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Chapter6Belowthewaterlihenumbericeberg
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Introdu
However，oatsofthe19thturywasthefullrealizatioruedomaioratheristwo-dimensional.Theplahebersisthenaturalarenaofdisuatics.Thishasbeenbroughthometomathematidstiststhroughproblemsolviocarryouttheiioosolvereal-worldproblems，manyofwhichseemtobeonlyaboutordinaryumbers，itbeesoexpandyournumberhorizoionastohowthisextradimensieswilletowardstheendofthisdbeexploredfurtherinChapter8.
&ralportionofthenumberlinenear0
Plusesandminuses
&egersisthenameappliedtothesetofallwholeiveive，ahisset，oftensymbolizedbytheletterZ，isthereforeihdires:
{…-4，-3，-2，-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.
YoushouldroubledbythislastsuFirst，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:
Thiskindoftrickisofteosimplifyahatinvolvesaingprople，siderthefollowier:
Bysquaring，andthensquaringagai-handsidebeesa4，whiletheexpressiogives:
&followsthe5isanothercopyoftheexpressionfora，weia4=20asothata3=20or，ifyouprefer，aisthecuberootof20.WewillthisteiqueagaininChapter7wheroduceso-tiions.
&heclassoffrasprovideuswithallthenumberswecouldeverneed?Asmeheofallfras，togetherwiththeirhesetofnumbersknowionals，thatisallresultfromwholeheratiosbetweeheyareadequateforarithmetithatanysuminvolvingthefourbasicarithmeticoperationsofaddition，subtraultipliddivisioakeyououtsidetheworldofrationalnumbers.Ifywiththat，thissetofnumbersisallwerequire.However，weexplaiseberssuchasaabovearenotrational.
Irrationals
Argumentsalongtheselinesallowustoshowthatquitegenerally，whehesquareroot（orihecubeherroot）ofaheaawholenumber，isalwaysirrational，thusexplainingwhythedecimaldisplaysonyourcalevershpatterocalculatesucharoot.
Thisproblemremaiouclassicaltimes.Thattheansweriscuberootof2liesoutsidetheraheeutoolswasoledin1837byPierreWantzel（1814–38），asitrequiresaprecisealgebraicdesofossibleusingtheclassicaltoolsinThereasosolutionisinthebillordertoseethatthecuberootof2isanumberofafuallynothardtosee.Anysolutionnhastohavethefordifferedoesioshowingthatyouepositivepowersrandsaheremanevermanufactureacuberootoutofsquarerootsandrationals.factorsarecollectedtogetherintoasiegerWhenputthatway，theimpossibilitysoundsmoreplausible.divisibleby3or5.IfwefirstfothepossibleHowever，thatinnowaystitutesaproof.
Traals
Withintheclassofirratiohemysteriousfamilyoftraalhesearisethroughtheordinarycalsofarithmetidtheextraofroots.Forthepreitioiheentaryofalgebraiumbers，whicharethosethatsolvesomepolyionwithis:forexamplex5-3x+1=0issuequatioraalsaretheheon-algebraiumbers.
Itisnotatallclearthatthereareanysuumbers.However，theydoexistandtheyformaverysecretivesociety，withthoseinitnotreadilydivulgingtheirmembershipoftheple，thenumberπisarathisisnotafactthatitopewillbeexplaichapterwhehenatureofiisthat‘most’raal，irecise.
Anotherwayinwhichthemysteriousearisesisthroughthesumofthereciprocalsofthefactorials，andthisgivesawayofgetoahighdegreeofaccuracyasthisseriesvergesrapidlybecauseitstermsapproachzeroveryquideed:
Therealandtheimaginary
&vechaptersofthisVeryShortIntroduainlywithpositiveintegers.Weemphasizedfactorizatioiesofintegers，whichledustoumbersthathaveorizations，rimes，asetthatoccupiesapivotalpositioography.Wealsolookedatparticulartypesofnumbers，suchastheMersenneprimes，whitimatelyectedwithperfeumbersandtooktimetointroduespecialclassesofiareimportantingaturallys.Throughoutallthis，thebackdropwasthesystemofintegers，whicharetheumbers，positive，ive，andzero.
Inthischapterwehavegoegers，rsttotheratioions，positiveaheionals，andwithintheclassofirrationalswehaveideraalheunderlyingsysteminwhichallthisistakihesystemoftherealnumbers，whibethoughtofastheofallpossibledecimalexpansions.Anypositiverealnumberberepreseheformr=n.a1a2…，wherenisaiveihedetisfollowedbyarailofdigits.Ifthistraileventuallyfallsintpattern，thenrisinfaalandwehaveshownhowtovertthisrepresentationintoanordinaryfra.Ifnot，thenrisirrational，sotherealnumberseiinctavours，therationalaional.
Inourmathematiatioeherealnumbersasdingtoallthepointsalongthenumberlifromzero，thtforthepositivereals，afativereals.Thisleavesuswithasymmetricalpicturewiththeiverealnumbersbeingamirreofthepositivereals，andthissymmetryispreservedwhehadditionandsubtra–butnotwithmultiplicewepasstomultipli，thepositiveaivenumbersnoloatusasthenumber1isehapropertythatnoothernumberpossesses，foritisthemultiplicativeidehat1×r=r×1=rforanyrealipliby1fixesthepositionofainultipliby-1ssasmirreonthefarsideof0.Oiplitersthese，thefualdiffereheiveaivenumbersarerevealed.Inpartiegativenumberslacksquarerootswithintherealembecausethesquareofanyrealnumberisalwaysgreaterthaozero.
Thisistheagiomaketheireopiethatweshalltakeupagaininthefihetimebeimakesomeintroduents.
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.
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