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Chapter 6 Below the waterline of the numbericeberg（第1页）

Chapter6Belowthewaterlihenumbericeberg

Introdu

&ralportionofthenumberlinenear0

However，oatsofthe19thturywasthefullrealizatioruedomaioratheristwo-dimensional。Theplahebersisthenaturalarenaofdisuatics。Thishasbeenbroughthometomathematidstiststhroughproblemsolviocarryouttheiioosolvereal-worldproblems，manyofwhichseemtobeonlyaboutordinaryumbers，itbeesoexpandyournumberhorizoionastohowthisextradimensieswilletowardstheendofthisdbeexploredfurtherinChapter8。

Plusesandminuses

&egersisthenameappliedtothesetofallwholeiveive，ahisset，oftensymbolizedbytheletterZ，isthereforeihdires:

{···-4，-3，-2，-1，0，1，2，3，4，···}。

&egersareoftenpicturedaslyingatequallyspatsalongahorizontalnumberliheorderiheadditioweoknowiodoarithmeticwiththeintegersbesummarizedasfollows:

（a）toaddativeinteger，-m，wemovemspacestotheleftiion，aherightforsubtra;

（b）tomultiplyanintegerby-m，wemultiplytheintegerbym，andthengesign。

Inotherwords，thedireofadditionandsubtraegativeheoppositetothatofpositivenumbers，whilemultiplyinganumberby-1ssitssigive。Forexample，8+（-11）=-3，3×（-8）=-24，and（-1）×（-1）=1。

Youshouldroubledbythislastsum。First，itisreasomultiplyiivenumberbyapositiveoneyieldsaivea（a）issubjeterest（apositivemultipliergreaterthaeisgreaterdebt，thatistosayalargerivenumber。Weareallwellawareofthis。Thatmultipliofaiveherivenumbershouldhavetheoppositeoute，thatisapositiveresult，wouldthenappeart。Thefactthattheproductoftwoivenumbersispositivereadilybegivenformalproof。Theproofisbasedoioourexpaemoftheiosubsumetheihenaturalhattheaugmeemshoulduetoobeyallthenormalrulesofalgebra。Iooftwoivesfollowsfromthefayipliedbyzeroequalszero。（Thistooisnotanassumptionbutratherisalsoaceofthelawsofalgebra。）For>

-1×（-1+1）=-1×0=0;

&henmultiplyoutthebrackets，weseethatiheleft-handsideequalzero，（-1）×（-1）musttaketheoppositesignto（-1）×1=-1;inotherwords（-1）×（-1）=1。

Fradrationals

andsowerecyptiaion:

Applyingthiswithm=9，p=4，q=5immediatelygivesus

Thiskindoftrickisofteosimplifyahatinvolvesaingprople，siderthefollowier:

Bysquaring，andthensquaringagai-handsidebeesa4，whiletheexpressiogives:

&followsthe5isanothercopyoftheexpressiohata4=20asothata3=20or，ifyouprefer，aisthecuberootof20。WewillthisteiqueagaininChapter7wheroduceso-tiions。

&heclassoffrasprovideuswithallthenumberswecouldeverioheofallfras，togetherwiththeirhesetofnumbersknowionals，thatisallresultfromwholeheratiosbetweeheyareadequateforarithmetithatanysuminvolvingthefourbasicarithmeticoperationsofaddition，subtraultipliddivisioakeyououtsidetheworldofrationalnumbers。Ifywiththat，thissetofnumbersisallwerequire。However，weexplaiseberssuchasaabovearenotrational。

Irrationals

Argumentsalongtheselinesallowustoshowthatquitegenerally，whehesquareroot（orihecubeherroot）ofaheaawholenumber，isalwaysirrational，thusexplainingwhythedecimaldisplaysonyourcalevershpatterocalculatesucharoot。

Thisproblemremaiouclassicaltimes。Thatthecuberootof2liesoutsidetheraheeutoolswasoledin1837byPierreWantzel（1814–38），asitrequiresaprecisealgebraicdesofossibleusingtheclassicaltoolsioseethatthecuberootof2isanumberofafuallydifferedoesioshowingthatyouevermanufactureacuberootoutofsquarerootsandratiohatway，theimpossibilitysoundsmoreplausible。However，thatinnowaystitutesaproof。

Traals

Withintheclassofirratiohemysteriousfamilyoftraalhesearisethroughtheordinarycalsofarithmetidtheextraofroots。Forthepreitioroduentaryofalgebraiumbers，whicharethosethatsolvesomepolyionwithis:forexamplex5-3x+1=0issuequatioraalsaretheheon-algebraiumbers。

Itisnotatallclearthatthereareanysuumbers。However，theydoexistandtheyformaverysecretivesociety，withthoseinitnotreadilydivulgingtheirmembershipoftheple，thenumberπisarathisisnotafactthatitopewillbeexplaichapterwhehenatureofiisthat‘most’raal，irecise。

Anotherwayinwhichthemysteriousearisesisthroughthesumofthereciprocalsofthefactorials，andthisgivesawayofgetoahighdegreeofaccuracyasthisseriesvergesrapidlybecauseitstermsapproachzeroveryquideed:

&heimaginary

&fivechaptersofthisVeryShortIntroduainlywithpositiveintegers。Weemphasizedfactorizatioiesofintegers，whichledustoumbersthathaveorizations，rimes，asetthatoccupiesapivotalpositioography。Wealsolookedatparticulartypesofnumbers，suchastheMersenneprimes，whitimatelyectedwithperfeumbersandtooktimetointroduespecialclassesofiareimportantingaturallys。Throughoutallthis，thebackdropwasthesystemofintegers，whicharetheumbers，positive，ive，andzero。

Inthischapterwehavegoegers，firsttotheratioions，positiveaheionals，andwithintheclassofirrationalswehaveideraalheunderlyingsysteminwhichallthisistakihesystemoftherealnumbers，whibethoughtofastheofallpossibledecimalexpansions。Anypositiverealnumberberepreseheformr=n。a1a2···，wherenisaiveihedetisfollowedbyarailofdigits。Ifthistraileventuallyfallsintpattern，thenrisinfaalandwehaveshownhowtovertthisrepresentationintoanordinaryfra。Ifnot，thenrisirrational，sotherealnumberseiinctflavours，therationalaional。

Inourmathematiatioeherealnumbersasdingtoallthepointsalongthenumberlifromzero，thtforthepositivereals，afativereals。Thisleavesuswithasymmetricalpicturewiththeiverealnumbersbeingamirreofthepositivereals，andthissymmetryispreservedwhehadditionandsubtra–butnotwithmultiplicewepasstomultipli，thepositiveaivenumbersnoloatusasthenumber1isehapropertythatnoothernumberpossesses，foritisthemultiplicativeidehat1×r=r×1=rforanyrealipliby1fixesthepositionofainultipliby-1ssasmirreonthefarsideof0。Oiplitersthese，thefualdiffereheiveaivenumbersarerevealed。Inpartiegativenumberslacksquarerootswithintherealembecausethesquareofanyrealnumberisalwaysgreaterthaozero。

ThisfirststrueihturywhenItaliaislearnthowtosolvedfreepolyionsinafashiohatusedtosolvequadratis。Theethod，asitcametobeknown，wouldofteninvolvesquarerootsofhoughthesolutioiourobepositiveiagesfromthispoint，theuseofbers，whicharethoseoftheforma+bi，whereaandbareordinaryrealnumbers，wasshowntofacilitateavarietyofmathematicalcals。Forexample，ihturyEulerrevealedaedthestunniioneiπ=-1，whiotfailtosurpriseaheirfirstenter。

Aroundthebeginnihtury，thegeometriterpretationofbersaspointsintheateplaandardsystemofxy-ates），wasiedbyWessellandArgand，fromwhittheuseofthe‘imaginary’becameacceptedasnormalmathematitifyingtheberx+iywiththepointwithates（x，y）allowsexaminationofthebehaviourofbersihebehaviourofpointsintheplahisprovestobeveryilluminatiheoryofso-plexvariables，whosesubjectmatterisrepresentedbyfunplexherthanjustrealnumbers，flourishedspectathehandsofAugustinCauchy（1789–1857）。Itisnowaathematiderpinsmuchnaltheory，airefieldofX-raydiffraisbuiltonbers。Thesenumbershaveprovedtohaverealmeaning，ahesystemispleteinthateverypolyionhasitsfullentofsolutionswithiemofbers。Weshallreturersinthefinalchapter。Befthat，however，weshallierlookmorecloselyattheiureoftherealnumberline。