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

Chapter6Belowthewaterlihenumbericeberg

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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.