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&ivennumbers，aandb，onletcbetheiraverage.Ifcisirrational，wehaveaherequiredkind（irrational）.Ifoherhand，cisrational，putd=c+t，wheretistheirratiohepreviraph.Bywhathasgonebefore，dwillalsobeirrational，aakenlargeenough，wealwaysedissoclosetecofthetwogivennumbersaandbthatitliesbetweehisway，weseethattheirratioooformadeand，aswiththerationalnumbers，weferthatthereareinnitelymanyirrationalnumberslyiwohenumberline.
Aofrationalsanditseofirrationalsareinonearable（theyarebothdehenumberline）andiherstsetistable，theseot）.
iddleThirdSet
Wenowhaveaclearerideaastohowtherationalaioerlatherealherationalnumbersformatableset，yetaredenselypaberlior’sMiddleThirdSetis，bywayoftrast，anuoftheuhelessissparselyspread.Itistheresultofthefollowingstru.
&ionofiddleThirdSettothe4thlevel
&wemakeanamaziion.Bytakiernaryexpansionofanynumberdreplageaceof2by1，weobtainthebinaryexpansionofsomenumbertheuhisgivesaoo-onedehthesetofallnumbersinI（writteninbinary）.ItfollowsthatthealityofCisthesameasthatofI，aerisanu（bytument），itfollowsthattheiddleThirdSetisnotonlyiuntable.
ThereforewehaveasetCthatisinonesensenegligibleinsize（hasmeasurezero），butbyanotherwayCishuge，asithasthesamealityasIahewholerealline.
Whatismore，farfrombeingdense，owheredebysayingthatasetliketherationalsisdewheakearealhereareratioobefoundinaervalsurroundinga，howeversmallthatintervalmightbe.Wesaythatanyneighbourhoodofaembersofthesetofratioorsethasquitetheoppositenature–inCmightlivetheirlivesinthereallieveringaembersofC，providedtheyetheirexperiehlocalityarouheylive.Toseethis，takeaisnotinC，sothatahasaternaryexpansionthatsatleastone1:a=0·....1.....，withthe1ihplace，say.Forasuffitlytinyintervalsurroundinga，thehatintervalhaveaternaryexpansionthatagreestopladthenth，andsoallofthemwillalsonotbemembersesetCastheirternaryexpansionswillalsoatleastoanceof1.
Oherhand，ahetorsetwillooisolated，forwhenalooksoutintoahatsurroundsitinthenumberline，howeversmall，awillfindneighboursfromthesetgalo（ainCaswell）.ecifyamemberbofthegivehatalsoliesingbtohaveaternaryexpansionthatagreeswithatoasuffitlylargenumberofplaces，butwithrybeinga1.Ihereareunanymembersof，theiddleThirdSetumerousasd，tothemembersoftheCclub，theirbrothersaobeseenallaroundthemwherevertheylook.TotheinC，however，Chardlyseemstoexistatall.NotonememberofCistobespottedintheirexeighbourhoods，aselfhasmeasurezero.Tothem，Cisalmostnothing.
Diophaions
Aiudyemerges，hoetaketheoppositetasistthatnotonlythetsofourequatioioobeintegers.Hereisaclassicexample.
Aboxsspidersalesand46legs.Howmanyofeadofcreaturearethere?Thislittlenumberpuzzlebesolvedeasilybytrial，butitisiethatfirst，itberepresentedbyaion:6b+8s=46，aweareoediainkindsofsolutionstothatequatiohosewheretheles（b）andspiders（s）areumbers.Ingeneral，asystemofequationsiscalledDiophantirigoursolutioospeumbertypes，typitegerorrationala
ThereisasimplemethlinearDiophaionssuchasthiso，dividethroughtheequatiohets，whithisd8sotheirhgthisonfactorof2weobtaiequation，thatistosayohesamesolutions:3b+4s=23.Iftheright-handsidewereegerafterperfthisdivision，thatwouldtellusthattherewerenointegralsolutioiohtthere.Theakeohets，thesmalleroneisnormallytheeasiest，andworki，inthiscase3.Ourequatioenas3b+3s+s=（3×7）+2;rearraains=（3×7）-3b-3s+2.Thepointofthisisthatitshowsthatshastheform3t+2forsomei.Substitutings=3t+2iionandmaki，weget
WehepletesolutioheDiophaion:b=5-4t，s=3t+2.ganyintegralvaluefiveasolution，andallsolutionsinintegersareofthisfor
12.Lattitsonthelineofaliion
inalproblem，however，wasfurtheredinthatbothbandshadtobeatleastzero，aslesaexist.Hehereareonlytwofeasiblevaluesoft，thosebeingt=0andt=1，giviwopossiblesolutiolesand2spiders，aleand5spiders.Ifweihepuzzleasmeaningthatthereisapluralityofbothtypesofcreature，wehavethetraditionalsolutiolesand2spiders.
Thistypeofproblemisearbecausethegraphoftheassociatedequationsistsofaninfis.TheDiophantihenistofiitsonthisline，oihatesareintegralor，ifositivesolutions，onlylattitsiivequadrantwilldo.
However，onceweallowsquaresandhigherpowersiioureofthedingproblemsaremuchmorevariedaing.AclassiofthistypethathasafullsolutionisthatoffindingallPythagoreaiveintegersa，b，andcsuchthata2+b2=c2.APythagoreantripleofcoursetakesitshefactthatitallowsyht-ariahsidesofthoseiheclassicexampleisthe（3，4，5）triahagoreantriple，weeratemoreofthemsimplybymultiplyingallthehetriplebyanypositivehePythagoreaionwilluetohold.Forexample，wedoublethepreviousexampletogetthe（6，8，10）triple.This，hivesasimilartrialythesameproportions，asthegeisonlyamatterofsotofshape.GiveriahesedPythagoreantriplesimplybymeasurihsofthesidesinunitsthatarehalfthesizeinalunits，therebydoublingthenumeriensions.Thereare，henuiriplessuchasththe（5，12，13）andthe（65，72，97）right-ariangles.
IodescribeallPythagoreaherefore，itisenoughtodothejobforalltriples（a，b，c）wherethehcfofthethreenumbersis1，asallothersaremerelyscaled-upversioherecipeisasfolloairofepositiveintegersmandn，withohemeveethelarger.Fivenbya=2mn，b=m2-n2and2.Thethreenumbersa，b，andgiveyouaPythagoreahealgebraiseasilydthethreenumbershavenoonfactor（alsonotdifficulttoverify）.Thethreeexamplesabovearisebytakingm=2andcase，m=3andn=2inthesed，whileforthelasttrianglewehavem=9，n=4.Ittakesmoreworktoverifytheverse:anysuchPythagoreantriplearisesinthisfashionforsuitablyvaluesofmandn，andwhatismore，therepresentationisutwodifferentpairs（m，n）otyieldthesametriple（a，b，c）.
Thediionfordhigherpowershasnosolutionatall:foraherearenopositiveiriplesx，y，andzsu+yn=zn.ThisisthefamousFermat'sLastTheorem，whifuturemightbeknowheoremasitrovedinthe1990sbySirAndrewWiles.Evenforthecaseofcubes，firstsolvedbyEuler，thisisaverydiffi.Itis，however，relativelyeasytoshowthatthesumoftwofourthpowersisneverasquare（aainlynotafourthpower）.Thisisenoughtoredutothecasewherenisaprimep（meaningthatifwesolvedtheproblemforallprimeexpohegewouldfollowatondiheproblemwassolvedforsularprimesiury.However，thefullsolutionwasonlyrealizedasaceofWilessettliioheShimura–Taniyamajecture.
VersionsofthePellequatioudiedbyDiophantushimselfaroundAD150buttheequationwassolvedbythegreatIndiaiBrahmagupta（AD628）ahodswereimproveduponbyBhaskaraII（AD1150），toewsolutionsfromaseedsolutiowasFermatwhoexhortedmathematistoturiontoPell’sequatioetheoryiscreditedtotherekhematiJoseph-Le（1736–1813）（theEnglishappellation‘Pell’isanhistorit）.
Fibonaduedfras
&hesequenbers，1，1，2，3，5，8，13，21，…discoveredbyFibonatroduChapter5.Takeapairofsuccessivetermsinthissequeethedingratioas1plusafra.Ifweiahisfrabyrepeatedlydividingtopandbottombythepatterake，forinstance
&ainamulti-flooredfrasistiirelyof1s，andeachpregratioofFibonaumbersappearsaswewindthroughtheusthappeime:bytheverywaythesenumbersaredefined，eaberislesstha，aofthedivisionwillleaveaquotientof1andtheremaihepregFibonaumber.YouwillrecallthattheratioofsuccessiveFibonaumbersapproachestheGoldenRatio，τ，andsothissuggeststhatτisthelimitiheuedfrasistiirelyof1s.
&ypeofuedfrasthatemergefromthisprocessareintrinsicallyimportant.roximateanirrationalnumberybyratiourallyturntothedecimalrepresentationofy.Thisisextfeneralcalsbut，beioaparticularbase，isiatural.Essentialtothenatureofyishowwellournumberybeapproximatedbyfraswithrelativelysmalldenominators.Isthereanywaytofindaseriesoffrasthatbestdealswiththegdemandsytoahighdegreeofaccuracywhilekeepiorsrelativelysmall?Theaheuedfrarepresentationofadoesthisthroughitstrunsateverlowerfloors.
ThespecialexampleaffordedbytheGoldenRatioopeotheideathatwemaybeabletorepreseiobyfiiions（whichareobviouslyjustratiobyinfihowistheuedfraberaproduced?Thereaderwilloleratealittlealgebraictriordertoseethisina，buthereishowitgoes.
&aiionsinapproximationofirrationalsbyrationalsisduetothesovergeion，whicharetherationalapproximationsiresultfrtherepresentatioandwoutthedingratioheserepreseapproximatiohenumberiiohataerapproximationwillhavealargerdenominatorthanthatofthets.ThetsoftheGoldenRatioaretheFibonacciratios.Siermiiioionofτis1，theceoftheseratiosisretardedasmuchasitpossiblycouldbe.Forthatreason，thereisnomorediffiumberthanτtoapproximatebyrationalsandtheFibonacciratiosarethebestyoudo.
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