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Chapter 1 How not to think about numbers（第1页）

Chapter1Hownottothinkaboutnumbers

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Weareallusedtoseeiendown，andtsomemeaningfromthem.However，anumeralsudtherepreseohing.InRomannumerals，forexample，ritethenumbersixasVI，butwerealizethatthisstandsforthesameiswrittenas6iion.Bothsymbolizesofthekiosixtallymarks:IIIIII.Weshallfirstspendalittletimegthedifferentresentandthinkaboutnumbers.

&imessolvenumberproblemsalmostwithit.Forexample，supposeyouaregameetingandyouwahateveryoherehasacopyoftheagenda.Youdealwiththisbylabellingeachcopyofthehandoutinturialsofeachofthosepresent.Aslongasyoudonotrunoutofcopiesbeforepletingthisprocess，youwillknowthatyouhaveasuffiumbertogoaround.Youhavethehisproblemwithtoarithmetidwithoutexpliting.Thereareworkforushereallthesameandtheyallowpreparisoionwithahoughthemembersthatmakeupthescouldhaveentirelydifferentcharacters，asisthecasehere，whereoisaofpeople，whiletheothersistsofpiecesofpaper.

Whatnumbersallowustodoistoparetherelativesizeofohanother.

Intheprevioussarioyouhertoanypeoplewerepresentasyoudidoknow–yourproblemwastodetermithenumberofcopiesoftheagendawasatleastasgreatasthenumberofpeople，andthevalueofthesenumberswasnotrequired.Youwill，however，akeaberpresentwhenyouorderlunchforfifteeaiestotottingupthebillforthatmeal，someonewillmakeuseofarithmetictoworkouttheexactifthesumsarealldoneonacalculator.

Ourmoderemallowsustoexpressnumbersianduniformmanner，whichmakesiteasytoberwithaoperformthearithmeticaloperationsthatarisethroughg.Io-dayworld，weemploybasetenforallourarithmetic，thatistosaywetbytefortheatalreasoendigitsonourhands.Whatmakesouremsoeffective，however，isnotourparticularchoiceofbasebutrathertheuseofpositionalvalueiniohevalueofanumeraldependsonitsplathering.Forexample，1984isshortfor4onesplus8lotsoftenplus9hundredsplus1thousand.

Itisimportaandheenumbersinparticularways.Inthischapter，wewillthinkaboutwhat，discoverdifferentapproaeetaveryimportaofheprimes），andintrodupleridingthem.

Howgwassortedout

Itiswafewmomentstoappreciatethattherearetwodistiheprocessofbuildingagsystembasedon，foriwobasictasksthatweimposeonarerememberinghowtorecitethealphabetandlearninghowtot.Theseprocessesaresuperficiallysimilarbutyethavefualdifferences.eisbasedoeralphabetand，roughlyspeakierdstoasouospeakwords.IislytruethattheEnglishlanguagehasdevelopedsothatitbewrittenusiof26symbols.However，ilediariesunlessweassigoouralphabet.Thereisnopartiaturalorderavailableawehavesettledonandallsinginschool，a，b，c，d，...seemsveryarbitraryiobesure，themorefrequeersgenerallyothefirsthalfofthealphabet，butthisisuideratherthaheoerssandt，forexample，soundingofflateintherollcall.Bytrast，theumbers，ornaturalheyarecalled，1，2，3，...etousinthatorder:forexample，thesymbol3ismeanttostahatfollows2andsohastobelistedasitssuccessor.toapoint，makeupafreshnameforeachsuumber.Sooer，however，wehavetogiveupandstartgroupiordertohaheunendingsequence.Groupingbytehefirststageofdevelopingasouem，andthisapproachhasbeenhroughouthistoryandacrosstheglobe.

Therewas，however，muchvariatioheRomansystemfavatheringbyfivesasmus，withspecialsymbols，VandL，forfiveandforfiftyrespectively.TheAemwassquarelybasedbytens.Theywouldusespecificletterstostandforimesdashedtotellthereaderthatthesymbolshouldbereadasaherthaerinsomeordinaryword.Forexample，πstoodfor80andγfor3，sotheymightwriteγπtodehismaylookequallyaseffidihesameasournotation，butitisnot.TheGreeksstillmissedthepoiiohevalueofeachoftheirsymbolswasfixed.Inparticular，γπcouldstillohesamenumber，3+80，whereasifweswitchtheorderofthedigitsihedifferentnumber38.

IntheHindu-Arabicsystem，thesedstageofionwasattaihebigideaistomakethevalueofasymboldepeupoothestring.Thisallowsustoexpressahjustafixedfamilyofsymbols.Wehavesettledoennumerals0，1，2，···，9，sothenormalemisdescribedasbaseten，butwecouldbuildouremupfrerorasmallerofbasicsymbols.Wemahasfewastwonumerals，0and1say，whichiswhatisknownasthebienusedinputing.Itisnotthechoiceofbasesize，however，thatwasrevolutionarybuttheideaofusingpositiorainformationabouttheidentityofyournumbers.

Forexample，riteanumberlike1905，thevalueofeachdigitdependsonitsplatherihereare5units，9lotsofonehundred（whichis10×10），aofohousand（whichis10×10×10）.Theuseofthezerosymbolisimportantasaplaceholder.Inthecaseof1905，notributiohe10’splace，butweorethatandjustwrite195ihatrepreseirelydifferentnumber.Iringofdigitsrepresentsadiffereisforthatreasonthathugenumbersmayberepresentedbyshs.Forinstanassigoeveryhumanbeihusingstringsendigitsandinthisersooeveryindividualbelongingtothishugeset.

&hepastsometimesuseddiffereheirwritingofhatismuchlesssignifithefaearlyallofthemlackedatruepositiohfulluseofazerosymbolasaplaceholder.InviewofhowveryahecivilizationofBabylon，itisremarkablethattheyamongthepeoplesoftheaworldcameclosesttoapositionalsystem.

&，however，fullyembracetheuseofthenot-so-naturalnumber0aheemptyregisterinthefinalpositionthewaywedotodistinguish，forexample，830from83.

&ualhurdlethathadtobeclearedwastherealizationthatzerowasindeedaedly，zeroisnotapositiveisahesameanduntilweiooureminafullytmanner，weremainhahisalstepwastakeninIndiainaboutthe6thturyAD.Ouremisdu-ArabicasitwasuniIndiatoEuropeviaArabia.

Livingwithandwithoutdecimals

Adoptingaparticularbaseforaemisalittlelikeplagaparticulargridsap.Itisnotintrinsictotheobjectbutisratherakintoasystemofatesimposedontopasaoftrol.Ourchoiceofbaseisarbitraryiheexclusiveuseofbasetensaddlesusallwithablihesetofumbers:1，2，3，4，···.Onlybyliftingtheveilweseeofaceforwhattheytrulyare.Wheionapartiumber，letussayforexampleforty-nine，allofushaveamentalpictureofthetwonumerals49.Thisissomewhatunfairtothenumberiionasweareimmediatelytypegforty-nineas（4×10）+9.Since49=（4×12）+1，itmayjustaseasilybethoughtofthatwayand，indeed，iy-hereforebewrittenas41，withthenumeral4nowstandingfor4lotsof12.Hivesthey-scharacteristhatitequalstheproduownasthesquareof7.Thisfacetofitspersonalityishighlightedihenthey-edas100，the1nowstandiof7×7.Wewouldbeequallyeouseanotherbase，suchastwelve，forourem:theMayayandtheBabyloy.Ihenumber60isagoodchoiceforagbaseas60hasmanydivisthesmallestnumberdivisiblebyallthenumbersfrhto6.Arelativelylargenumbersuchas60hasthedisadvatouseitasabasewouldrequireustointroduce60separatesymbolstostahenumbersfromzerouptofifty-nine.

Onenumberisafaotherifthefirstnumberdividesintotheseberoftimes.Forexample，6isafactorof42=6×7but8isnotafactorof28as8iimeswitharemaihepropertyofhavingmanyfactorsisahaohaveforthebaseofyourem，elvemayhavebeeerchoitenforournumberbaseas12has1，2，3，4，6，and12asitslistoffactorswhile10isdivisibleonlyby1，2，5，and10.

&ivenessandsheerfamiliarityofouremembuesuswithafalsedwithsomeinhibitions.ierwithasihanwithaicalexpression.Forexample，mostpeoplewouldrathertalkabout5969than47×127，althoughthetwoexpressiohesamething.outtheanswer’，5969，dowefeelthatwe‘have’thenumberandlookitihereis，however，aofdelusioninthisaswehaveohenumberasasumofpowersoften.Thegeneralshapeoftheherpropertiesferredmorefromthealternativeformwherethenumberisbrokendoroductoffactors.Tobesure，thisstandardform，5969，doesallowdireparisonwithotherareexpressediitdoeshefullhenumber.InChapter4，youwillseeonereasoorizedformofanumberbemuchmorepreitsbaseteion，vitalfactorshidden.

&agethattheasdidhaveoverusisthattheywererappedwithiylemiberpatterns，itwasnaturalforthemtothinkintermsofspeetricpropertiesthatapartiumbermayormaynotenjoy.Forexample，numberssuchas10ariangular，somethingthatisvisibleththetriangleofpinsinten-pinbowliriangularrackoffifteenredballsihisishatindfromthebasetendisplaysofthesenumbersalohefreedomtheasenjoyedbydefaultturebygasideourbasetenprejudidtelliwearefreetothinkofnumbersinquitedifferent>

Haviedourselvesinthisway，wemightchoosetofofactorizationsofaistosaythewaythenumberberodualleripliedtogether.Factorizatiohingofthenumber’siure.Ifwesuspeofthinkingofnumberssimplyasservantsofsderdtakealittletimetostudythemintheirhtwithoutreferehingelse，muchisrevealedthatotherwisewouldremaihenaturesofindividualnumbersifestthemselvesiernsinnature，moresubtletharianglesahespiralheadofasunflower，whichrepresentsaso-calledFibonaumber，ahatwillbeintroduChapter5.

Aglaheprimenumbersequence

&hegloriesofnumbersissoself-evidentthatitmayeasilybeoverlooked–everyohemisunique.Eaumberhasitsownstructure，itsowncharacterifyoulike，ayofindividualnumbersisimportantbeapartiumberarises，itsnaturehascesforthestructureofthetowhiumberapplies.Therearealsorelatioweerevealthemselveswhenwecarryoutthefualionsofadditionandmultipli.yumbergreaterthan1beexpressedasthesumofsmallernumbers.Hoestartmultiplyiher，wesooherearesomeurnupastheaooursums.Theseheprimesahebuildingbloultipli.

Aprimenumberisanumberlike7or23or103，whichhasexactlytwofactors，thosenecessarilybeing1andtheself.（Theworddivisorisalsousedasaivewordforfactor.）Wedonott1asaprimeasithasoor.Thefirstprimethenis2，whichistheonlyevehefollowingtrioofoddnumbers3，5，and7areallprime.erthan1thatarenotprimearepositeastheyareallerhenumber4=2×2=22isthefirstber;9isthefirstoddber，and9=32isalsoasquare.Withthenumber6=2×3，wehavethefirsttrulyberinthatitisposedoftwodifferentfactorsthataregreaterthan1butsmallerthantheself，while8=23isthefirstpropercube，whichisthewordthatmeansthatthenumberisequaltosomenumberraisedtothepower3.