continued-fraction-expansions 连续分数简介

CONTINUEDFRACTIONEXPANSIONSContents1.Introduction22.IntegralContinuedFractionExpansionofaRealNumber23.FiniteContinuedFraction34.QuadraticIrrationalsandPeriodicContinuedFraction44.1.GL2(Z)actionontheProjectiveLine44.2.QuadraticIrrationals4References612CONTINUEDFRACTIONEXPANSIONS1.IntroductionInclass,wediscussedacontinuedfractionexpressionofrealnumbers.Ageneralcontinuedfractionisanexpressionoftheform:x=a0+b1a1+b2a2+b3a3+


Theai'sandbi's,whichgenerallycanberealorcomplexnumbers,arecalledthecoecientsortermsofthecontinuedfraction,andxisthevalueofthecontinuedfraction.Iftheexpressioncontainsanitenumberofterms,itiscalledafinitecontinuedfraction.Iftheexpressioncontainsaninnitenumberofterms,itiscalledaninfinitecontinuedfraction.Inthispaper,wemainlyfocusonsimplecontinuedfractions,whichmeansallbi'sare1,i.e.x=a0+1a1+1a2+1a3+


Ifallai'sareintegers,wewillcallitintegralcontinuedfraction.2.IntegralContinuedFractionExpansionofaRealNumberTheArchimedeanpropertygivesthatforeveryrealnumberx,thereisauniqueintegernsuchthatnxn+1.Theintegerniscalledthefloorortheintegerpartofx,andisdenotedbyn=[x].Thenumberu=fxg:=x[x]2[0;1)iscalledthedecimalpartofx;particularly,fxg=0ifandonlyifxisaninteger.Thusforgivenrealnumberx,thereisauniquedecompositionx=n+uwherenisanintegeranduisintheunitinterval.Thisdecompositioniscalledmodonedecomposition.Givenxonebeginswiththemodonedecomposition:x=n1+u1.Ifu1=0,therecursiveprocessendshere.Ifu10,then1=u11,thenapplythemodonedecompositionto1=u1:1=u1=n2+u2.Ifu2=0weendtheprocess;ifnot,wecontinuethesameprocess.Afterksteps,wemaywritex=n1+1n2+1n3+1


+1nk+ukCONTINUEDFRACTIONEXPANSIONS3Weshouldalsointroducethenotation[a0]=a0,[a0;a1]=a0+1a1,[a0;a1;a2]=a0+1a1+1a2,etc.Thuswecanwriten1+1n2+1


+1nk+uk=[n1;n2;


;nk+uk]:Ifuk=0theprocessendsafterksteps;otherwisetheprocesswouldcontinueatleastonemorestepwith1=uk=nk+1+uk+1.Inthiswayxisassignedwithasequenceofintegersn1;n2;


,whichcouldbeniteorinnite.Thissequenceistheintegralcontinuedfractionexpansionofx.Bythenatureoftheprocess,weknowtheexpansionisunique.3.FiniteContinuedFractionTheorem3.1.Theintegralcontinuedfractionexpansionofarealnumberisnite(i.e.theprocessendsinnitenumberofsteps)ifandonlyiftherealnumberisrational.Proof.Letx=a=b;b0bearepresentationofarationalnumberxasaquotientofintegersaandb.Themodonedecompositionab=n1+u1;u1=an1bbshowsthatu1=r1=b,wherer1istheremainderfordivisionofabyb.Thecasewhereu1=0isthecasewherexisaninteger.Otherwiseu10,andthemodonedecompositionof1=u1givesbr1=n2+u2;u2=bn2r1r1Thisshowsthatu2=r2=r1,wherer2istheremainderfordivisionofbbyr1.Thus,thesuccessivequotientsinEuclid'salgorithmaretheintegersn1,n2,:::.occurringinthecontinuedfraction.Euclid'salgorithmterminatesafteranitenumberofstepswiththeappearanceofazeroremainder.Hence,thecontinuedfractionexpansionofeveryrationalnumberisnite.Conversely,Let[a1;a2;


;an]beanitecontinuedfraction.Wewanttoshowthatitisarational.Denerealfunctionfai:x7!1=(xai),sinceaiisaninteger,weknowthatfaiisabijectivemaponitsdomain.Itmapsrationalstorationals,andirrationalstoirrationals.Nowwecanseethatan=fan1fan2


fa1([a1;a2;


;an]):Sincetherearenitelycompositions,wecanconclude[a1;a2;


;an]isrational.Corollary3.2.Theintegralcontinuedfractionexpansionofarealnumberisin-nite(i.e.theprocessneverends)ifandonlyiftherealnumberisirrational.4CONTINUEDFRACTIONEXPANSIONS4.QuadraticIrrationalsandPeriodicContinuedFraction4.1.GL2(Z)actionontheProjectiveLine.Leta;b;c;d2Randadbc6=0,andM=abcd:WedenetheactionofMonz,forz2R,by:M
z:=az+bcz+d:anditisconvenienttosetM
(d=c):=1,andM
1:=a=c(Thismakessensebytakinglimits).Lemma4.1.If[a1;a2;


]isanycontinuedfraction,then[a1;a2;


;ar;ar+1;


]=M
[ar+1;


]:(4.2)whereM=a1110


ar110:Proof.Wecanseeai110
x=ai+1x=[ai;x],whichshiftthecontinuedfraction1digittotheright,andaddaiontheleft.SowecanseeMaddar;ar1;


;a1onebyonetotheleft.4.2.QuadraticIrrationals.Anirrationalrealnumberiscalledquadraticirra-tionalifitisirrationalandsatisestheequation:x2+x+=0where;2Q.Wesay0isconjugateofifitistheotherrootofthesameequation.Generallyaquadraticirrationalisoftheforma+bpmcwherea;b;c,andmareallintegerswithm0,c6=0andmnotaperfectsquare.Theorem4.3.EveryPeriodiccontinuedfractionisthecontinuedfractionexpan-sionofaquadraticirrational.Proof.Numbersoftheforma+bpmcwithxedmbutvaryingintegersa,b,andc6=0maybeadded,subtracted,multiplied,anddividedwithoutleavingtheclassofsuchnumbers.(Thestatementhereaboutdivisionbecomesclearifoneremembersalwaystorationalizedenominators.)Consequently,forMinGL2(Z)thenumberM
zwillbeanumberofthisformor1ifandonlyifzisinthesameclass.Sinceaperiodiccontinuedfractionisrationallyequivalenttoapurelyperiodiccontinuedfraction,thequestionofwhetheranyperiodiccontinuedfractionisaqua-draticirrationalityreducestothequestionofwhetherapurelyperiodiccontinuedfractionissuch.Letx=[a1;


;ar;a1;


;ar;


]beapurelyperiodiccontinuedfraction.Bythelemma4.1,wehavex=M
x.SinceMisaproductofmatricesinGL2(Z),wehaveM=abcd:CONTINUEDFRACTIONEXPANSIONS5ThenWehavex=ax+bcx+d;equivalentlyxisthesolutionof