmetictexts,inthe15thcentury.[11] "Sum"and"summand"derivefromtheLatinnounsumma"thehighest,thetop"andassociatedverbsummare.Thisisappropriatenotonlybecausethesumoftwopositivenumbersisgreaterthaneither,butbecauseitwascommonfortheancientGreeksandRomanstoaddupward,contrarytothemodernpracticeofaddingdownward,sothatasumwasliterallyhigherthantheaddends.[12] AddereandsummaredatebackatleasttoBoethius,ifnottoearlierRomanwriterssuchasVitruviusandFrontinus;Boethiusalsousedseveralothertermsfortheadditionoperation.ThelaterMiddleEnglishterms"adden"and"adding"werepopularizedbyChaucer.[13] Theplussign"+"(Unicode:U+002B;ASCII:+)isanabbreviationoftheLatinwordet,meaning"and".[14]Itappearsinmathematicalworksdatingbacktoatleast1489.[15] Interpretations[edit] Additionisusedtomodelmanyphysicalprocesses.Evenforthesimplecaseofaddingnaturalnumbers,therearemanypossibleinterpretationsandevenmorevisualrepresentations. Combiningsets[edit] Possiblythemostfundamentalinterpretationofadditionliesincombiningsets: Whentwoormoredisjointcollectionsarecombinedintoasinglecollection,thenumberofobjectsinthesinglecollectionisthesumofthenumbersofobjectsintheoriginalcollections. Thisinterpretationiseasytovisualize,withlittledangerofambiguity.Itisalsousefulinhighermathematics(fortherigorousdefinitionitinspires,see§ Naturalnumbersbelow).However,itisnotobvioushowoneshouldextendthisversionofadditiontoincludefractionalnumbersornegativenumbers.[16] Onepossiblefixistoconsidercollectionsofobjectsthatcanbeeasilydivided,suchaspiesor,stillbetter,segmentedrods.[17]Ratherthansolelycombiningcollectionsofsegments,rodscanbejoinedend-to-end,whichillustratesanotherconceptionofaddition:addingnottherodsbutthelengthsoftherods. Extendingalength[edit] Anumber-linevisualizationofthealgebraicaddition2+4=6.Atranslationby2followedbyatranslationby4isthesameasatranslationby6. Anumber-linevisualizationoftheunaryaddition2+4=6.Atranslationby4isequivalenttofourtranslationsby1. Asecondinterpretationofadditioncomesfromextendinganinitiallengthbyagivenlength: Whenanoriginallengthisextendedbyagivenamount,thefinallengthisthesumoftheoriginallengthandthelengthoftheextension.[18] Thesuma+bcanbeinterpretedasabinaryoperationthatcombinesaandb,inanalgebraicsense,oritcanbeinterpretedastheadditionofbmoreunitstoa.Underthelatterinterpretation,thepartsofasuma+bplayasymmetricroles,andtheoperationa+bisviewedasapplyingtheunaryoperation+btoa.[19]Insteadofcallingbothaandbaddends,itismoreappropriatetocallatheaugendinthiscase,sinceaplaysapassiverole.Theunaryviewisalsousefulwhendiscussingsubtraction,becauseeachunaryadditionoperationhasaninverseunarysubtractionoperation,andviceversa. Properties[edit] Commutativity[edit] 4+2=2+4withblocks Additioniscommutative,meaningthatonecanchangetheorderofthetermsinasum,butstillgetthesameresult.Symbolically,ifaandbareanytwonumbers,then a+b=b+a. Thefactthatadditioniscommutativeisknownasthe"commutativelawofaddition"or"commutativepropertyofaddition".Someotherbinaryoperationsarecommutative,suchasmultiplication,butmanyothersarenot,suchassubtractionanddivision. Associativity[edit] 2+(1+3)=(2+1)+3withsegmentedrods Additionisassociative,whichmeansthatwhenthreeormorenumbersareaddedtogether,theorderofoperationsdoesnotchangetheresult. Asanexample,shouldtheexpressiona+b+cbedefinedtomean(a+b)+cora+(b+c)?Giventhatadditionisassociative,thechoiceofdefinitionisirrelevant.Foranythreenumbersa,b,andc,itistruethat(a+b)+c=a+(b+c).Forexample,(1+2)+3=3+3=6=1+5=1+(2+3). Whenadditionisusedtogetherwithotheroperations,theorderofoperationsbecomesimportant.Inthestandardorderofoperations,additionisalowerprioritythanexponentiation,nthroots,multiplicationanddivision,butisgivenequalprioritytosubtraction.[20] Identityelement[edit] 5+0=5withbagsofdots Addingzerotoanynumber,doesnotchangethenumber;thismeansthatzeroistheidentityelementforaddition,andisalsoknownastheadditiveidentity.Insymbols,foreverya,onehas a+0=0+a=a. ThislawwasfirstidentifiedinBrahmagupta'sBrahmasphutasiddhantain628 AD,althoughhewroteitasthreeseparatelaws,dependingonwhetheraisnegative,positive,orzeroitself,andheusedwordsratherthanalgebraicsymbols.LaterIndianmathematiciansrefinedtheconcept;aroundtheyear830,Mahavirawrote,"zerobecomesthesameaswhatisaddedtoit",correspondingtotheunarystatement0+a=a.Inthe12th century,Bhaskarawrote,"Intheadditionofcipher,orsubtractionofit,thequantity,positiveornegative,remainsthesame",correspondingtotheunarystatementa+0=a.[21] Successor[edit] Withinthecontextofintegers,additionofonealsoplaysaspecialrole:foranyintegera,theinteger(a+1)istheleastintegergreaterthana,alsoknownasthesuccessorofa.[22]Forinstance,3isthesuccessorof2and7isthesuccessorof6.Becauseofthissuccession,thevalueofa+bcanalsobeseenasthebthsuccessorofa,makingadditioniteratedsuccession.Forexample,6+2is8,because8isthesuccessorof7,whichisthesuccessorof6,making8the2ndsuccessorof6. Units[edit] Tonumericallyaddphysicalquantitieswithunits,theymustbeexpressedwithcommonunits.[23]Forexample,adding50 millilitersto150 millilitersgives200 milliliters.However,ifameasureof5 feetisextendedby2 inches,thesumis62 inches,since60 inchesissynonymouswith5 feet.Ontheotherhand,itisusuallymeaninglesstotrytoadd3 metersand4 squaremeters,sincethoseunitsareincomparable;thissortofconsiderationisfundamentalindimensionalanalysis. Performingaddition[edit] Innateability[edit] Studiesonmathematicaldevelopmentstartingaroundthe1980shaveexploitedthephenomenonofhabituation:infantslooklongeratsituationsthatareunexpected.[24]AseminalexperimentbyKarenWynnin1992involvingMickeyMousedollsmanipulatedbehindascreendemonstratedthatfive-month-oldinfantsexpect1+1tobe2,andtheyarecomparativelysurprisedwhenaphysicalsituationseemstoimplythat1+1iseither1or3.Thisfindinghassincebeenaffirmedbyavarietyoflaboratoriesusingdifferentmethodologies.[25]Another1992experimentwitholdertoddlers,between18and35 months,exploitedtheirdevelopmentofmotorcontrolbyallowingthemtoretrieveping-pongballsfromabox;theyoungestrespondedwellforsmallnumbers,whileoldersubjectswereabletocomputesumsupto5.[26] Evensomenonhumananimalsshowalimitedabilitytoadd,particularlyprimates.Ina1995experimentimitatingWynn's1992result(butusingeggplantsinsteadofdolls),rhesusmacaqueandcottontoptamarinmonkeysperformedsimilarlytohumaninfants.Moredramatically,afterbeingtaughtthemeaningsoftheArabicnumerals0through4,onechimpanzeewasabletocomputethesumoftwonumeralswithoutfurthertraining.[27]Morerecently,Asianelephantshavedemonstratedanabilitytoperformbasicarithmetic.[28] Childhoodlearning[edit] Typically,childrenfirstmastercounting.Whengivenaproblemthatrequiresthattwoitemsandthreeitemsbecombined,youngchildrenmodelthesituationwithphysicalobjects,oftenfingersoradrawing,andthencountthetotal.Astheygainexperience,theylearnordiscoverthestrategyof"counting-on":askedtofindtwoplusthree,childrencountthreepasttwo,saying"three,four,five"(usuallytickingofffingers),andarrivingatfive.Thisstrategyseemsalmostuniversal;childrencaneasilypickitupfrompeersorteachers.[29]Mostdiscoveritindependently.Withadditionalexperience,childrenlearntoaddmorequicklybyexploitingthecommutativityofadditionbycountingupfromthelargernumber,inthiscase,startingwiththreeandcounting"four,five."Eventuallychildrenbegintorecallcertainadditionfacts("numberbonds"),eitherthroughexperienceorrotememorization.Oncesomefactsarecommittedtomemory,childrenbegintoderiveunknownfactsfromknownones.Forexample,achildaskedtoaddsixandsevenmayknowthat6+6=12andthenreasonthat6+7isonemore,or13.[30]Suchderivedfactscanbefoundveryquicklyandmostelementaryschoolstudentseventuallyrelyonamixtureofmemorizedandderivedfactstoaddfluently.[31] Differentnationsintroducewholenumbersandarithmeticatdifferentages,withmanycountriesteachingadditioninpre-school.[32]However,throughouttheworld,additionistaughtbytheendofthefirstyearofelementaryschool.[33] Table[edit] Childrenareoftenpresentedwiththeadditiontableofpairsofnumbersfrom0to9tomemorize.Knowingthis,childrencanperformanyaddition. + 0 1 2 3 4 5 6 7 8 9 0 0 1 2 3 4 5 6 7 8 9 1 1 2 3 4 5 6 7 8 9 10 2 2 3 4 5 6 7 8 9 10 11 3 3 4 5 6 7 8 9 10 11 12 4 4 5 6 7 8 9 10 11 12 13 5 5 6 7 8 9 10 11 12 13 14 6 6 7 8 9 10 11 12 13 14 15 7 7 8 9 10 11 12 13 14 15 16 8 8 9 10 11 12 13 14 15 16 17 9 9 10 11 12 13 14 15 16 17 18 Decimalsystem[edit] Theprerequisitetoadditioninthedecimalsystemisthefluentrecallorderivationofthe100single-digit"additionfacts".Onecouldmemorizeallthefactsbyrote,butpattern-basedstrategiesaremoreenlighteningand,formostpeople,moreefficient:[34] Commutativeproperty:Mentionedabove,usingthepatterna+b=b+areducesthenumberof"additionfacts"from100to55. Oneortwomore:Adding1or2isabasictask,anditcanbeaccomplishedthroughcountingonor,ultimately,intuition.[34] Zero:Sincezeroistheadditiveidentity,addingzeroistrivial.Nonetheless,intheteachingofarithmetic,somestudentsareintroducedtoadditionasaprocessthatalwaysincreasestheaddends;wordproblemsmayhelprationalizethe"exception"ofzero.[34] Doubles:Addinganumbertoitselfisrelatedtocountingbytwoandtomultiplication.Doublesfactsformabackboneformanyrelatedfacts,andstudentsfindthemrelativelyeasytograsp.[34] Near-doubles:Sumssuchas6+7=13canbequicklyderivedfromthedoublesfact6+6=12byaddingonemore,orfrom7+7=14butsubtractingone.[34] Fiveandten:Sumsoftheform5+xand10+xareusuallymemorizedearlyandcanbeusedforderivingotherfacts.Forexample,6+7=13canbederivedfrom5+7=12byaddingonemore.[34] Makingten:Anadvancedstrategyuses10asanintermediateforsumsinvolving8or9;forexample,8+6=8+2+4=10+4=14.[34] Asstudentsgrowolder,theycommitmorefactstomemory,andlearntoderiveotherfactsrapidlyandfluently.Manystudentsnevercommitallthefactstomemory,butcanstillfindanybasicfactquickly.[31] Carry[edit] Mainarticle:Carry(arithmetic) Thestandardalgorithmforaddingmultidigitnumbersistoaligntheaddendsverticallyandaddthecolumns,startingfromtheonescolumnontheright.Ifacolumnexceedsnine,theextradigitis"carried"intothenextcolumn.Forexample,intheaddition27+59 ¹ 27 +59 ———— 86 7+9=16,andthedigit1isthecarry.[b]Analternatestrategystartsaddingfromthemostsignificantdigitontheleft;thisroutemakescarryingalittleclumsier,butitisfasteratgettingaroughestimateofthesum.Therearemanyalternativemethods. Decimalfractions[edit] Decimalfractionscanbeaddedbyasimplemodificationoftheaboveprocess.[35]Onealignstwodecimalfractionsaboveeachother,withthedecimalpointinthesamelocation.Ifnecessary,onecanaddtrailingzerostoashorterdecimaltomakeitthesamelengthasthelongerdecimal.Finally,oneperformsthesameadditionprocessasabove,exceptthedecimalpointisplacedintheanswer,exactlywhereitwasplacedinthesummands. Asanexample,45.1+4.34canbesolvedasfollows: 45.10 +04.34 ———————————— 49.44 Scientificnotation[edit] Mainarticle:Scientificnotation§ Basicoperations Inscientificnotation,numbersarewrittenintheform x = a × 10 b {\displaystylex=a\times10^{b}} ,where a {\displaystylea} isthesignificandand 10 b {\displaystyle10^{b}} istheexponentialpart.Additionrequirestwonumbersinscientificnotationtoberepresentedusingthesameexponentialpart,sothatthetwosignificandscansimplybeadded. Forexample: 2.34 × 10 − 5 + 5.67 × 10 − 6 = 2.34 × 10 − 5 + 0.567 × 10 − 5 = 2.907 × 10 − 5 {\displaystyle2.34\times10^{-5}+5.67\times10^{-6}=2.34\times10^{-5}+0.567\times10^{-5}=2.907\times10^{-5}} Non-decimal[edit] Mainarticle:Binaryaddition Additioninotherbasesisverysimilartodecimaladdition.Asanexample,onecanconsideradditioninbinary.[36]Addingtwosingle-digitbinarynumbersisrelativelysimple,usingaformofcarrying: 0+0→0 0+1→1 1+0→1 1+1→0,carry1(since1+1=2=0+(1×21)) Addingtwo"1"digitsproducesadigit"0",while1mustbeaddedtothenextcolumn.Thisissimilartowhathappensindecimalwhencertainsingle-digitnumbersareaddedtogether;iftheresultequalsorexceedsthevalueoftheradix(10),thedigittotheleftisincremented: 5+5→0,carry1(since5+5=10=0+(1×101)) 7+9→6,carry1(since7+9=16=6+(1×101)) Thisisknownascarrying.[37]Whentheresultofanadditionexceedsthevalueofadigit,theprocedureisto"carry"theexcessamountdividedbytheradix(thatis,10/10)totheleft,addingittothenextpositionalvalue.Thisiscorrectsincethenextpositionhasaweightthatishigherbyafactorequaltotheradix.Carryingworksthesamewayinbinary: 11111(carrieddigits) 01101 +10111 ————————————— 100100=36 Inthisexample,twonumeralsarebeingaddedtogether:011012(1310)and101112(2310).Thetoprowshowsthecarrybitsused.Startingintherightmostcolumn,1+1=102.The1iscarriedtotheleft,andthe0iswrittenatthebottomoftherightmostcolumn.Thesecondcolumnfromtherightisadded:1+0+1=102again;the1iscarried,and0iswrittenatthebottom.Thethirdcolumn:1+1+1=112.Thistime,a1iscarried,anda1iswritteninthebottomrow.Proceedinglikethisgivesthefinalanswer1001002(3610). Computers[edit] Additionwithanop-amp.SeeSummingamplifierfordetails. Analogcomputersworkdirectlywithphysicalquantities,sotheiradditionmechanismsdependontheformoftheaddends.Amechanicaladdermightrepresenttwoaddendsasthepositionsofslidingblocks,inwhichcasetheycanbeaddedwithanaveraginglever.Iftheaddendsaretherotationspeedsoftwoshafts,theycanbeaddedwithadifferential.AhydraulicaddercanaddthepressuresintwochambersbyexploitingNewton'ssecondlawtobalanceforcesonanassemblyofpistons.Themostcommonsituationforageneral-purposeanalogcomputeristoaddtwovoltages(referencedtoground);thiscanbeaccomplishedroughlywitharesistornetwork,butabetterdesignexploitsanoperationalamplifier.[38] Additionisalsofundamentaltotheoperationofdigitalcomputers,wheretheefficiencyofaddition,inparticularthecarrymechanism,isanimportantlimitationtooverallperformance. PartofCharlesBabbage'sDifferenceEngineincludingtheadditionandcarrymechanisms Theabacus,alsocalledacountingframe,isacalculatingtoolthatwasinusecenturiesbeforetheadoptionofthewrittenmodernnumeralsystemandisstillwidelyusedbymerchants,tradersandclerksinAsia,Africa,andelsewhere;itdatesbacktoatleast2700–2300 BC,whenitwasusedinSumer.[39] BlaisePascalinventedthemechanicalcalculatorin1642;[40]itwasthefirstoperationaladdingmachine.Itmadeuseofagravity-assistedcarrymechanism.Itwastheonlyoperationalmechanicalcalculatorinthe17th century[41]andtheearliestautomatic,digitalcomputer.Pascal'scalculatorwaslimitedbyitscarrymechanism,whichforceditswheelstoonlyturnonewaysoitcouldadd.Tosubtract,theoperatorhadtousethePascal'scalculator'scomplement,whichrequiredasmanystepsasanaddition.GiovanniPolenifollowedPascal,buildingthesecondfunctionalmechanicalcalculatorin1709,acalculatingclockmadeofwoodthat,oncesetup,couldmultiplytwonumbersautomatically. "Fulladder"logiccircuitthataddstwobinarydigits,AandB,alongwithacarryinputCin,producingthesumbit,S,andacarryoutput,Cout. Addersexecuteintegeradditioninelectronicdigitalcomputers,usuallyusingbinaryarithmetic.Thesimplestarchitectureistheripplecarryadder,whichfollowsthestandardmulti-digitalgorithm.Oneslightimprovementisthecarryskipdesign,againfollowinghumanintuition;onedoesnotperformallthecarriesincomputing999+1,butonebypassesthegroupof9sandskipstotheanswer.[42] Inpractice,computationaladditionmaybeachievedviaXORandANDbitwiselogicaloperationsinconjunctionwithbitshiftoperationsasshowninthepseudocodebelow.BothXORandANDgatesarestraightforwardtorealizeindigitallogicallowingtherealizationoffulladdercircuitswhichinturnmaybecombinedintomorecomplexlogicaloperations.Inmoderndigitalcomputers,integeradditionistypicallythefastestarithmeticinstruction,yetithasthelargestimpactonperformance,sinceitunderliesallfloating-pointoperationsaswellassuchbasictasksasaddressgenerationduringmemoryaccessandfetchinginstructionsduringbranching.Toincreasespeed,moderndesignscalculatedigitsinparallel;theseschemesgobysuchnamesascarryselect,carrylookahead,andtheLingpseudocarry.Manyimplementationsare,infact,hybridsoftheselastthreedesigns.[43][44]Unlikeadditiononpaper,additiononacomputeroftenchangestheaddends.Ontheancientabacusandaddingboard,bothaddendsaredestroyed,leavingonlythesum.TheinfluenceoftheabacusonmathematicalthinkingwasstrongenoughthatearlyLatintextsoftenclaimedthatintheprocessofadding"anumbertoanumber",bothnumbersvanish.[45]Inmoderntimes,theADDinstructionofamicroprocessoroftenreplacestheaugendwiththesumbutpreservestheaddend.[46]Inahigh-levelprogramminglanguage,evaluatinga+bdoesnotchangeeitheraorb;ifthegoalistoreplaceawiththesumthismustbeexplicitlyrequested,typicallywiththestatementa=a+b.SomelanguagessuchasCorC++allowthistobeabbreviatedasa+=b. //Iterativealgorithm intadd(intx,inty){ intcarry=0; while(y!=0){ carry=AND(x,y);//LogicalAND x=XOR(x,y);//LogicalXOR y=carry<<1;//leftbitshiftcarrybyone } returnx; } //Recursivealgorithm intadd(intx,inty){ returnxif(y==0)elseadd(XOR(x,y),AND(x,y)<<1); } Onacomputer,iftheresultofanadditionistoolargetostore,anarithmeticoverflowoccurs,resultinginanincorrectanswer.Unanticipatedarithmeticoverflowisafairlycommoncauseofprogramerrors.Suchoverflowbugsmaybehardtodiscoveranddiagnosebecausetheymaymanifestthemselvesonlyforverylargeinputdatasets,whicharelesslikelytobeusedinvalidationtests.[47]TheYear2000problemwasaseriesofbugswhereoverflowerrorsoccurredduetouseofa2-digitformatforyears.[48] Additionofnumbers[edit] Toprovetheusualpropertiesofaddition,onemustfirstdefineadditionforthecontextinquestion.Additionisfirstdefinedonthenaturalnumbers.Insettheory,additionisthenextendedtoprogressivelylargersetsthatincludethenaturalnumbers:theintegers,therationalnumbers,andtherealnumbers.[49](Inmathematicseducation,[50]positivefractionsareaddedbeforenegativenumbersareevenconsidered;thisisalsothehistoricalroute.[51]) Naturalnumbers[edit] Furtherinformation:Naturalnumber Therearetwopopularwaystodefinethesumoftwonaturalnumbersaandb.Ifonedefinesnaturalnumberstobethecardinalitiesoffinitesets,(thecardinalityofasetisthenumberofelementsintheset),thenitisappropriatetodefinetheirsumasfollows: LetN(S)bethecardinalityofasetS.TaketwodisjointsetsAandB,withN(A)=aandN(B)=b.Thena+bisdefinedas N ( A ∪ B ) {\displaystyleN(A\cupB)} .[52] Here,A∪BistheunionofAandB.AnalternateversionofthisdefinitionallowsAandBtopossiblyoverlapandthentakestheirdisjointunion,amechanismthatallowscommonelementstobeseparatedoutandthereforecountedtwice. Theotherpopulardefinitionisrecursive: Letn+bethesuccessorofn,thatisthenumberfollowingninthenaturalnumbers,so0+=1,1+=2.Definea+0=a.Definethegeneralsumrecursivelybya+(b+)=(a+b)+.Hence1+1=1+0+=(1+0)+=1+=2.[53] Again,thereareminorvariationsuponthisdefinitionintheliterature.Takenliterally,theabovedefinitionisanapplicationoftherecursiontheoremonthepartiallyorderedsetN2.[54]Ontheotherhand,somesourcesprefertousearestrictedrecursiontheoremthatappliesonlytothesetofnaturalnumbers.Onethenconsidersatobetemporarily"fixed",appliesrecursiononbtodefineafunction"a +",andpastestheseunaryoperationsforallatogethertoformthefullbinaryoperation.[55] ThisrecursiveformulationofadditionwasdevelopedbyDedekindasearlyas1854,andhewouldexpanduponitinthefollowingdecades.[56]Heprovedtheassociativeandcommutativeproperties,amongothers,throughmathematicalinduction. Integers[edit] Furtherinformation:Integer Thesimplestconceptionofanintegeristhatitconsistsofanabsolutevalue(whichisanaturalnumber)andasign(generallyeitherpositiveornegative).Theintegerzeroisaspecialthirdcase,beingneitherpositivenornegative.Thecorrespondingdefinitionofadditionmustproceedbycases: Foranintegern,let|n|beitsabsolutevalue.Letaandbbeintegers.Ifeitheraorbiszero,treatitasanidentity.Ifaandbarebothpositive,definea+b=|a|+|b|.Ifaandbarebothnegative,definea+b=−(|a|+|b|).Ifaandbhavedifferentsigns,definea+btobethedifferencebetween|a|and|b|,withthesignofthetermwhoseabsolutevalueislarger.[57]Asanexample,−6+4=−2;because−6and4havedifferentsigns,theirabsolutevaluesaresubtracted,andsincetheabsolutevalueofthenegativetermislarger,theanswerisnegative. Althoughthisdefinitioncanbeusefulforconcreteproblems,thenumberofcasestoconsidercomplicatesproofsunnecessarily.Sothefollowingmethodiscommonlyusedfordefiningintegers.Itisbasedontheremarkthateveryintegeristhedifferenceoftwonaturalintegersandthattwosuchdifferences,a–bandc–dareequalifandonlyifa+d=b+c. So,onecandefineformallytheintegersastheequivalenceclassesoforderedpairsofnaturalnumbersundertheequivalencerelation (a,b)~(c,d)ifandonlyifa+d=b+c. Theequivalenceclassof(a,b)containseither(a–b,0)ifa≥b,or(0,b–a)otherwise.Ifnisanaturalnumber,onecandenote+ntheequivalenceclassof(n,0),andby–ntheequivalenceclassof(0,n).Thisallowsidentifyingthenaturalnumbernwiththeequivalenceclass+n. Additionoforderedpairsisdonecomponent-wise: ( a , b ) + ( c , d ) = ( a + c , b + d ) . {\displaystyle(a,b)+(c,d)=(a+c,b+d).} Astraightforwardcomputationshowsthattheequivalenceclassoftheresultdependsonlyontheequivalencesclassesofthesummands,andthusthatthisdefinesanadditionofequivalenceclasses,thatisintegers.[58]Anotherstraightforwardcomputationshowsthatthisadditionisthesameastheabovecasedefinition. Thiswayofdefiningintegersasequivalenceclassesofpairsofnaturalnumbers,canbeusedtoembedintoagroupanycommutativesemigroupwithcancellationproperty.Here,thesemigroupisformedbythenaturalnumbersandthegroupistheadditivegroupofintegers.Therationalnumbersareconstructedsimilarly,bytakingassemigroupthenonzerointegerswithmultiplication. ThisconstructionhasbeenalsogeneralizedunderthenameofGrothendieckgrouptothecaseofanycommutativesemigroup.Withoutthecancellationpropertythesemigrouphomomorphismfromthesemigroupintothegroupmaybenon-injective.Originally,theGrothendieckgroupwas,morespecifically,theresultofthisconstructionappliedtotheequivalencesclassesunderisomorphismsoftheobjectsofanabeliancategory,withthedirectsumassemigroupoperation. Rationalnumbers(fractions)[edit] Additionofrationalnumberscanbecomputedusingtheleastcommondenominator,butaconceptuallysimplerdefinitioninvolvesonlyintegeradditionandmultiplication: Define a b + c d = a d + b c b d . {\displaystyle{\frac{a}{b}}+{\frac{c}{d}}={\frac{ad+bc}{bd}}.} Asanexample,thesum 3 4 + 1 8 = 3 × 8 + 4 × 1 4 × 8 = 24 + 4 32 = 28 32 = 7 8 {\displaystyle{\frac{3}{4}}+{\frac{1}{8}}={\frac{3\times8+4\times1}{4\times8}}={\frac{24+4}{32}}={\frac{28}{32}}={\frac{7}{8}}} . Additionoffractionsismuchsimplerwhenthedenominatorsarethesame;inthiscase,onecansimplyaddthenumeratorswhileleavingthedenominatorthesame: a c + b c = a + b c {\displaystyle{\frac{a}{c}}+{\frac{b}{c}}={\frac{a+b}{c}}} ,so 1 4 + 2 4 = 1 + 2 4 = 3 4 {\displaystyle{\frac{1}{4}}+{\frac{2}{4}}={\frac{1+2}{4}}={\frac{3}{4}}} .[59] Thecommutativityandassociativityofrationaladditionisaneasyconsequenceofthelawsofintegerarithmetic.[60]Foramorerigorousandgeneraldiscussion,seefieldoffractions. Realnumbers[edit] Addingπ2/6andeusingDedekindcutsofrationals. Furtherinformation:Constructionoftherealnumbers AcommonconstructionofthesetofrealnumbersistheDedekindcompletionofthesetofrationalnumbers.ArealnumberisdefinedtobeaDedekindcutofrationals:anon-emptysetofrationalsthatiscloseddownwardandhasnogreatestelement.Thesumofrealnumbersaandbisdefinedelementbyelement: Define a + b = { q + r ∣ q ∈ a , r ∈ b } . {\displaystylea+b=\{q+r\midq\ina,r\inb\}.} [61] Thisdefinitionwasfirstpublished,inaslightlymodifiedform,byRichardDedekindin1872.[62] Thecommutativityandassociativityofrealadditionareimmediate;definingtherealnumber0tobethesetofnegativerationals,itiseasilyseentobetheadditiveidentity.Probablythetrickiestpartofthisconstructionpertainingtoadditionisthedefinitionofadditiveinverses.[63] Addingπ2/6andeusingCauchysequencesofrationals. Unfortunately,dealingwithmultiplicationofDedekindcutsisatime-consumingcase-by-caseprocesssimilartotheadditionofsignedintegers.[64]Anotherapproachisthemetriccompletionoftherationalnumbers.ArealnumberisessentiallydefinedtobethelimitofaCauchysequenceofrationals,lim an.Additionisdefinedtermbyterm: Define lim n a n + lim n b n = lim n ( a n + b n ) . {\displaystyle\lim_{n}a_{n}+\lim_{n}b_{n}=\lim_{n}(a_{n}+b_{n}).} [65] ThisdefinitionwasfirstpublishedbyGeorgCantor,alsoin1872,althoughhisformalismwasslightlydifferent.[66] Onemustprovethatthisoperationiswell-defined,dealingwithco-Cauchysequences.Oncethattaskisdone,allthepropertiesofrealadditionfollowimmediatelyfromthepropertiesofrationalnumbers.Furthermore,theotherarithmeticoperations,includingmultiplication,havestraightforward,analogousdefinitions.[67] Complexnumbers[edit] Additionoftwocomplexnumberscanbedonegeometricallybyconstructingaparallelogram. Complexnumbersareaddedbyaddingtherealandimaginarypartsofthesummands.[68][69]Thatistosay: ( a + b i ) + ( c + d i ) = ( a + c ) + ( b + d ) i . {\displaystyle(a+bi)+(c+di)=(a+c)+(b+d)i.} Usingthevisualizationofcomplexnumbersinthecomplexplane,theadditionhasthefollowinggeometricinterpretation:thesumoftwocomplexnumbersAandB,interpretedaspointsofthecomplexplane,isthepointXobtainedbybuildingaparallelogramthreeofwhoseverticesareO,AandB.Equivalently,XisthepointsuchthatthetriangleswithverticesO,A,B,andX,B,A,arecongruent. Generalizations[edit] Therearemanybinaryoperationsthatcanbeviewedasgeneralizationsoftheadditionoperationontherealnumbers.Thefieldofabstractalgebraiscentrallyconcernedwithsuchgeneralizedoperations,andtheyalsoappearinsettheoryandcategorytheory. Abstractalgebra[edit] Vectors[edit] Mainarticle:Vectoraddition Inlinearalgebra,avectorspaceisanalgebraicstructurethatallowsforaddinganytwovectorsandforscalingvectors.Afamiliarvectorspaceisthesetofallorderedpairsofrealnumbers;theorderedpair(a,b)isinterpretedasavectorfromtheoriginintheEuclideanplanetothepoint(a,b)intheplane.Thesumoftwovectorsisobtainedbyaddingtheirindividualcoordinates: ( a , b ) + ( c , d ) = ( a + c , b + d ) . {\displaystyle(a,b)+(c,d)=(a+c,b+d).} Thisadditionoperationiscentraltoclassicalmechanics,inwhichvectorsareinterpretedasforces. Matrices[edit] Mainarticle:Matrixaddition Matrixadditionisdefinedfortwomatricesofthesamedimensions.Thesumoftwom×n(pronounced"mbyn")matricesAandB,denotedbyA+B,isagainanm×nmatrixcomputedbyaddingcorrespondingelements:[70][71] A + B = [ a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋱ ⋮ a m 1 a m 2 ⋯ a m n ] + [ b 11 b 12 ⋯ b 1 n b 21 b 22 ⋯ b 2 n ⋮ ⋮ ⋱ ⋮ b m 1 b m 2 ⋯ b m n ] = [ a 11 + b 11 a 12 + b 12 ⋯ a 1 n + b 1 n a 21 + b 21 a 22 + b 22 ⋯ a 2 n + b 2 n ⋮ ⋮ ⋱ ⋮ a m 1 + b m 1 a m 2 + b m 2 ⋯ a m n + b m n ] {\displaystyle{\begin{aligned}\mathbf{A}+\mathbf{B}&={\begin{bmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\a_{21}&a_{22}&\cdots&a_{2n}\\\vdots&\vdots&\ddots&\vdots\\a_{m1}&a_{m2}&\cdots&a_{mn}\\\end{bmatrix}}+{\begin{bmatrix}b_{11}&b_{12}&\cdots&b_{1n}\\b_{21}&b_{22}&\cdots&b_{2n}\\\vdots&\vdots&\ddots&\vdots\\b_{m1}&b_{m2}&\cdots&b_{mn}\\\end{bmatrix}}\\&={\begin{bmatrix}a_{11}+b_{11}&a_{12}+b_{12}&\cdots&a_{1n}+b_{1n}\\a_{21}+b_{21}&a_{22}+b_{22}&\cdots&a_{2n}+b_{2n}\\\vdots&\vdots&\ddots&\vdots\\a_{m1}+b_{m1}&a_{m2}+b_{m2}&\cdots&a_{mn}+b_{mn}\\\end{bmatrix}}\\\end{aligned}}} Forexample: [ 1 3 1 0 1 2 ] + [ 0 0 7 5 2 1 ] = [ 1 + 0 3 + 0 1 + 7 0 + 5 1 + 2 2 + 1 ] = [ 1 3 8 5 3 3 ] {\displaystyle{\begin{bmatrix}1&3\\1&0\\1&2\end{bmatrix}}+{\begin{bmatrix}0&0\\7&5\\2&1\end{bmatrix}}={\begin{bmatrix}1+0&3+0\\1+7&0+5\\1+2&2+1\end{bmatrix}}={\begin{bmatrix}1&3\\8&5\\3&3\end{bmatrix}}} Modulararithmetic[edit] Mainarticle:Modulararithmetic Inmodulararithmetic,thesetofintegersmodulo12hastwelveelements;itinheritsanadditionoperationfromtheintegersthatiscentraltomusicalsettheory.Thesetofintegersmodulo 2hasjusttwoelements;theadditionoperationitinheritsisknowninBooleanlogicasthe"exclusiveor"function.Ingeometry,thesumoftwoanglemeasuresisoftentakentobetheirsumasrealnumbersmodulo 2π.Thisamountstoanadditionoperationonthecircle,whichinturngeneralizestoadditionoperationsonmany-dimensionaltori. Generaltheory[edit] Thegeneraltheoryofabstractalgebraallowsan"addition"operationtobeanyassociativeandcommutativeoperationonaset.Basicalgebraicstructureswithsuchanadditionoperationincludecommutativemonoidsandabeliangroups. Settheoryandcategorytheory[edit] Afar-reachinggeneralizationofadditionofnaturalnumbersistheadditionofordinalnumbersandcardinalnumbersinsettheory.Thesegivetwodifferentgeneralizationsofadditionofnaturalnumberstothetransfinite.Unlikemostadditionoperations,additionofordinalnumbersisnotcommutative.Additionofcardinalnumbers,however,isacommutativeoperationcloselyrelatedtothedisjointunionoperation. Incategorytheory,disjointunionisseenasaparticularcaseofthecoproductoperation,andgeneralcoproductsareperhapsthemostabstractofallthegeneralizationsofaddition.Somecoproducts,suchasdirectsumandwedgesum,arenamedtoevoketheirconnectionwithaddition. Relatedoperations[edit] Addition,alongwithsubtraction,multiplicationanddivision,isconsideredoneofthebasicoperationsandisusedinelementaryarithmetic. Arithmetic[edit] Subtractioncanbethoughtofasakindofaddition—thatis,theadditionofanadditiveinverse.Subtractionisitselfasortofinversetoaddition,inthataddingxandsubtractingxareinversefunctions. Givenasetwithanadditionoperation,onecannotalwaysdefineacorrespondingsubtractionoperationonthatset;thesetofnaturalnumbersisasimpleexample.Ontheotherhand,asubtractionoperationuniquelydeterminesanadditionoperation,anadditiveinverseoperation,andanadditiveidentity;forthisreason,anadditivegroupcanbedescribedasasetthatisclosedundersubtraction.[72] Multiplicationcanbethoughtofasrepeatedaddition.Ifasingletermxappearsinasumntimes,thenthesumistheproductofnandx.Ifnisnotanaturalnumber,theproductmaystillmakesense;forexample,multiplicationby−1yieldstheadditiveinverseofanumber. Acircularsliderule Intherealandcomplexnumbers,additionandmultiplicationcanbeinterchangedbytheexponentialfunction:[73] e a + b = e a e b . {\displaystylee^{a+b}=e^{a}e^{b}.} Thisidentityallowsmultiplicationtobecarriedoutbyconsultingatableoflogarithmsandcomputingadditionbyhand;italsoenablesmultiplicationonasliderule.Theformulaisstillagoodfirst-orderapproximationinthebroadcontextofLiegroups,whereitrelatesmultiplicationofinfinitesimalgroupelementswithadditionofvectorsintheassociatedLiealgebra.[74] Thereareevenmoregeneralizationsofmultiplicationthanaddition.[75]Ingeneral,multiplicationoperationsalwaysdistributeoveraddition;thisrequirementisformalizedinthedefinitionofaring.Insomecontexts,suchastheintegers,distributivityoveradditionandtheexistenceofamultiplicativeidentityisenoughtouniquelydeterminethemultiplicationoperation.Thedistributivepropertyalsoprovidesinformationaboutaddition;byexpandingtheproduct(1+1)(a+b)inbothways,oneconcludesthatadditionisforcedtobecommutative.Forthisreason,ringadditioniscommutativeingeneral.[76] Divisionisanarithmeticoperationremotelyrelatedtoaddition.Sincea/b=a(b−1),divisionisrightdistributiveoveraddition:(a+b)/c=a/c+b/c.[77]However,divisionisnotleftdistributiveoveraddition;1/(2+2)isnotthesameas1/2+1/2. Ordering[edit] Log-logplotofx+1andmax(x,1)fromx=0.001to1000[78] Themaximumoperation"max(a,b)"isabinaryoperationsimilartoaddition.Infact,iftwononnegativenumbersaandbareofdifferentordersofmagnitude,thentheirsumisapproximatelyequaltotheirmaximum.Thisapproximationisextremelyusefulintheapplicationsofmathematics,forexampleintruncatingTaylorseries.However,itpresentsaperpetualdifficultyinnumericalanalysis,essentiallysince"max"isnotinvertible.Ifbismuchgreaterthana,thenastraightforwardcalculationof(a+b)−bcanaccumulateanunacceptableround-offerror,perhapsevenreturningzero.SeealsoLossofsignificance. Theapproximationbecomesexactinakindofinfinitelimit;ifeitheraorbisaninfinitecardinalnumber,theircardinalsumisexactlyequaltothegreaterofthetwo.[79]Accordingly,thereisnosubtractionoperationforinfinitecardinals.[80] Maximizationiscommutativeandassociative,likeaddition.Furthermore,sinceadditionpreservestheorderingofrealnumbers,additiondistributesover"max"inthesamewaythatmultiplicationdistributesoveraddition: a + max ( b , c ) = max ( a + b , a + c ) . {\displaystylea+\max(b,c)=\max(a+b,a+c).} Forthesereasons,intropicalgeometryonereplacesmultiplicationwithadditionandadditionwithmaximization.Inthiscontext,additioniscalled"tropicalmultiplication",maximizationiscalled"tropicaladdition",andthetropical"additiveidentity"isnegativeinfinity.[81]Someauthorsprefertoreplaceadditionwithminimization;thentheadditiveidentityispositiveinfinity.[82] Tyingtheseobservationstogether,tropicaladditionisapproximatelyrelatedtoregularadditionthroughthelogarithm: log ⁡ ( a + b ) ≈ max ( log ⁡ a , log ⁡ b ) , {\displaystyle\log(a+b)\approx\max(\loga,\logb),} whichbecomesmoreaccurateasthebaseofthelogarithmincreases.[83]Theapproximationcanbemadeexactbyextractingaconstanth,namedbyanalogywithPlanck'sconstantfromquantummechanics,[84]andtakingthe"classicallimit"ashtendstozero: max ( a , b ) = lim h → 0 h log ⁡ ( e a / h + e b / h ) . {\displaystyle\max(a,b)=\lim_{h\to0}h\log(e^{a/h}+e^{b/h}).} Inthissense,themaximumoperationisadequantizedversionofaddition.[85] Otherwaystoadd[edit] Incrementation,alsoknownasthesuccessoroperation,istheadditionof1toanumber. Summationdescribestheadditionofarbitrarilymanynumbers,usuallymorethanjusttwo.Itincludestheideaofthesumofasinglenumber,whichisitself,andtheemptysum,whichiszero.[86]Aninfinitesummationisadelicateprocedureknownasaseries.[87] Countingafinitesetisequivalenttosumming1overtheset. Integrationisakindof"summation"overacontinuum,ormorepreciselyandgenerally,overadifferentiablemanifold.Integrationoverazero-dimensionalmanifoldreducestosummation. Linearcombinationscombinemultiplicationandsummation;theyaresumsinwhicheachtermhasamultiplier,usuallyarealorcomplexnumber.Linearcombinationsareespeciallyusefulincontextswherestraightforwardadditionwouldviolatesomenormalizationrule,suchasmixingofstrategiesingametheoryorsuperpositionofstatesinquantummechanics. Convolutionisusedtoaddtwoindependentrandomvariablesdefinedbydistributionfunctions.Itsusualdefinitioncombinesintegration,subtraction,andmultiplication.Ingeneral,convolutionisusefulasakindofdomain-sideaddition;bycontrast,vectoradditionisakindofrange-sideaddition. Seealso[edit] Lunararithmetic Mentalarithmetic Paralleladdition(mathematics) Verbalarithmetic(alsoknownascryptarithms),puzzlesinvolvingaddition Notes[edit] ^"Addend"isnotaLatinword;inLatinitmustbefurtherconjugated,asinnumerusaddendus"thenumbertobeadded". ^Someauthorsthinkthat"carry"maybeinappropriateforeducation;VandeWalle(p.211)callsit"obsoleteandconceptuallymisleading",preferringtheword"trade".However,"carry"remainsthestandardterm. Footnotes[edit] ^FromEnderton(p.138):"...selecttwosetsKandLwithcardK=2andcardL=3.Setsoffingersarehandy;setsofapplesarepreferredbytextbooks." ^"Addition".www.mathsisfun.com.Retrieved2020-08-25. ^Devineetal.p.263 ^Mazur,Joseph.EnlighteningSymbols:AShortHistoryofMathematicalNotationandItsHiddenPowers.PrincetonUniversityPress,2014.p.161 ^DepartmentoftheArmy(1961)ArmyTechnicalManualTM11-684:PrinciplesandApplicationsofMathematicsforCommunications-Electronics.Section5.1 ^abShmerko,V.P.;Yanushkevich[Ânuškevič],SvetlanaN.[SvitlanaN.];Lyshevski,S.E.(2009).Computerarithmeticsfornanoelectronics.CRCPress.p. 80. ^abSchmid,Hermann(1974).DecimalComputation(1st ed.).Binghamton,NY:JohnWiley&Sons.ISBN 0-471-76180-X.andSchmid,Hermann(1983)[1974].DecimalComputation(reprintof1st ed.).Malabar,FL:RobertE.KriegerPublishingCompany.ISBN 978-0-89874-318-0. ^abWeisstein,EricW."Addition".mathworld.wolfram.com.Retrieved2020-08-25. ^Hosch,W.L.(Ed.).(2010).TheBritannicaGuidetoNumbersandMeasurement.TheRosenPublishingGroup.p.38 ^abSchwartzmanp.19 ^Karpinskipp.56–57,reproducedonp.104 ^Schwartzman(p.212)attributesaddingupwardstotheGreeksandRomans,sayingitwasaboutascommonasaddingdownwards.Ontheotherhand,Karpinski(p.103)writesthatLeonardofPisa"introducesthenoveltyofwritingthesumabovetheaddends";itisunclearwhetherKarpinskiisclaimingthisasanoriginalinventionorsimplytheintroductionofthepracticetoEurope. ^Karpinskipp.150–153 ^Cajori,Florian(1928)."Originandmeaningsofthesigns+and-".AHistoryofMathematicalNotations,Vol.1.TheOpenCourtCompany,Publishers. ^"plus".OxfordEnglishDictionary(Online ed.).OxfordUniversityPress. (Subscriptionorparticipatinginstitutionmembershiprequired.) ^SeeViro2001foranexampleofthesophisticationinvolvedinaddingwithsetsof"fractionalcardinality". ^Addingitup(p.73)comparesaddingmeasuringrodstoaddingsetsofcats:"Forexample,inchescanbesubdividedintoparts,whicharehardtotellfromthewholes,exceptthattheyareshorter;whereasitispainfultocatstodividethemintoparts,anditseriouslychangestheirnature." ^Mosley,F.(2001).Usingnumberlineswith5–8yearolds.NelsonThornes.p.8 ^Li,Y.,&Lappan,G.(2014).Mathematicscurriculuminschooleducation.Springer.p.204 ^Bronstein,IljaNikolaevič;Semendjajew,KonstantinAdolfovič(1987)[1945]."2.4.1.1.".InGrosche,Günter;Ziegler,Viktor;Ziegler,Dorothea(eds.).TaschenbuchderMathematik(inGerman).1.TranslatedbyZiegler,Viktor.Weiß,Jürgen(23 ed.).ThunandFrankfurtamMain:VerlagHarriDeutsch(andB.G.TeubnerVerlagsgesellschaft,Leipzig).pp. 115–120.ISBN 978-3-87144-492-0. ^Kaplanpp.69–71 ^Hempel,C.G.(2001).ThephilosophyofCarlG.Hempel:studiesinscience,explanation,andrationality.p.7 ^R.Fierro(2012)MathematicsforElementarySchoolTeachers.CengageLearning.Sec2.3 ^Wynnp.5 ^Wynnp.15 ^Wynnp.17 ^Wynnp.19 ^Randerson,James(21August2008)."Elephantshaveaheadforfigures".TheGuardian.Archivedfromtheoriginalon2April2015.Retrieved29March2015. ^F.Smithp.130 ^Carpenter,Thomas;Fennema,Elizabeth;Franke,MeganLoef;Levi,Linda;Empson,Susan(1999).Children'smathematics:Cognitivelyguidedinstruction.Portsmouth,NH:Heinemann.ISBN 978-0-325-00137-1. ^abHenry,ValerieJ.;Brown,RichardS.(2008)."First-gradebasicfacts:Aninvestigationintoteachingandlearningofanaccelerated,high-demandmemorizationstandard".JournalforResearchinMathematicsEducation.39(2):153–183.doi:10.2307/30034895.JSTOR 30034895. ^ Beckmann,S.(2014).Thetwenty-thirdICMIstudy:primarymathematicsstudyonwholenumbers.InternationalJournalofSTEMEducation,1(1),1-8. Chicago ^Schmidt,W.,Houang,R.,&Cogan,L.(2002)."Acoherentcurriculum".AmericanEducator,26(2),1–18. ^abcdefgFosnotandDolkp.99 ^RebeccaWingard-Nelson(2014)DecimalsandFractions:It'sEasyEnslowPublishers,Inc. ^DaleR.Patrick,StephenW.Fardo,VigyanChandra(2008)ElectronicDigitalSystemFundamentalsTheFairmontPress,Inc.p.155 ^P.E.BatesBothman(1837)Thecommonschoolarithmetic.HenryBenton.p.31 ^TruittandRogerspp.1;44–49andpp.2;77–78 ^Ifrah,Georges(2001).TheUniversalHistoryofComputing:FromtheAbacustotheQuantumComputer.NewYork:JohnWiley&Sons,Inc.ISBN 978-0-471-39671-0.p.11 ^JeanMarguin,p.48(1994) ;QuotingRenéTaton(1963) ^SeeCompetingdesignsinPascal'scalculatorarticle ^FlynnandOvermanpp.2,8 ^FlynnandOvermanpp.1–9 ^Yeo,Sang-Soo,etal.,eds.AlgorithmsandArchitecturesforParallelProcessing:10thInternationalConference,ICA3PP2010,Busan,Korea,May21–23,2010.Proceedings.Vol.1.Springer,2010.p.194 ^Karpinskipp.102–103 ^Theidentityoftheaugendandaddendvarieswitharchitecture.ForADDinx86seeHorowitzandHillp.679;forADDin68kseep.767. ^JoshuaBloch,"Extra,Extra–ReadAllAboutIt:NearlyAllBinarySearchesandMergesortsareBroken"Archived2016-04-01attheWaybackMachine.OfficialGoogleResearchBlog,June2,2006. ^Neumann,PeterG."TheRisksDigestVolume4:Issue45".TheRisksDigest.Archivedfromtheoriginalon2014-12-28.Retrieved2015-03-30. ^Endertonchapters4and5,forexample,followthisdevelopment. ^AccordingtoasurveyofthenationswithhighestTIMSSmathematicstestscores;seeSchmidt,W.,Houang,R.,&Cogan,L.(2002).Acoherentcurriculum.Americaneducator,26(2),p.4. ^Baez(p.37)explainsthehistoricaldevelopment,in"starkcontrast"withthesettheorypresentation:"Apparently,halfanappleiseasiertounderstandthananegativeapple!" ^Beglep.49,Johnsonp.120,Devineetal.p.75 ^Endertonp.79 ^Foraversionthatappliestoanyposetwiththedescendingchaincondition,seeBergmanp.100. ^Enderton(p.79)observes,"Butwewantonebinaryoperation+,notalltheselittleone-placefunctions." ^Ferreirósp.223 ^K.Smithp.234,SparksandReesp.66 ^Endertonp.92 ^SchyrletCameron,andCarolynCraig(2013)AddingandSubtractingFractions,Grades5–8MarkTwain,Inc. ^TheverificationsarecarriedoutinEndertonp.104andsketchedforageneralfieldoffractionsoveracommutativeringinDummitandFootep.263. ^Endertonp.114 ^Ferreirósp.135;seesection6ofStetigkeitundirrationaleZahlenArchived2005-10-31attheWaybackMachine. ^Theintuitiveapproach,invertingeveryelementofacutandtakingitscomplement,worksonlyforirrationalnumbers;seeEndertonp.117fordetails. ^Schubert,E.Thomas,PhillipJ.Windley,andJamesAlves-Foss."HigherOrderLogicTheoremProvingandItsApplications:Proceedingsofthe8thInternationalWorkshop,volume971of."LectureNotesinComputerScience(1995). ^Textbookconstructionsareusuallynotsocavalierwiththe"lim"symbol;seeBurrill(p.138)foramorecareful,drawn-outdevelopmentofadditionwithCauchysequences. ^Ferreirósp.128 ^Burrillp.140 ^Conway,JohnB.(1986),FunctionsofOneComplexVariableI,Springer,ISBN 978-0-387-90328-6 ^Joshi,KapilD(1989),FoundationsofDiscreteMathematics,NewYork:JohnWiley&Sons,ISBN 978-0-470-21152-6 ^Lipschutz,S.,&Lipson,M.(2001).Schaum'soutlineoftheoryandproblemsoflinearalgebra.Erlangga. ^Riley,K.F.;Hobson,M.P.;Bence,S.J.(2010).Mathematicalmethodsforphysicsandengineering.CambridgeUniversityPress.ISBN 978-0-521-86153-3. ^Thesetstillmustbenonempty.DummitandFoote(p.48)discussthiscriterionwrittenmultiplicatively. ^Rudinp.178 ^Leep.526,Proposition20.9 ^Linderholm(p.49)observes,"Bymultiplication,properlyspeaking,amathematicianmaymeanpracticallyanything.Byadditionhemaymeanagreatvarietyofthings,butnotsogreatavarietyashewillmeanby'multiplication'." ^DummitandFootep.224.Forthisargumenttowork,onestillmustassumethatadditionisagroupoperationandthatmultiplicationhasanidentity. ^Foranexampleofleftandrightdistributivity,seeLoday,especiallyp.15. ^CompareViroFigure1(p.2) ^Endertoncallsthisstatementthe"AbsorptionLawofCardinalArithmetic";itdependsonthecomparabilityofcardinalsandthereforeontheAxiomofChoice. ^Endertonp.164 ^Mikhalkinp.1 ^Akianetal.p.4 ^Mikhalkinp.2 ^Litvinovetal.p.3 ^Virop.4 ^Martinp.49 ^Stewartp.8 References[edit] History Ferreirós,José(1999).LabyrinthofThought:AHistoryofSetTheoryandItsRoleinModernMathematics.Birkhäuser.ISBN 978-0-8176-5749-9. Karpinski,Louis(1925).TheHistoryofArithmetic.RandMcNally.LCC QA21.K3. Schwartzman,Steven(1994).TheWordsofMathematics:AnEtymologicalDictionaryofMathematicalTermsUsedinEnglish.MAA.ISBN 978-0-88385-511-9. Williams,Michael(1985).AHistoryofComputingTechnology.Prentice-Hall.ISBN 978-0-13-389917-7. Elementarymathematics Sparks,F.;ReesC.(1979).ASurveyofBasicMathematics.McGraw-Hill.ISBN 978-0-07-059902-4. Education Begle,Edward(1975).TheMathematicsoftheElementarySchool.McGraw-Hill.ISBN 978-0-07-004325-1. CaliforniaStateBoardofEducationmathematicscontentstandardsAdoptedDecember1997,accessedDecember2005. Devine,D.;Olson,J.;Olson,M.(1991).ElementaryMathematicsforTeachers(2e ed.).Wiley.ISBN 978-0-471-85947-5. NationalResearchCouncil(2001).AddingItUp:HelpingChildrenLearnMathematics.NationalAcademyPress.doi:10.17226/9822.ISBN 978-0-309-06995-3. VandeWalle,John(2004).ElementaryandMiddleSchoolMathematics:Teachingdevelopmentally(5e ed.).Pearson.ISBN 978-0-205-38689-5. Cognitivescience Fosnot,CatherineT.;Dolk,Maarten(2001).YoungMathematiciansatWork:ConstructingNumberSense,Addition,andSubtraction.Heinemann.ISBN 978-0-325-00353-5. Wynn,Karen(1998)."Numericalcompetenceininfants".TheDevelopmentofMathematicalSkills.Taylor&Francis.ISBN 0-86377-816-X. Mathematicalexposition Bogomolny,Alexander(1996)."Addition".InteractiveMathematicsMiscellanyandPuzzles(cut-the-knot.org).ArchivedfromtheoriginalonApril26,2006.Retrieved3February2006. Dunham,William(1994).TheMathematicalUniverse.Wiley.ISBN 978-0-471-53656-7. Johnson,Paul(1975).FromSticksandStones:PersonalAdventuresinMathematics.ScienceResearchAssociates.ISBN 978-0-574-19115-1. Linderholm,Carl(1971).MathematicsMadeDifficult.Wolfe.ISBN 978-0-7234-0415-6. Smith,Frank(2002).TheGlassWall:WhyMathematicsCanSeemDifficult.TeachersCollegePress.ISBN 978-0-8077-4242-6. Smith,Karl(1980).TheNatureofModernMathematics(3rd ed.).Wadsworth.ISBN 978-0-8185-0352-8. Advancedmathematics Bergman,George(2005).AnInvitationtoGeneralAlgebraandUniversalConstructions(2.3 ed.).GeneralPrinting.ISBN 978-0-9655211-4-7. Burrill,Claude(1967).FoundationsofRealNumbers.McGraw-Hill.LCC QA248.B95. Dummit,D.;Foote,R.(1999).AbstractAlgebra(2 ed.).Wiley.ISBN 978-0-471-36857-1. Enderton,Herbert(1977).ElementsofSetTheory.AcademicPress.ISBN 978-0-12-238440-0. Lee,John(2003).IntroductiontoSmoothManifolds.Springer.ISBN 978-0-387-95448-6. Martin,John(2003).IntroductiontoLanguagesandtheTheoryofComputation(3 ed.).McGraw-Hill.ISBN 978-0-07-232200-2. Rudin,Walter(1976).PrinciplesofMathematicalAnalysis(3 ed.).McGraw-Hill.ISBN 978-0-07-054235-8. Stewart,James(1999).Calculus:EarlyTranscendentals(4 ed.).Brooks/Cole.ISBN 978-0-534-36298-0. Mathematicalresearch Akian,Marianne;Bapat,Ravindra;Gaubert,Stephane(2005)."Min-plusmethodsineigenvalueperturbationtheoryandgeneralisedLidskii-Vishik-Ljusterniktheorem".INRIAReports.arXiv:math.SP/0402090.Bibcode:2004math......2090A. Baez,J.;Dolan,J.(2001).MathematicsUnlimited–2001andBeyond.FromFiniteSetstoFeynmanDiagrams.p. 29.arXiv:math.QA/0004133.ISBN 3-540-66913-2. Litvinov,Grigory;Maslov,Victor;Sobolevskii,Andreii(1999).Idempotentmathematicsandintervalanalysis.ReliableComputing,Kluwer. Loday,Jean-Louis(2002)."Arithmetree".JournalofAlgebra.258:275.arXiv:math/0112034.doi:10.1016/S0021-8693(02)00510-0. Mikhalkin,Grigory(2006).Sanz-Solé,Marta(ed.).ProceedingsoftheInternationalCongressofMathematicians(ICM),Madrid,Spain,August22–30,2006.VolumeII:Invitedlectures.TropicalGeometryanditsApplications.Zürich:EuropeanMathematicalSociety.pp. 827–852.arXiv:math.AG/0601041.ISBN 978-3-03719-022-7.Zbl 1103.14034. Viro,Oleg(2001).Cascuberta,Carles;Miró-Roig,RosaMaria;Verdera,Joan;Xambó-Descamps,Sebastià(eds.).EuropeanCongressofMathematics:Barcelona,July10–14,2000,VolumeI.DequantizationofRealAlgebraicGeometryonLogarithmicPaper.ProgressinMathematics.201.Basel:Birkhäuser.pp. 135–146.arXiv:math/0005163.Bibcode:2000math......5163V.ISBN 978-3-7643-6417-5.Zbl 1024.14026. Computing Flynn,M.;Oberman,S.(2001).AdvancedComputerArithmeticDesign.Wiley.ISBN 978-0-471-41209-0. Horowitz,P.;Hill,W.(2001).TheArtofElectronics(2 ed.).CambridgeUP.ISBN 978-0-521-37095-0. Jackson,Albert(1960).AnalogComputation.McGraw-Hill.LCC QA76.4 J3. Truitt,T.;Rogers,A.(1960).BasicsofAnalogComputers.JohnF.Rider.LCC QA76.4 T7. Marguin,Jean(1994).HistoiredesInstrumentsetMachinesàCalculer,TroisSièclesdeMécaniquePensante1642–1942(inFrench).Hermann.ISBN 978-2-7056-6166-3. Taton,René(1963).LeCalculMécanique.QueSais-Je ?n°367(inFrench).PressesuniversitairesdeFrance.pp. 20–28. Furtherreading[edit] Baroody,Arthur;Tiilikainen,Sirpa(2003).TheDevelopmentofArithmeticConceptsandSkills.Twoperspectivesonadditiondevelopment.Routledge.p. 75.ISBN 0-8058-3155-X. Davison,DavidM.;Landau,MarshaS.;McCracken,Leah;Thompson,Linda(1999).Mathematics:Explorations&Applications(TE ed.).PrenticeHall.ISBN 978-0-13-435817-8. Bunt,LucasN.H.;Jones,PhillipS.;Bedient,JackD.(1976).TheHistoricalrootsofElementaryMathematics.Prentice-Hall.ISBN 978-0-13-389015-0. Poonen,Bjorn(2010)."Addition".Girls'AngleBulletin.3(3–5).ISSN 2151-5743. Weaver,J.Fred(1982)."AdditionandSubtraction:ACognitivePerspective".AdditionandSubtraction:ACognitivePerspective.InterpretationsofNumberOperationsandSymbolicRepresentationsofAdditionandSubtraction.Taylor&Francis.p. 60.ISBN 0-89859-171-6. vteElementaryarithmetic    +Addition(+) −Subtraction(−) ×Multiplication(×or·) ÷Division(÷or∕) vteHyperoperationsPrimary Successor(0) Addition(1) Multiplication(2) Exponentiation(3) Tetration(4) Pentation(5) Inverseforleftargument Subtraction(1) Division(2) RootExtraction(3) Super-root(4) Inverseforrightargument Subtraction(1) Division(2) Logarithm(3) Super-logarithm(4) Relatedarticles Ackermannfunction Conwaychainedarrownotation Grzegorczykhierarchy Knuth'sup-arrownotation Steinhaus–Mosernotation AuthoritycontrolGeneral IntegratedAuthorityFile(Germany) Nationallibraries France(data) UnitedStates Retrievedfrom"https://en.wikipedia.org/w/index.php?title=Addition&oldid=1051931847" Categories:AdditionElementaryarithmeticMathematicalnotationHiddencategories:CS1German-languagesources(de)WebarchivetemplatewaybacklinksArticleswithshortdescriptionShortdescriptionmatchesWikidataGoodarticlesCS1French-languagesources(fr)ArticleswithGNDidentifiersArticleswithBNFidentifiersArticleswithLCCNidentifiersArticleswithexampleCcode Navigationmenu Personaltools NotloggedinTalkContributionsCreateaccountLogin Namespaces ArticleTalk Variants expanded collapsed Views ReadEditViewhistory More expanded collapsed Search Navigation MainpageContentsCurrenteventsRandomarticleAboutWikipediaContactusDonate Contribute HelpLearntoeditCommunityportalRecentchangesUploadfile Tools WhatlinkshereRelatedchangesUploadfileSpecialpagesPermanentlinkPageinformationCitethispageWikidataitem Print/export DownloadasPDFPrintableversion Inotherprojects WikimediaCommonsWikiversity Languages AlemannischالعربيةAragonésঅসমীয়াAsturianuAzərbaycancaتۆرکجهবাংলাBân-lâm-gúБашҡортсаБеларускаяБеларуская(тарашкевіца)BikolCentralБългарскиབོད་ཡིགBosanskiBrezhonegБуряадCatalàЧӑвашлаČeštinaChiShonaCymraegDanskDeutschEestiΕλληνικάEspañolEsperantoEuskaraفارسیFrançaisGàidhligGalego贛語ગુજરાતી客家語/Hak-kâ-ngîХальмг한국어Հայերենहिन्दीHrvatskiIdoBahasaIndonesiaInterlinguaᐃᓄᒃᑎᑐᑦ/inuktitutIñupiakIsiXhosaÍslenskaItalianoעבריתJawaಕನ್ನಡҚазақшаKiswahiliKriyòlgwiyannenКыргызчаLatinaLatviešuLietuviųLa.lojban.LombardMagyarМакедонскиമലയാളംमराठीمصرىBahasaMelayuMìng-dĕ̤ng-ngṳ̄မြန်မာဘာသာNāhuatlNaVosaVakavitiNederlandsनेपालभाषा日本語NorskbokmålNorsknynorskNovialOccitanଓଡ଼ିଆਪੰਜਾਬੀپښتوPatoisPolskiPortuguêsRomânăRunaSimiРусскийSängöᱥᱟᱱᱛᱟᱲᱤShqipSicilianuSimpleEnglishسنڌيSlovenčinaSlovenščinaŚlůnskiکوردیСрпски/srpskiSrpskohrvatski/српскохрватскиSuomiSvenskaTagalogதமிழ்తెలుగుไทยTürkçeУкраїнськаئۇيغۇرچە/UyghurcheVahcuenghVènetoTiếngViệtWinaray吴语ייִדישYorùbá粵語中文 Editlinks