千米小说网

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

Chapter1Hownottothinkaboutnumbers

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。

&hesinumbers，wehaveourberbase10=2×5，whichisspeethelessbeingtriangularinthat10=1+2+3+4（rememberten-pinbowlihenhaveapairoftwinprimesin11and13，whicharetwosebersthatarebothprime，separatedbythenumber12，whitrasthasmanyfactorsforitssize。Ihefirstso-calledabundahethesumofitsproperfactors，thoselessthantheself，exceedsthenumberiion:1+2+3+4+6=16。Thenumber14=2×7maylookundisti，astheparadoxicalquipgoes，beiundistinguishednumbermakesitdistierall。In15=3×5，wehaveariangularisthefirstoddistheproductoftwoproperfactors。Ofcourse，16=24isnotonlyasquarebutthefirstfourthpower（after1），makingitveryspedeed。Thepair17aherpairoftwinprimes，ahereadertomaketheirowionsaboutthepeatureofthenumbers18，20，andsoon。Foreaakeae。