number

Birthanddevelopment

"Number"istheconceptofmeasuringthings.Itistheconsciousexpressionofthequantitythatexistsobjectively."Numbers"originatedfromthesymbolsusedbyprimitivehumanstocountandformnaturalnumbers"numbers".Itisoneofthegreatestinventionsofmankind,andisthebasisforhumanstoaccuratelydescribethings.Inthelonghistoryofmankind,

1°obtainednumbersbycountingrealthings;

2°numberscanbecalculatedincertainways;

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3°Whenthenumberisrelatedtospacethings,itcanindicatethenumberofthesethings.(Fromtheoriginalnumbertheoryofnaturalnumbers)

Afewyearsago,inordertosurvive,humanancestorsoftenlivedingroupswithdozensofpeopletogether.Theyworktogetherduringtheday,huntingforbeasts,birds,orcollectingfruitfood;atnight,theyliveincavesandsharetheirlaborincome.Inthelong-termjointlaborandlife,theygraduallyreachedapointwheretheyhadtosaysomething,andlanguagecameintobeing.Theycanusesimplelanguagemixedwithgesturestoexpressfeelingsandexchangeideas.Withthedevelopmentoflaborcontent,theirlanguagehasalsocontinuedtodevelop,finallysurpassingthelanguageofallotheranimals.Oneofthemainsignsisthatlanguagecontainsthecolorofarithmetic.

Theyhuntedandreturned,andtheirpreywaseitherwithorwithout,sothereweretwoconceptsof"being"and"nothing".Forafewdaysinarow,ifthereisnoanimaltocatch,therewillbenomeattoeat,andtheconceptsof“have”and“nothing”willgraduallydeepen.

Later,gregariousnessdevelopedintotribes.Thetribeismadeupoffamilieswithfewmembers.Theso-called"being"isdividedintofourtypes:"one","two","three",and"multiple"(sometribesdon'tevenhave"three").Anynumbergreaterthan"three",theyareunderstoodas"many"or"abunch"or"agroup".Althoughsomechiefsareelders,theycan’ttellhowmanykindsofbeastshehascapturedandhowmanykindsoftreeshehasseen.Chantitout.However,nomatterwhat,theycanalreadyusetheirhandstosaysomethinglikethis(withonefingerpointingtothedeer,threefingerspointingtothearrow):"Toexchangeformeadeer.Youhavetogivemethreearrows."Thisisanarithmeticthattheydidn'thaveatthetime.knowledge.

About10,000yearsago,theglacierretreated.SomeStoneAgehunterswhowereengagedinnomadismstartedanewwayoflifeinthemountainsoftheMiddleEast-farminglife.Theyencounteredproblemssuchashowtorecordthedate,season,andhowtocountthenumberofgrainsandseedsinthecollection.EspeciallywhenmorecomplexagriculturalsocietiesweredevelopedintheNileValley,TigrisandEuphratesriverbasins,theyalsoencounteredtheproblemofpayingtaxes.Thisrequiresthenumbertohaveaname.Andthecountingmustbemoreaccurate,only"one","two","three","more"isfarfromenough.

BetweentheTigrisandtheEuphratesandaroundthetworivers,itiscalledMesopotamia.Therewasaculturethat,liketheEgyptianculture,isalsooneoftheoldestintheworld.AlthoughtheMesopotamiansandtheEgyptianswerefarapart,theyestablishedtheearliestsystemofwritingnaturalnumbersinthesameway-makingmarksontreesorstonestorecordthepassingdays.Althoughtheshapesofnumbersaredifferent,theyhavesomethingincommon.Theyallusesinglestrokestorepresent"one".

Later(especiallyaftertheysettledinthevillage),theygraduallyreplacedthenotcheswithsymbols,thatis,onesymbolrepresentsonething,twosymbolsrepresenttwothings,andsoon.Thecountingmethodlastedforalongtime.About5000yearsago,Egyptianpriestshadwrittennumbersymbolsonakindofstrawpapermadeofreeds,whileMesopotamianpriestshadwrittenthemonsoftclaytablets.Inadditiontostillusingsinglestrokestorepresent"-",theyalsouseothersymbolstorepresent"+"orlargernaturalnumbers;theyrepeatedlyusethesesinglestrokesandsymbolstorepresenttherequirednumbers.

In1500BC,thePeruvianInca(apartoftheIndians)inSouthAmericawereusedto"knotcounting"-everytimeabundleofcropswerecollected,theytiedaknotontheropeandusedaknot.Torecordtheharvest."Knot"hasthesameeffectasmarks,andisalsousedtoexpressnaturalnumbers.Accordingtotherecordoftheancientbook"IChing"inmycountry,theChineseinancienttimesalso"ruledbyknottingtherope",whichusedthemethodoftyingknotsontheropetorecordthenumberofevents.Later,itwaschangedto"bookdeed",thatis,aknifewasusedtomakemarksonbambooorwoodtocountnumbers.Useastroketorepresent"one".Tothisday,weChinesestilloftenusetheword"zheng"tocountnumbers.Eachstrokerepresents"one".Ofcourse,theword"zheng"alsocontainsthemeaningof"everyfiveadvancesintoone".

Naturalnumbers

Whencountingthings,thenumbers0,1,2,3,4,5,6,7,8,9,...arecallednaturalnumbers.

Naturalnumbershavetwomeanings:quantityandorder,whicharedividedintocardinalnumbersandordinalnumbers.Thebasicunitis1,andthecountingunitsareone,ten,onehundred,onethousand,tenthousand,etc.

Classification

Accordingto"canbedivisibleby2",itcanbedividedinto:oddandeven.

Accordingtothe"numberoffactors",itcanbedividedinto:primenumbersandcompositenumbers.

Usedtomeasurethenumberofthingsorthenumberthatindicatestheorderofthings.Thatis,thenumbersrepresentedbythenumbers0,1,2,3,4,...Naturalnumbersstartfrom0,oneafteranother,forminganinfinitegroup.Thenaturalnumbersethasadditionandmultiplicationoperations.Theresultofaddingormultiplyingtwonaturalnumbersisstillanaturalnumber,anditcanalsobesubtractedordivided,buttheresultsofsubtractionanddivisionarenotnecessarilynaturalnumbers,sothesubtractionanddivisionoperationsareinthenaturalnumbersetItisnotalwayspossible.Naturalnumbersarethemostbasiccategoryofallnumbersthatpeopleknow.Inordertomakethenumbersystemhavearigorouslogicalfoundation,mathematiciansinthe19thcenturyestablishedtwoequivalenttheoriesofnaturalnumbers:theordinalnumbertheoryandthecardinalnumbertheoryofnaturalnumbers.Theconcept,operationandrelatedpropertiesofnaturalnumbersarestrictlydiscussed.

Ordinalnumbertheory

ItwasproposedbytheItalianmathematicianGiuseppePeano.Hesummarizedthepropertiesofnaturalnumbersandgavethefollowingdefinitionsofnaturalnumbersusingaxioms.

NaturalnumbersetNreferstothesetthatmeetsthefollowingconditions:

INcontainsoneelement,denotedas1.

IINeveryelementacanfindanelementinNasitssuccessor,rememberAsa'.

Ⅲ0'=1.

IV0isnotthesuccessorofanyelement.

ⅤDifferentelementshavedifferentsuccessors.

Ⅵ(Inductionaxiom)ForanysubsetMofN,if1∈M,andaslongasacanbelaunchedinManda'isalsoinM,thenM=N.

Cardinalnumbertheory

Thecardinalnumbertheorydefinesnaturalnumbersasthecardinalnumbersofafiniteset.Thistheoryproposesthattwofinitesetsthatcanestablishaone-to-onecorrespondencebetweenelementshaveacommonThisfeatureiscalledcardinality.Inthisway,allsingleelementsets{x},{y},{a},{b},etc.Havethesamebase,denotedas1.Similarly,anysetthatcanestablishaone-to-onecorrespondencewithtwofingershasthesamebase,whichisrecordedas2,andsoon.Theadditionandmultiplicationofnaturalnumberscanbedefinedinordinalorcardinalnumbertheory,andtheoperationsunderthetwotheoriesareconsistent.

Naturalnumbersplayaveryimportantroleindailylife,andpeopleusenaturalnumbersextensively.Naturalnumbersaretheearliestnumbersinhumanhistory.Naturalnumbershaveawiderangeofapplicationsincountingandmeasurement.Peopleoftenusenaturalnumberstolabelorrankthings,suchascitybusroutes,housenumbers,andzipcodes.

Whether"0"isincludedinnaturalnumbersiscontroversial.Somepeoplethinkthatnaturalnumbersarepositiveintegers,thatis,countingfrom1;whileothersthinkthatnaturalnumbersarenon-negativeintegers,thatis,countingfrom0.Thereisnoconsensusonthisissue.However,innumbertheory,theformerismoreadopted;insettheory,thelatterismoreadopted.Inourcountry’sprimaryandsecondaryschooltextbooks,0isclassifiedasanaturalnumber.

Naturalnumbersareintegers,butnotallintegersarenaturalnumbers.

Forexample:-1,-2,-3,...areintegers,notnaturalnumbers.

Inshort,anaturalnumberisanintegergreaterthanorequalto0.

Thesetofallnon-negativeintegersiscalledthesetofnon-negativeintegers(thatis,thesetofnaturalnumbers).

Classificationofnumbers

Wecombineallnon-negativeintegerssuchas0,1,2,3,4,5,6,7,8,9,10,etc.Called"naturalnumbers".Expand1,2,3,...,9,10forwardtoobtainpositiveintegers1,2,3,...,9,10,11,...,andexpanditbackwardtoobtainnegativeintegers...,-11,-10,-9,...,-3,-2,-1,the"0"betweenthepositiveandnegativeintegersisaneutralnumber;putthemtogethertoget...,-11,-10,-9,...,-3,-2,-1,0,1,2,3,...,9,10,11,...arecalledintegers.Fouroperationscanbeperformedonintegers:addition,subtraction,multiplication,anddivision,whicharecalledthefourarithmeticoperations.Integersformaclosedsetofnumbersforaddition,subtraction,andmultiplicationoperations,andaretheobjectofresearchintheancientbranchofmathematics"numbertheory".ThefamousGermanmathematicianGausssaid:"Mathematicsisthequeenofscience,andnumbertheoryisthecrownofmathematics."Divisionoperations,suchas7/11=0.636363…,11/7=1.5714285…,arenolongerintegers,whichmeansthatintegersarenotclosedtodivisionoperations.Inordertomakethenumbersetclosedforthefourarithmeticoperationsofaddition,subtraction,multiplication,anddivision,newnumbersmustbeadded,suchas7/11,11/7,whicharetheratiooftwointegers,calledcomparablenumbers,fractions,andnowCommonlyknownasrationalnumbers.

Summarizeandorganizethepropertiesofnumbers,thefourarithmeticoperationsbetweennumbersandtheexperienceintheapplicationprocesstoformtheoldestmathematics-arithmetic.Thesetofrationalnumbersformsaclosedsetofnumbersforthefourarithmeticoperationsofaddition,subtraction,multiplication,anddivision,whichseemstobeverycomplete.Morethan2500yearsago,manypeople,andevensomemathematiciansatthattime,thoughtthisway.

Inthe5thcenturyBC,thePythagoreanschoolatthattimeattachedgreatimportancetointegersandwantedtouseittoexplaineverything."Numbersarethefoundationofallthings"becametheirphilosophy.ThediscoveryofirrationalnumberswasafatalblowtothePythagoreanphilosophybasedonintegers.Inthehistoryofmathematics,thisincidentwascalledthe"firstmathematicscrisis."Afterthat,manyirrationalnumberswerediscovered,andthepiratioisthemostimportantone.Inthe15thcentury,thefamousItalianpainterLeonardocalleditthe"irrationalnumber".Now,peoplecombinerationalnumbersandirrationalnumberstogetherandcallthem"realnumbers".Fromthiswegettwosolutions:sum,aretheystillsolutionsof(2)?Ifyouthinkitisnot,thereisnosolutionto(2),andsolvingtheequationislikeenteringadeadend.Inordertosolvethisproblem,mathematicianshavetoexpandtherangeofnumbersagain,introducingthesymbol"i"torepresent"squarerootof-1",thatis,,whichiscalledimaginarynumber;andthenrealnumbera,bandimaginarynumbersarecombinedtoformaformofnumber,called"pluralnumber".Foralongperiodoftime,peoplecouldnotfindthequantityrepresentedbyimaginaryandpluralnumbersinreallife,whichmadepeoplefeelabitillusory.Withthedevelopmentofscience,imaginarynumbershavebeenwidelyusedinthefieldsofhydraulics,graphics,andaviation.Inthisway,thefamilyofnumbersisfurtherexpandedtoincludethetwocategoriesofrealnumbersandimaginarynumbers,andtheaddition,subtraction,multiplication,anddivisionareextendedtoincludepowersandsquareroots,forminganewbranchofmathematics"algebra".Algebrahasfurtherdevelopedintwoaspects.Oneistostudyasystemoflinearequationswithmoreunknowns,andtointroducesymbolsandconceptssuchasmatrices,vectors,andspacestoform"linearalgebra";theotheristostudyhigher-orderequationswithhigherdegreesofunknownstoform"PolynomialAlgebra"(alsocalled"PolynomialTheory").Inthisway,theobjectofalgebraresearchisnotonlynumbers,butalsomatrices,vectors,vectorspacesandtheirtransformations.Theycanallbe"arithmetic",althoughitisalsocalledadditionormultiplication,butthebasicoperationlawsaboutnumbersaresometimesnolongervalid.Therefore,thecontentofalgebracanbesummarizedasasetofsomealgebraicstructureswithoperations,suchasgroups,rings,fields,etc.,aswellasmanybranchessuchasabstractalgebra,Booleanalgebra,relationalalgebra,andcomputeralgebra.DuetothedevelopmentofscienceandtechnologyAsneeded,therangeofnumberscontinuestoexpand,frompositiveintegers,naturalnumbers,integers,realnumberstocomplexnumbers,andthentovectors,tensors,matrices,groups,rings,domains,andsoon.Forthesakeofdistinction,peoplecallrealnumbersandcomplexnumbers"narrownumbers",andvectors,tensors,andmatricesarecalled"generalizednumbers".Althoughpeoplestillhavedifferentopinionsonhowtoclassifynumbers,theyalladmitthattheconceptofnumberswillcontinuetoexpandanddevelop.

Thestorageformatofnumbers

Thestorageformatofnumbersisthestorageorderofnumbers.Whenexpressingthesizeofthevalue,onebyte(byte)canonlyexpress255(0xFF)atmost,whichisfarfromenough.Inordertomeettheactualneedsofuse,usually2,4or8bytesareusedtorepresentthesizeofthevalue.Inthecaseofusingmultiplebytestorepresentvalues,thereisanorderproblem.Therearetwostoragesequencesfornumbers-Big-endianformatandLittle-endianformat.

Theunitofnumber

TheYellowEmperoristhelaw,andnumberhastens.It'sveryuseful,buttherearethreeYan.Tenth-class,100million,trillion,Jing,Gai,Zi,soil,ditch,Jian,Zheng,Zai.Thethirdclassiscalledupper,middleandlower.Thenextnumberwillbechangedintentoten.Iftentrillionisabillion,abillionisatrillion,andtentrillionisadayinBeijing.Inthemiddleofthenumber,everythingchanges,ifyousayamillionisabillion,atrillionisatrillion,andamillionisacapital.Thosewhoareinthetopcountwillchangeiftheyarepoor.Frombillionstoyears,finallygreatdevelopment.Thecountissimpleandshort,buttheplanisendless.Onthenumberofmacros,theworldisnotavailable.Therefore,FuYe,preferredtothemiddlenumberears.

YuShiasked:"Mister'swords,thenumberofpeoplewhoarepoorwillchange.Nowthatthecloudfinallyevolves,theevolvementislimited,sowhyisit[infinite]poor?"

Thegentlemanlaughedandsaid:"Gaiweithinksaboutears.Countingforuse,emphasisonwordswillchange,makesmallandbig,addcirculation.Theprincipleofcirculationisendless.Almost."

Forthosewhoarebothsmallandbig,pleaseaddDong's"ThreeGradesofShushu".Addingandupdatingisannoying,so[omitted]Yan.

ChineseclassicsfromtheHanDynastytotheQingDynastyhavealwaysbeeninthemillion-digitsystem,butJapan's"DustandTribulation"adoptsthemillion-digitsystem.TherearealsodifferencesbetweenChinaandJapan:China’slargestisinfinitenumbers,andthesmallestisnet.Japandoesnothaveinfinitenumbers.Instead,itisinfinite,whichisthelargestItislargenumber,andthesmallestisangstroms.InChina,therearenomulti-syllablenumeralssuchasempty,clean,andalaya.ZhuShijie,amathematicianintheYuanDynastyofChina,andhis"EnlightenmentofMathematicalStudies"creatively(perhapsreferringtothe"HuaYanJing"and"TheLawofMonks")inheritedtheMathematicianXuYueoftheEasternHanDynastyandtheTangandSongDynasties.ThemathematicianXieChawei’s"FaMengShuJing"expandstheChineselargenumbersanddecimalsto10^±128(10¹⁰⁰meansWanGangSha,10^-100meanswanxu).InXuYue's"SuccessofNumbers","numbersareused,theemphasisofwordschanges,thesmallandthebig,andthecycleisadded.Theprincipleofcycle,thereisnolimit."AndtheexpansionofChina'slargenumbersanddecimalsatthesametimeIt's10^±∞.ThesearebeyondthereachofJapan's"DustTribulation".