addition addition plus symbol basic operations division addition numbers total quantities combined example picture right combination making total observation equivalent mathematical expression add equal counting fruits addition represent combining physical objects systematic addition defined abstract quantities rational numbers real numbers complex numbers abstract objects rules addition involving negative numbers devised addition studied addition important properties meaning order matter meaning adds numbers order addition performed matter repeated addition counting addition change number addition predictable rules concerning related operations performing addition simplest numerical tasks addition small numbers accessible basic task performed young months non human animals primary education students taught add numbers system starting single progressively difficult problems mechanical aids range ancient modern computer research efficient addition continues day terminology addition written plus sign terms result expressed equals sign example formula plus equals formula plus equals formula plus equals formula formula situations addition understood symbol appears column numbers number column usually indicates numbers column added sum written number number followed immediately fraction indicates sum called mixed number example cause confusion contexts denotes instead sum series related numbers expressed capital denotes iteration example formula numbers objects added general addition collectively referred terms terminology carries multiple terms distinguished factors multiplied authors fact renaissance authors consider today property addition rarely terms generally called terminology derives latin addition add english words derived latin turn compound dare root add results thing added likewise increase gets thing increased sum derive latin highest associated appropriate sum positive numbers greater common ancient greeks romans add upward contrary modern practice adding downward sum literally higher date earlier roman writers terms addition operation later middle english terms adding popularized plus sign latin word meaning appears mathematical works dating interpretations addition model countless physical processes simple case adding natural numbers possible interpretations visual representations combining sets possibly fundamental interpretation addition lies combining sets collections combined single collection number objects single collection sum number objects original collections interpretation easy little danger ambiguity useful higher mathematics rigorous definition natural numbers obvious extend version addition include numbers negative numbers possible fix consider collections objects easily divided better rods combining collections segments rods joined end end illustrates conception addition adding rods lengths rods extending length second interpretation addition comes extending initial length given length original length extended given final length sum original length length extension sum interpreted operation combines sense interpreted addition units interpretation parts sum play roles operation viewed applying operation instead calling appropriate case plays passive role view useful discussing addition operation operation vice versa properties addition change order terms sum result symbolically numbers fact addition known law addition phrase suggests laws example law operations division misleading speak law addition adding numbers order operations matter example expression defined mean addition tells choice definition irrelevant numbers true example addition operations order operations important standard order operations addition lower priority roots division given equal priority identity element adding zero number quantity change zero identity element addition known identity symbols law identified wrote separate laws depending negative positive zero words symbols later indian refined concept year wrote zero added corresponding statement 12th century wrote addition quantity positive negative remains corresponding statement successor context addition plays special role greater known successor instance successor successor succession value seen formula successor making addition succession examples successor successor making 2nd successor units numerically add physical quantities units expressed common units example adding gives measure feet extended inches sum inches inches synonymous feet hand usually meaningless try add meters square meters units sort consideration fundamental dimensional analysis performing addition innate ability studies mathematical development starting 1980s exploited phenomenon look longer situations unexpected seminal experiment involving dolls manipulated screen demonstrated month old expect comparatively surprised physical situation imply finding affirmed variety laboratories different experiment older months exploited development motor control allowing retrieve balls box youngest responded small numbers older subjects able sums animals limited ability add particularly experiment imitating result instead dolls performed similarly human dramatically taught meanings arabic able sum training recently asian elephants demonstrated ability perform basic learning addition children typically children master counting given problem requires items items combined young children model situation physical objects fingers drawing count total gain experience learn discover strategy counting asked find plus children count past saying usually fingers arriving strategy universal children easily pick peers teachers discover independently additional experience children learn add quickly exploiting addition counting larger number case starting counting eventually children begin recall certain addition facts number bonds experience facts committed memory children begin derive unknown facts known ones example child asked add seven know reason derived facts found quickly elementary school students eventually rely mixture derived facts add different nations introduce numbers different ages countries teaching addition pre school world addition taught end year elementary school addition table children presented addition table pairs numbers knowing perform addition system addition system fluent recall single digit addition facts facts pattern based strategies people efficient property mentioned pattern reduces number addition facts adding basic task accomplished counting ultimately zero zero identity adding zero trivial nonetheless teaching students introduced addition process increases word problems help exception zero doubles adding number related counting doubles facts form backbone related facts students find relatively easy grasp near doubles sums quickly derived doubles fact adding sums form usually early facts example derived adding making advanced strategy uses intermediate sums involving example students grow older commit facts memory learn derive facts rapidly students commit facts memory find basic fact quickly carry standard adding numbers align vertically add columns starting ones column right column exceeds extra digit carried column example addition digit carry alternate strategy starts adding significant digit left route makes carrying little faster getting rough estimate sum alternative methods addition added simple modification process point location necessary add trailing shorter length longer finally performs addition process point placed answer exactly placed example solved follows scientific scientific numbers written form formula formula formula addition requires numbers scientific represented simply added example formula addition bases addition bases similar addition example consider addition adding single digit numbers relatively simple form carrying carry adding produces digit added column similar happens certain single digit numbers added result equals exceeds value digit left carry carry known carrying result addition exceeds value digit procedure carry excess divided left adding value correct position weight higher factor equal carrying works way carried example added row shows carry bits starting column carried left written column second column right added carried written column time carried written row proceeding like gives final answer computers analog computers work directly physical quantities addition mechanisms depend form mechanical represent positions sliding blocks case added averaging rotation speeds shafts added differential hydraulic add pressures chambers exploiting newton second law balance forces assembly common situation general purpose analog computer add referenced ground accomplished roughly network better design exploits operational addition fundamental operation digital computers efficiency addition particular carry mechanism important limitation overall performance called counting frame calculating tool use centuries adoption written modern system widely merchants traders asia africa dates invented mechanical operational adding machine use gravity assisted carry mechanism operational mechanical 17th century earliest automatic digital computers limited carry mechanism forced wheels turn way add operator use complement required steps addition followed building second functional mechanical calculating clock wood setup numbers automatically execute addition electronic digital computers usually simplest architecture carry follows standard multi digit slight improvement carry skip design following human perform carries computing bypasses group answer practice addition achieved logical operations conjunction operations shown gates straightforward realize digital logic allowing realization circuits turn combined complex logical operations modern digital computers addition typically fastest instruction largest impact performance floating point operations basic tasks address generation memory access instructions branching increase speed modern designs parallel schemes names carry select carry fact hybrids designs unlike addition paper addition computer changes ancient adding board destroyed leaving sum influence mathematical thinking strong early latin texts claimed process adding number number numbers modern times add instruction replaces sum preserves high level programming language evaluating change goal replace sum explicitly requested typically statement languages allow abbreviated computer result addition large store overflow occurs resulting incorrect answer overflow fairly common cause program errors overflow bugs hard discover manifest large input data sets likely tests especially notable error bug overflow errors digit format years caused significant computer problems addition numbers prove usual properties addition define addition context question addition defined natural numbers set theory addition extended progressively larger sets include natural numbers rational numbers real numbers mathematics education positive added negative numbers considered historical route natural numbers popular ways define sum natural numbers defines natural numbers finite sets set number elements set appropriate define sum follows let set sets defined formula union alternate version definition allows possibly overlap takes union mechanism allows common elements separated counted twice popular definition let successor number following natural numbers define define general sum minor variations definition literature taken literally definition application partially ordered set hand sources prefer use restricted applies set natural numbers considers temporarily fixed applies define function operations form operation addition developed early expand following decades proved properties mathematical induction simplest conception consists absolute value natural number sign generally positive negative zero special case positive negative corresponding definition addition proceed cases let absolute value let zero treat identity positive define negative define different signs define difference sign term absolute value larger example different signs absolute values negative term larger answer negative definition useful concrete problems far complicated produce elegant general cases consider convenient conception group construction essential observation expressed uniquely difference natural numbers define difference natural numbers addition defined compatible given natural numbers define rational numbers addition rational numbers common simpler definition involves addition define formula example sum formula addition simpler case simply add leaving formula formula rational addition easy consequence laws rigorous general discussion field real numbers common construction set real numbers completion set rational numbers real number defined cut non set closed downward greatest element sum real numbers defined element element define formula definition published slightly modified form real addition immediate defining real number set negative easily seen identity probably construction pertaining addition definition unfortunately dealing cuts time consuming case case process similar addition signed approach metric completion rational numbers real number essentially defined limit sequence addition defined term term define formula definition published slightly different prove operation defined dealing sequences task properties real addition follow immediately properties rational numbers furthermore operations including straightforward analogous definitions complex numbers complex numbers added adding real imaginary parts formula complex numbers complex plane addition following geometric interpretation sum complex numbers interpreted points complex plane point obtained building point operations viewed addition operation real numbers field abstract centrally concerned operations appear set theory category theory addition abstract addition linear space structure allows adding scaling familiar space set ordered pairs real numbers ordered pair interpreted origin plane point plane sum obtained adding individual coordinates addition operation central classical mechanics interpreted forces matrix addition matrix addition defined dimensions sum pronounced denoted matrix adding corresponding elements formula example formula set elements addition operation central musical set theory set elements addition operation known logic exclusive function geometry sum angle measures taken sum real numbers amounts addition operation circle turn addition operations dimensional general addition general theory abstract allows addition operation operation set basic structures addition operation include groups addition set theory category theory far reaching addition natural numbers addition numbers cardinal numbers set theory different addition natural numbers unlike addition operations addition numbers addition cardinal numbers operation closely related union operation category theory union seen particular case operation general abstract addition direct sum wedge sum named evoke connection addition related operations addition division considered basic operations elementary thought kind addition addition sort addition adding functions given set addition operation define corresponding operation set set natural numbers simple example hand operation uniquely determines addition operation operation identity reason group described set closed thought repeated addition single term appears sum times sum product natural number product sense example yields number real complex numbers addition function identity allows carried consulting table computing addition hand enables slide rule formula good order broad context lie groups relates group elements addition associated lie addition general operations distribute addition requirement definition ring contexts addition existence identity uniquely determine operation property provides information addition expanding product ways concludes addition forced reason ring addition general division operation remotely related addition division right addition division left addition ordering maximum operation max operation similar addition fact numbers different orders magnitude sum approximately equal maximum extremely useful applications mathematics example taylor series presents perpetual difficulty numerical analysis essentially max greater straightforward calculation accumulate unacceptable round error returning zero loss significance exact kind infinite limit infinite cardinal number cardinal sum exactly equal greater accordingly operation infinite cardinals like addition furthermore addition preserves ordering real numbers addition max way addition max max reasons tropical geometry replaces addition addition context addition called tropical called tropical addition tropical identity negative infinity authors prefer replace addition identity positive infinity tying observations tropical addition approximately related regular addition log max log log accurate base increases exact constant named analogy constant quantum mechanics taking classical limit tends zero formula sense maximum operation version addition ways add known successor operation addition number describes addition numbers usually includes idea sum single number sum zero infinite delicate procedure known series counting finite set equivalent set integration kind precisely generally integration zero dimensional reduces linear combinations combine sums term usually real complex number linear combinations especially useful contexts straightforward addition violate rule mixing strategies game theory states quantum mechanics add independent random defined distribution functions usual definition combines integration general useful kind domain addition contrast addition kind range addition
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