The NLCS Journal of Pure and Applied Mathematics

Credits

Firstofall,wewouldliketoexpressoursinceregratitudetothefollowingcontributorstoVolume3Issue1 ofNLCSJejuAnnualJournalofMathematics.

TheMathematics Society

Chair

AarenJoonseokKang(12)

Co-Chair

JamesDaewoongKang(12)

PublicityOfficer

TerryTaehoonKim(11)

Secretary

EmmaChaeeunChung(11)

Members

JooyoungKim(12)

AlvinSehunJeong(12)

JeanKim(11)

JuneKim(11)

SophieJisuLee(11)

AidenHyunminPark(10)

AlbertJinyongShin(10)

DerekYejunYoo(10)

JamesJiminLee(10)

SangyoonLee(10)

DanielTaehyeonJun(10)

JungseoPark(9)

LinkTeachers

MrWilliamHebbron MsDuyguBulut

TheMathematics PublicationCCA

ProgramLeader AarenJoonseokKang(12) JamesDaewoongKang(12)

Writers AidenHyunminPark(10) JungseoPark(9)

LinkTeacher MsDugyuBulut

LATEXEditors&Managers

AarenJoonseokKang(12)

JooHyunKim(12)

TaeheeKim(12)

EmmaChaeeunChung(11)

VictoriaSeoyunJu(12)

DanielSoohyukChang(12)

Also,wewouldliketothankthefollowingcontributorsthatarenotpartofthemathematicssocieties,but haveassistedusinvariousways.

MarketingDepartment forhelpinguswiththeprintingandpublicizingourjournal. MrTamlyn forcoordinatingsocietiesinourschool.

© CopyrightbytheNLCSJejuJournalofPureandAppliedMathematics

FormattingbyAarenJoonseokKang

EditedbyJPAMEditingTeam

Editor’sNote

IamveryexcitedtoannounceourfirstissueoftheNLCSJournalofPure andAppliedMathematics(JPAM)ofthe2022-23academicyear.Thisyear, itwasevidentthattalentedmathematiciansinourschoolhavedevotedtheir besteffortsandprecioustimetoproduceaqualityarticle.Allthecontributorshaveresearchedaboutaspecificdisciplineinmathematicstheyaremost interestedinandhaveputintheirutmosteffortinvariouswaysbysolving problems,programming,anddraftingtheirarticles.Itisdoubtlessthatthe academicrigourofthisjournalisatahighlevelregardlessofthefieldand thecomplexityofthetopic.

ThisversionoftheJPAMprovidesvaluableinsightintomathematicsin twodifferentsections:PuremathematicsandAppliedmathematics.Pure mathematicsinvolvesthediscussionoftheoreticalandmethodologicalaspectsofmathematicsinvariousmathematicalideasandproofs,divingfurtherintothebeautyofthepurenessinmathematics.Ontheotherhand, Appliedmathematicsapproachesmathematicsinaninterdisciplinaryaspect displayinghowthebeautyofmathematicsisappliedindisciplinessuchas computerscience,physics,socialsciences,andsportssciences.

Asthechiefeditorofthisjournal,Ienjoyedreadingandeditingevery singlepieceofarticleinthisjournal.MymanythankstothefellowJPAM editorsintheteam-JooHyunKim,VictoriaJu,EmmaChung,Daniel Chang,andTaeheeKim-whohavehelpedspeeduptheprocesstremendously.Itwasthefirstyeartohaverecruitedaformalteamofeditors,but wehaveworkedveryefficientlyasateamtoproduceahighqualityjournal intermsofformatandlayout.Also,anotherlargethankstoHyeminLyoo, ourdesignerofthecoverpageofthisissue,fordesigningamathematical,yet artisticcoverpage.

Onafinalnote,IwouldliketoexpressmygratitudetoallthecontributorsandreaderswhoareinterestedinourpublicationsofNLCSJejuJournal ofPureandAppliedMathematics.

AarenJoonseokKang-ChairoftheMathematicsSociety,Leader oftheJPAMEditingTeam

MathematicalProofandParadox

EmmaChaeeunChung

Year11Noro

SecretaryofMathematicsSociety

MemberoftheJPAMEditingTeam

Email:cechung25@pupils.nlcsjeju.kr

Editor

EmmaChaeeunChung ∗

RecommendedYearLevel:KS4andabove

Keywords:Proof,Paradox,ProofbyContradiction, ProofbyInduction

1Introduction

“IfonlyIhadtheTheorems!ThenIshouldfind theproofseasilyenough”,saidBernhardRiemann,the renownedGermanmathematician.Indeedmathematical‘proof’maynotbedifficulttowrite-thatis,fallacies thatareoftentimesdeceivingbecausetheyinitiallyseem derivedfromobviouslogic.Inthispaper,methodsof appropriatemathematicalproofwillbediscussed,along withcommonmathematicalerrors.

2Definitions

Proofplaysacentralroleindeveloping,establishing andcommunicatingmathematicalknowledge.Amathematicalproofisdefinedasan‘inferentialargument’ thatlogicallyguaranteestheconclusion,constructedby originalassumptionscalledaxoims.Asfirstprinciples, orunprovablerulesthatareacceptedastrue,axioms serveasastartingpointforreasoning.

Belowisanexampleofalogicalaxiom-Axiomof Equality-inwhichastatementisusedtodeduceother logicallyvalidformulas:

x = x (1)

Anon-logicalaxiom,ontheotherhand,aretheoryspecificassumptions.Inotherwords,theyshouldbe tetheredtotheapplicationbeforeusingtheminproofs. (Rocha,2019).

∗ MemberofJPAMEditingTeam

‘QED’,theinitialismoftheLatinphrase,‘quoderat demonstrandum(thusithasbeendemonstrated)’ora Halmostombstone(❚)isoftenwrittenattheendofa proof.Theyannouncetheendoftheproof,actingas avisualcueandremindsthereaderofthepreposition thatwasproven,especiallyusefulinextensivenested proofs(Weisstein,n.d.).

3TypesofMathematicalProof

3.1ProofbyContradiction

ProofbyContradictionestablishesthevalidityofa statementbyshowingthatassumingthestatementto befalseleadstocontradiction.Proofbycontradictionis alsoknownas‘indirectproof’,inwhichthealternatives ofastatementisdeniedandtheoriginalstatementis impliedvialogicaldeductions.Followingisaproofby contradictionthat √2 isanirrationalnumber.

(i)Suppose √2 isarationalnumberexpressedas a b whereaandbarenon-zerointegerswithnocommon factor.

(ii)b√2 =a,2b2 = a2 a2 iseven,whichimpliesthatamustalsobeeven.Let a=2c,wherecisaninteger.Substitute2cinto2b2 = a2 andweget2b2 =4c2 .Dividingbothsidesby2yields b2 =2c2 ,implyingthatbisdivisibleby2.

(iii)aandbarebotheven,with2astheircommon factor.Thiscontradictsthepreviousstatementthata andbsharenocommonfactors.

(iv)Assuming √2 tobearationalnumberleadstoa contradiction,thereforetheoppositemustbetrue. √2 isanirrationalnumber.

However,itistobenotedthatinsomemathematicalcircles,directprovingmethodsarepreferredover proofbycontradiction.Mathematicalconstructivists mayconsidersomecasesofproofbycontradictionin-

validbecausetheprovingstatementsareindependentof axiomsandtheproofsarelowinfecundityasthereare nointermediaryconclusions(Korner,2011).

4ProofbyMathematicalInduction

Mathematicalinductionisamethodusedtoprove athatastatementistrueforallnaturalnumbers.

ProofbyInductionconsistsoftwosteps:firsta‘base case’isprovesthestatementtrueinasinglecase, usuallywhenn=1.Thesecondstepistheinduction stepwhichprovesthatifthestatementistrueforn=k, itisalsotrueforn=k+1(Pierce,2017).Belowproves thatallpositiveintegersintheform2n-1areodd.Let P(n)represent‘2n-1isodd’.

(i)Forn=1,2n-1=1and1isodd.P(1)istrue.

(ii)Foranyn,if2n-1isodd,(2n-1)+2mustalsobe odd,becauseadding2toanoddnumbergivesanodd number.

(2n-1)+2=2n+1=2(n+1)-1 P(n)impliesP(n+1).

Thus2n-1isoddforallpositiveintegers n(zimmer.csufresno.edu,n.d.).

FirstraisedbyGeorgePólyain1954,the‘Horse Paradox’isawell-knownexampleofaflaweduseof mathematicalinduction.TheHorseParadoxstates that‘allhorsesarethesamecolour’,asillustrated below:

(i)Thebasecaseisonehorse.Withonlyonehorse, obviouslyallhorsesinthisgrouparethesamecolour.

(ii)AssumethatNhorsesarealwaysthesamecolour. ConsiderN+1horses:ifthelasthorseisexcluded, theremainingNhorseswillbethesamecoloursince Nhorsesarethesamecolour.Byexcludingthefirst horse,theotherNhorseswillbethesamecolourby thesamereasoning.

ThebasecaseprovedthestatementtrueforN=1.The inductivestepproveditvalidforN=2,3,andsoon. Thusinanygroupofhorses,allhorsesmustbethe samecolour.

TheHorseParadoxdoesn’tcontainanycontradictions,butargumentsthemselvesareincorrect.TheinductivestepiserroneousgoingfromN=1toN=2becausethefirstandlasthorsehavenohorsesincommon andmaybeindifferentcolours(Benjamin,2019).

4.1ProofbyContraposition

InProofbyContraposition,theconclusion‘ifA, thenB’isinferredbyproving‘ifnotB,thennotA’.The twostatementsarelogicallyequivalentbecausethey arecontrapositives;astatementisformedbynegating bothtermsandreversingthedirectionofinference. Proofbycontrapositionisoneofthetwotypesof

indirectproofalongwithproofbycontradiction.

Afamouscaseofproofbycontrapositionisthe provingofthestatement:‘If x2 iseven,xiseven.’

(i)Thecontrapositiveofthestatementis:‘Ifxisnot even,then x2 isnoteven’.

(ii)Thelatterstatementcanbeproven.Ifxisnoteven, i.e.,odd,then x2 isalsooddbecausetheproductoftwo oddnumbersisodd.

(iii)Thereforetheoriginalstatementistrue. If x2 iseven,xiseven.

5VisualProof

Visualproofisavisualdemonstrationofapreposition.Utilisingcomputergraphicscouldbeaneffective meansofshowingverity,hencevisualproofissometimesreferredtoas‘proofwithoutwords’.

Provingtheformulafortheareaofacircleisoftendemonstratedbytheimageabove.Whenacircle isdividedintoinfinitelysmallsectorsandplacedalternatelyforminganalmost-rectangle,thelengthishalf theperimeter(= π r)andthewidthistheradius.

TheimageaboveprovesthePythagoreantheorem. Theareaoftheoutersquareis (a + b)2 andthesumof thetriangles’areagives(2ab). (a + b)2 -(2ab)gives a2 + b2 ,whichisequaltothesubtractionofthetriangles fromtheoutersquareyieldingthewhiteinnersquare; c2

Fig.1:LogicStatements:TruthTable

Fig.2:Areaofacircle

Alternately,thePythagoreantheoremcanbe provenbyrearrangingthetrianglesasbelow,showing twosquaresofarea a2 and b2 ,equaltotheprevious c2

Ontheotherhand,visualproofisnotaformal methodofmathematicalproof.Illusionsareeasily created,perplexingtheviewers.

Thefollowingvisualproofprovesthatthesum ofnsquaresis 1 3 n(n+1)(n+ 1 2 ).Threeblockspieces representingthesumofnsquaresmergetocreatea cuboidwithsidesn,n+1,andn+ 1 2

Thesetwotrianglesappeartobeconsistingof identicalsmallerpieces,althoughthesecondtriangle hasamissingblock.Thefirsttriangle’shypotenuse slightlycurvesoutwardsandthesecondcurvesinwards, resultingintheinaccuracy.

TheAristotle’sWheelParadoxisanotherdeceptive caseofvisualproof.Awheelisdepictedastwocircles tangentialtothesurface/ground.Thesmallerinner circleisfixedtothelargeroutercircleandhasthesame center.Assumingthelargercirclerollswithoutslipping foronefullrevolution,thedistancestravelledbyeach circleisequaltothelargercircle’scircumference, therebycreatingaparadox.

Thelawofcosinescanalsobeprovenwithvisual proof.Intheimagebelow,(2acosθ -b)b=(a-c)(c+a) andrearrangingthisequationresultsin c2 = a2 + b2

Galileoanalysedtheproblembysubstitutingthe circleswithhexagons.Considerrollingthehexagon, andonemaynoticetheinnerhexagon’slittle‘skips’of spacewitheachrevolutionoftheouterhexagon.Since acirclecanbemadeincreasingthenumberofsidesofa polygon,these‘skips’areinfinitelysmallandinfinitely

Fig.3:ProvingthePythagoreanTheorem(1)

Fig.4:ProvingthePythagoreanTheorem(2)

Fig.5:SumofNsquares

-2abcosθ (Doyle,2014).

Fig.6:LawofCosines

Fig.7:Anillusioninvisualproof

Fig.8:Aristotle’sWheelParadox many(orunquanitfiable)inacircularwheel.

https://www.zmescience.com/other/featurepost/10-beautiful-visual-mathematical-proofselegance-and-simplicity/.

5. Rocha,H.(2019).Mathematicalproof: frommathematicstoschoolmathematics. PhilosophicalTransactionsoftheRoyalSocietyA:Mathematical,PhysicalandEngineeringSciences,377(2140),p.20180045. doi:https://doi.org/10.1098/rsta.2018.0045.

6. Weisstein,E.W.(n.d.).Axiom.[online]mathworld.wolfram.com.Availableat: https://mathworld.wolfram.com/Axiom.html.

7. zimmer.csufresno.edu.(n.d.).MathematicalInduction.[online]Availableat: http://zimmer.csufresno.edu/larryc/proofs/proofs.mathinduction.html.

Drawingapoint’stravelpathduringtherevolution corroboratestheerrorintheoriginalstatement.As seenbelow,thepointonthesmallerinnercircletravels ashorterpathwaythanthepointonthelargerouter circle,thereforethedistancetravelledisnotequivalent tothesmallercircle’scircumference(Puiu,2016).

6Bibliography

1. Doyle,T.(2014).ProofsWithoutWords andBeyond-PWWsandMathematicalProof|MathematicalAssociationof America.[online]www.maa.org.Available at:https://www.maa.org/press/periodicals/ convergence/proofs-without-words-and-beyondpwws-and-mathematical-proof.

2. Korner,K.(2011).AnIntroductiontoProofby Contradiction.[online]nrich.maths.org.Available at:https://nrich.maths.org/4717.

3. Pierce,R.(2017).MathematicalInduction.[online]Mathsisfun.com.Availableat: https://www.mathsisfun.com/algebra/mathematicalinduction.html.

4. Puiu,T.(2016).10BeautifulVisualMathematicalProofs:EleganceandSimplicity.[online]ZMEScience.Availableat:

Fig.9:Substitutingcircleswithhexagons

Fig.10:Drawingapoint’stravelpath

Goldbach’sConjecture

TerryTaehoonKim Year11Mulchat

PublicityOfficeroftheMathematicsSociety

Email:th3kim25@pupils.nlcsjeju.kr

Editor JooHyunKim∗

RecommendedYearLevel:KS4,KS5

Keywords:NumberTheory,OldestConjuncture, Unproved,PrimeNumber

1Introduction

Goldbach’sconjecture,suggestedbyrenownedGermanmathematician,ChristianGoldbach,isregardedas oneofthemostcomplextheoremsinmodernmathematics.TheGoldbachconjecturestatesthat“everyeven naturalnumbergreaterthan2isthesumoftwoprime numbers ”.Consideringtherelativelyshortlengthofthis conjecture,onecanbemisguidedthatitisrelatively straightforwardintermsoftheequationandproving. However,this“simpleconjecture”remainsamystery despitetheprodigiousadvancementsinmodernmathematicssinceitsemergencein1742.Thestudiesbelow willmainlyfocusonthefactorwhichmakesthisconjectureinexplicablefor250years.

2TheConjecture

“Everyevennaturalnumbergreaterthan2isthe sumoftwoprimenumbers.”

3ResearchAim

(a) AnalysisoftheConjecture

(b) ApproachestotheConjecture

(c) Analysisontheapproaches

(d) CurrentProgress

(e) RelatedProblems

4Analysis-Conjecture

Theconjecturestates, “Everyevennaturalnumbergreaterthan2isthesum oftwoprimenumbers.”

Thisstatementmeansthatanyevennumbergreater thantwoiscomposedofthesumoftwoprimenumbers.

Figure1(Fig.1)showsthegeometricalapplicationofthisconjecture.

Fig.1

5ApproachestotheConjecture

5.1ComputationalApproach

Oneapproachtothisconjectureissimplysubstitutingeveryprimenumberforthesumofevennumbers. Withtheadventofcomputers,manymorevaluesof n havebeenchecked;T.OliveiraeSilvaranadistributed computersearchthatdemonstratedtheconjecturefor n ≤ 4 × 1018 asof2013.Onerecordfromthissearchis that3325581707333960528isthesmallestnumberthat cannotbewrittenasasumoftwoprimeswhereoneis smallerthan9781.

5.2HeuristicApproach

Theprimenumbertheoremstatesthatrandomly selectedinteger m wouldhaveroughly 1 ln(m) chanceof beingprime.Hence,since n inthisconjecturerefersto anevenintegerbetween3and n 2 ,theprobabilityofboth mm n beingaprimenumberwouldbe 1 ln(m)ln(m n) Thetotalnumberofmethodstoexpressalargeeven integerasthesumoftwooddprimenumberswouldbe:

6In-depthAnalysis

In-depth,however,thisheuristicapproachisrather inaccurate.Thefollowingmethoddoesnotputcorrelationsbetweenthelikelihoodof m and n m being prime.Thiscouldresultinvariousconsequences.For instance,ifweconsider m asodd,then n m would alsobeodd,andif m iseven,then n m iseven,a non-trivialrelationbecause(besides2)onlyoddnumberscanbeprime.Exploringthisstyleofanalysisin depth,HardyandLittlewoodin1923conjecturedthat foranyfixed c ≥ 2,thenumberofrepresentationsofa largeinteger n asthesumof c primes:

Thequantitywouldincreasetoinfinityas n increaseseverylargeevenintegerhasnotjustonerepresentationasthesumoftwoprimesbutinfactvery manysuchrepresentations.

Shouldbeasymptoticallyequalto:

wheretheproductisoverallprimes p and γc,p (n) is thenumberofsolutionstotheequation n = q1 + + qc ( mod p) inmodulararithmetics,subjecttotheconstraints q1 , ,qc =0(mod p) Thisformulahasbeen provedtobevalidbyVinogradovonrangeof c =2.In laterissue,theformulaabovesimplifiesto0when n is oddandto:

When n iseven,where 2 isthetwinprimesconstant:

7Goldbach’sWeakConjecture

Goldbach’s“weak”conjecturestatesamore moderatetermforGoldbach’sconjecture.Theweak conjecturestates: 12

Fig.2:PossibilityCalculationbyprogram

“Everyoddnumbergreaterthan5canbeexpressedas thesumofthreeprimes.(Aprimemaybeusedmore thanonceinthesamesum.)”

Thisconjectureiscalled"weak"becauseitdependson itsvalidityontheGoldbach’sstrongconjecture(Ifthe strongconjectureistruethenthiswouldalsobetrue). Forifeveryevennumbergreaterthan4isthesumof twooddprimes,adding3toeachevennumbergreater thanfourwillproducetheoddnumbersgreaterthan7 (andsevenitselfisequalto 2+2+3).Thisisproven byHaraldHelfgottin2019throughthemajorand minorarcestimates,whichweresufficientenoughto provetheGoldbach’sweakconjecture.Unfortunately, theexplanationofthemajorandminorarcisexcessive intermsoflength,whichwouldmakeitunavailablefor thisstudytoexplorefurther.

8Goldbach’sComet

OneofthemostwellknowninfluencesofGoldbach’sconjecturewouldbeGoldbach’sfunction,also knownasGoldbach’scometduetoitsdistinctshape. Itassociatestoeachevenintegerthenumberofwaysit canbedecomposedintoasumoftwoprimes.

Thecoloringofpointsintheaboveimageisbased onthevalueof E 2 mod3 withredpointscorrespondingto 0mod3,bluepointscorrespondingto 1mod3 andgreenpointscorrespondingto 2mod3.Moreover, thegraphalsosuggeststhatthetightupperandlower boundsonthenumberofrepresentationsofaneven numberasthesumoftwoprimes,andalsothatthe numberoftheserepresentationsheavilydependonthe valuemodulo3ofthenumber.

Fig.3:Goldbach’sComet

GeometricalInterpretationofComplexNumbers

JooyoungKim

Year12Halla

MemberofMathematicsSociety

Email:jy2kim24@pupils.nlcsjeju.kr

Editors

VictoriaSeoyunJu∗

RecommendedYearLevel:KS5andabove

Keywords:ComplexAnalysis,ComplexNumbers, Cauchy-RiemannEquation,HolomorphicFunction

InthefieldofcomplexanalysistheCauchyRiemannequationsfortwopartialdifferentialequationswhosesatisfactionareaconditiononwhether ornotacomplexfunctionisholomorphic.Itsname comesfromtwomathematiciansAugustinCauchyand BernhardRiemann.Acomplexfunctionisconsidered holomorphicwhenitisdifferentiableateverypoint. Thiscanonlybethecaseifandonlyifthethe functionsatisfiestheCauchy-Riemannequationsas wellasanothercondition:if f (z )= u(x,y )+ iv (x,y ), thenuandvmustberealdifferentiable.Thisstems fromthedefinitionofaholomorphicfunction:A function f (z ): C → C issaidtobeholomorphic inanopenset A ⊂ C ifitisdifferentiableateach pointoftheset A.Inthisentry,IwillbegivingabasicguidetotheproofoftheCauchy-Riemannequations.

Akeythemehereistheseparationofrealandimaginarypartsofthecomplexfunction.Wewilldothisby usingvariousprinciplesofmultivariablecalculus.

Proof

Considerthefunction f (z ): C → C,z ∈ C Bythefirstprinciple

f ′ (z )=lim h→ 0( f (z + h) f (z ) h )

Wecandothisbythinkingof h asapoint (a,b) andthe originas (0, 0)

Thus:

f ′ (x + iy )=lim h→0( f (x + iy + h) f (x + iy ) h )

⇒ f ′ (x + iy )

=lim (a,b)→(0,0)( f (x + iy + a + bi) f (x + iy ) a + bi )

(2)

Thislimitisonlytakingintoconsiderationthe casewhereboth a and b areapproachingzero.Inthe idealcasewewouldwanttoconsidertwolimitsfor easeofsimplification: a → 0 and b =0,and a =0 and b → 0,whichwouldyielduspartialderivatives.While thetwolimitsexpressedabovearenotthesameas (a,b) → (0, 0) aseachasaservererestriction,butifthe derivativefor f (z ) existstheneachofthetwolimits wouldbeequaltolimit (a,b) → (0, 0).Howeverforthis tohappen,wemustbeabletoexpress f (z ) intermsof realnumbers.

⇒ f ′ (x + iy )

=lim h→ 0( f (x + iy + h) f (x + iy ) h ) (1)

Thoughthelimitisfor h → 0,wecancontinuewith furtherseparatingtherealandimaginarypartsof f (z )

∗

Consideranewfunction F : R2 → C where F (x,y )= f (x + iy ).Thisnewfunctionallowsus toconsider f (z ) intermsofrealvariablewhileclearly distingushingitfromthefunctionofasinglecomplex variable f (z ).

Backtotheequationwecannowwrite f ′ (z ) in thetermsoftworealvariablesthus:

Bythesetwoequations,wecanseethatif f (z ) iscomplexdifferentiablethen:

Byequatingtherealandimaginarypartsofthis equality,wegettwoequationsthatrelateboththepartialderivativesofrealandimaginarypartsof f (z ):

Itisimportanttonotethat ∂ notationisnotthesame as d,astheformerisapartialderivative,whichisthe functionoftwoormorevariableswithrespecttoone variable,theother(s)beingtreatedasconstants.In ourcase, a and b areourtwovariables,andineach caseoneortheeitherweresettoequal0.

Wecancontinueseparatingtherealandimaginary partsofthefunction, f ,byfurtherdefiningbyreal functions:

u : R2 → R,v : R2 → RF (x,y )= u(x,y )+ iv (x,y )

Whilethismightseemconfusingatfirst,itis importanttounderstandthatacomplexfunctionhas4 componentstoconsider:therealpartsoftheinputand output,andthesamefortheimaginaryparts.This hastheinterestingeffectofmakingcomplexfunctions technicallyrequiring4dimensionstoproperlygraph. Whatfunctionsuandvdohereisbypullingoutthe realandimaginaryoutputsof F (x,y ).Anotherwayof puttingisthat u and v aretherealandimaginaryparts ofthecomplexfunctionofasinglecomplexvariable, with u(x,y ) beingtherealvalueof f (z ) and iv (x,y ) beingthesamefortheimaginarypart.

Bydefining F intermsof u and v wecanturnthe rewritetheequation

Puttingtheseequationsintowords

Theequationontheleftsaysthatthechangein realoutput(∂u)bytherealpartof f isequaltothe changeoftheimaginaryoutputbythechangeinthe imaginarypartof f

Thesecondequationchangeinrealoutputby changeintheimaginarypartof f isnegativeofthe changeinimaginaryoutputbuthechangeinrealpart of f

Asstatedbefore,afunctionisholomorphicwhen theCuachy-Riemannequationsandanotherproperty issatisfied,whichisthattherealandimaginaryparts of f (z ),functions u and v ,arereal-differentiable.

Example

Considerthat f (z )= z 2 isdifferentiableatanypoint z = x + iy inthecomplexplane:

Ifwedefine u(x,y ) and v (x,y ) astherealandimaginarypartsof f (z ),then:

andtheirpartialderivativesare

Since u and v arerealdifferentiableandtheirpartial derivativessatisfytheCuachy-Riemannequations,we cansaythat f (z ) isaholomorphicfunction.

TheValidityofConsideringtheSignificanceofan ImaginaryNumberinRelationtoReality

AidenHyunminPark Year10Sarah

MemberoftheMathematicsSociety Email:hmpark26@pupils.nlcsjeju.kr

Editor

AarenJoonseokKang ∗ VictoriaSeoyunJu †

RecommendedYearLevel:KS4

Keywords:ImaginaryNumbers,ComplexNumbers

1Introduction

1.1Whatdomathstandfor

Considerasimplequestion.“Whatisthemostbasicreasonforusdoingmath?”Tosomepeople,itisbecausemathhelpsthemtounderstandtheworldbetter andtosomeothers,mathcouldbejustahasslefortheir grades.However,thefundamentalmeaningofmathematicsissomethingtodowithquantity.Inotherwords, dealingwiththingsintherealworld.Startingfromsimplycountingthingswithfingers,mathematicsbasically allowedhumanstoperformsomuchstuff,suchasmeasuringlandscapes,derivingnumerouslawsandformulas fromasingleright-angledtriangle,andevendeducing theinstantaneousspeedbetweenveryshortperiods.In thisway,animaginarynumberwhichdoesn’texistin thefirstplace,seemsobscureandextraneoustolearn inregardstothebasicmeaningofmathematics.To giveananswertothisobscurity,itmightbehelpfulto discussabitmoreaboutthenumbersthemselves.

1.2Evolutionofnumbers

Behindtheaspectofmathematicscurrentlystandingasoneofthebasicsubjects,mathematicsencounterednumerousaltercationsandsupplementationsover time,startingfromprimitiveagestomoderndays.Likelyinthefuturetoo.-Humansstartedmathematics

∗ ChairoftheMathematicsSociety,LeaderofJPAMEditing Team

† MemberofJPAMEditingTeam

bymerelycountingthings;however,ascivilizationadvanced,itsconcepts,calculationsandapplicationsbecamesimultaneouslymoresophisticatedandcomplex. Theearlyhistoryofmathisextremelymurkytotell theclearorigin.Inotherwords,nobodycantellexactlywhenhumanbeingsbeganusingnumbersorsimplycountingstuff.Althoughtheoriginhasconsigned tooblivion,historianssuspectthatthepeoplebackthen wouldhavebeenunabletodothingsmorethansimply countingstuff.Ifweweretodrawanumberlinewith theinterpretationofnumbersatthattime,thenumber linewouldbeascalar;onlyallowedtomovetowards theright-Thisisbecausesincethenegativenumber wasn’tpresentbackthen,theonlywaynumberscould befurtheredfromthezeroisbyincreasing.

1.3Fractions

ThetrendstayedthesameuntilEgyptians(ancient egypt1740B.C.)cameupwiththefirstinnovation about4000yearsago,theconceptoffractions.

Althoughnowadays,theegyptianfractionsareno longercurrentandhavebeensupersededwithdecimal notationsandvulgarfractions(theformoffractionused nowadays),themeaningofitsusage,whichledusto anotherlevelofmathshouldbefocusedprimarily.

1.4Zeroandnegativenumbers

Thenextinnovativestepofmathematics,followed byfractions,wastheallowanceofzeroandnegative

Fig.1:Theprimitiveconceptofnumbers

numbers.Unlikethesedayswhereuseofnegativenumbersandzerohasbecomevirtuallynecessaryinmath, thesetwoconceptswereoccasionallymetupwithskepticismamongpeoplebackthen.Overall,earliercultures seemedtoprohibittheuseofzeroandnumberssmaller asanumber.Amongvariousreasonsbehind,itturns outthattheprimaryfactorwhichdeterminedtheacceptabilityofthesenumbersbackthenwas“Howpeople viewedtheconnectionbetweenmathematicsandreality.”(WelchLabs,2015)Mainly,itisthesubtletyofconsideringthevoidandsomethingevenlessthanemptyas numberswhichhaspreventedtheuseofzeroandnegativenumbers.Clearly,itwasnotbecausepeopleinthe pastwerelessintelligenttoperceivetheabstractconcept,but,theconceptsthemselveswerefartoosubtle tobeimplementedasnumbers.Tonameafew,ancient culturesliketheAncientEgypt(1740B.C),Babylonians(300B.C),andOlmecs(400B.C)haveseemedto admittheconceptofvoidastheplaceholderofnumber lines,however,stilltheuseofzeroandnegativenumberswerefollowedbyaskepticism.-Itisambiguous totelltheexactoriginofzero:Somehistoriansclaim thatitwasfirstintroducedbyBrahmaguptatheIndian mathematician,whereastheclaimthatitwasfirstused byMesopotamia(around3B.C.E)isoftenbroughtup too.-AroundB.C200,negativenumberswerefinally imposedaspropernumbersbyChina.Manyhistorians expecttheChinesetohaveusednegativenumbersfor recordingdebts.

StartingfromancientChina,countriesaroundthe worldstartedtoacknowledgetheconceptofnegative numbersandzeroinrelationtotheirpuredefinitionin mathdespitetheirmeaningsbeingsubtleinnature.

1.5Subtletyofnegativenumbers

Itisconceivablethatthischapterseemstrivialto some.Itlookslikethewholediscussionhasseguedinto thelonghistoryofmathwhilethemainfocusisimaginarynumbers.However,notonlythehistoryofimaginarynumbershasalottodowithpeople’sperception ofnumbersjustspoken,butweshouldalsofocusonthe factthatwearetakingnegativenumbersforgranted whilsttheyarestillsubtletermstorelatewithreal-

Fig.3:TheancientChina’snegativenumbers

itylikeimaginarynumbers.Althoughtheyareused forquantifyingobjectssuchasthebasementfloorsand debts,thoseobjectsarestilltechnicallycountedwith positivenumberswhereweconsiderthemasnegative sincetheyareoppositefromthestandard.Generally, negativenumbersarealsonotpresentinnaturewhere westillconsiderthemasnumbersforgranted.Asamatteroffact,theallowanceofnegativenumbersandzero hashittedwithbigchangestothenumbergrid.The acceptance0andnegativenumbersallowedthenumberlinebeingavector,meaningthatnumberswereno longermerelyincreasingbutalsodecreasingtoinfinity fromthen.Infact,thewayhowimaginarynumbersare consideredsignificantintermsofmathisveryanalogous tothewayhowthenegativenumberswere.Inspiteof thembeingsubtleintermsofthenature,themeanings theyinheremademathtobeconsistentwithitslaw.

2HistoryofImaginaryNumbers

2.1Avoidingnegativenumbers

Thisisnowthepointwherethingsspokenarefinallyembeddingwiththeimaginarynumbers.AlthoughculturesstartedtoacknowledgenegativenumbersaspropernumbersinancientChina,theoccasional appearanceoftheminequationsandformulaswerestill largelyignoredandavoidedevenuntilthe16thcentury, andthistrendhaseventuallyledtothediscoveryof imaginarynumbers.Generally,mathematiciansatthat time(Europe,16thcentury)tendedtoavoidanysort ofequations,havingthelikelihoodofnegativenumbers gettinginvolved.Inparticular,consideraquadratic equation,seencommonlyinclass:

Ofcoarse,asexpected,wehave2solutionswhere x=3couldbetheoneAndx=-5couldbetheother. However,unlikethesedayswhereequationslikethis arenosuchproblem,mathematiciansinthe16thcenturyweresoaversetonegativenumberssincenegative

Fig.2:TheEgyptianunitfractions

Fig.4:ExamplesofKhayyam’sdepressedcubic

numbersclearlydidn’tmakeanysensewithreality.In thisway,mathematiciansmanipulatedandalteredpolynomialequationstobekeptinapreferableformwhere anypresenceofnegativenumberswaseliminated.Thus, quadraticequationsandanyotherpolynomialwithadegreeofevennumbersweretotallyavoided,sinceingeneral,theyinevitablyhavenegativenumbersinvolved.

2.2Depressedcubics

Thesameapproachwasdonetocubicequations, wherethepersianmathematician,OmarKhayyamidentifieddifferentformsofcubicequationsknownasthe “depressedcubic”:

Asdemonstratedinfigure4,depressedcubicisa formofcubicequationwhereanynegativecoefficients andthetermofx2areabsentforthesakeofeliminatingnegativenumbers.-Anypolynomialofdegree nissaidtobedepressedifthetermof(n-1)ismissing.-Infact,cubicfunctionsandimaginarynumbers, twoconceptswhichseemmutuallyrelationless,share agreatconnection-“Complexnumbersarosefromthe needtosolvecubicequations,andnot(asitiscommonly believed)quadraticequations.”(AShortHistoryOf ComplexNumbers,2006)-.Thejourneyofimaginary numbersstartedfromanItalianmathematician,ScipionedelFerro(professoroftheuniversityofbologna).-It isoftenknownthatthefrenchscholar,ReneDescartes hasinventedtheimaginarynumber,howeveritistechnicallynottruesincetherewerealreadyothermathematicianspredatedtheconceptbutDescartesonlyhad formallydefinedthetermas“imaginarynumber”.

Inthe11thcenturywhenKhayyamidentifiedthose depressedcubics,thoseequationswerecomposedpreciselyandcleverlyenough,howeveritwasanotherissue formathematiciansatthattimetosolvethem.Atlast, bythetimearound1510,about5centurieslater,Del Ferrodiscoveredamethodsolvingdepressedcubicsin suchaconvenientandreliablemanner.Byperformingseveralcalculations,hecameupwithhiscompletely originalformulawhichmadehimabletosolveanydepressedcubicbysimplyplugginginvaluesintothefor-

mula.-Youcouldthinkofitlikeaquadraticformulafor cubicequations.-Sincethisisaverysignificanttopicin regardstothehistoryofimaginarynumbers,wewould talkaboutitfurther,startingwiththederivation.

ordertodeducetheformula,wesimplystartwith adepressedcubic:

Sincetheformulashouldworkforanycircumstance, wecansetthepremisethat3uv+c=0.Inthisway,

Fig.5:ScipionedelFerro

Considertheequationisquadraticandsubstitute valuesintothequadraticformula:

Inthisway,DelFerrodeducedtheultimatesolution totheconundrumwhichbotheredmathematiciansfor centuries.However,surprisingly,thissolutionwasnever revealedtothepublicevenafterDelFerro’sdeath.The reasonwhyDelFerroharboredhisinnovativesolution hastodowiththecustomamongmathematiciansin the1500s.Amathematicianwouldoftenchallengeanothermathematicianfortheirpositionwhichcouldbe saidasmathduels.Thiswasdonebyeachparticipant submittingaseriesofmathproblemstotheopponent andsimplytheonewhogotmoresolvingscorrectseized thetriumph.-wheretheloserwasoftencondemnedto

humiliation-.SinceDelFerrowastheonlyonewhohad thesolutiontothosedepressedcubics,hekeptitsecret toguaranteehisjobsecurity.Aftertwodecades(1526), whileonhisdeathbed,DelFerrofinallyhandeddown thesecretmethodtohisstudent,AntonioMariaFior. AfterDelFerro’sdeath,Fiorwasconceivedofhismathematicalprowessasheknewtheformula.Hewasboastfulandthoughthewouldbeinvincibleinmathduelsso thathechallengedawaymoreskilledmathematicianat thattime,NiccoloFontanaTartaglia.

OnFebruary12,1535,FiorchallengedTartaglia andtheygave30problemstoeachother,allofwhich weredepressedcubics.Surprisingly,eventhoughFior knewtheformula,hestruggledtosolveevenasingle questionwhereasTartagliacompletedsolvinginless than2hoursandprocessedtocompletelydominate Fior.Evenfurther,Tartagliafoundtheformulasolving depressedcubicshimselfashesuspectedFiorwasgiven downasecretmethodfromsomeothergreatmathematiciansincehehadtheboastfulnessbutdidn’tseem tobecapableoffindinghisown.Aftertheduelwhere Tartagliahadalsobecomeawareoftheformula,healso kepttheformulasecrettothepublicjustlikeDelFerro didforthesamereason.

Afterall,Tartagliabecamerenownedforhisvictory andmathematiciansreveredhimandstrivedtolearnthe know-howofsolvingdepressedcubics.WhileTartaglia kepthimselfspeechless,oneItalianpolymath,named GerolamoCardano-knownasoneofthemostinfluentialmathematiciansduringRenaissance-wasparticularlydesperatetolearnthemethod.

CardanopersistentlysentletterstoTartaglia,sometimessoundingflattery,butsometimesalsothreatening. Atlast,March25,1539,-inMilan-Tartagliarevealed hismethodtoCardonosecretlyunderasolemnoathfor Cardanotomaintainthesecrecythusnotpublishingit andtoonlywritethemethodinacypher.

Bygoingthroughfurthercalculations,Cardano

Fig.6:NiccolòFontanaTartaglia

foundawaytoturnanyformofcubicequationintoa depressedcubicandthussolveinTartaglia’sway.The followingshowsthemethod:

Considerastandardcubicequationinaform: y = ax 3 + bx2 + cx + d (12)

Tofindtherootsof x,thiscouldbeturnedas:

ax 3 + bx2 + cx + d (13)

Atthispoint,substituteanothervariable, z to x where

Inthisway,theequationwillbewrittenas:

add b 3a to x sincetheinitialpremisewasthat x b 3a Cardanowasdelightedbythefactthatfinallytheissue thathadstumpedmathematiciansovercenturieswas solvedandhadthedesiretopublishthesecretmethod. Simply,whatcountedthemosttoaforward-looking physicianwasthecreditthatwouldhavebeengiven byrevealingthetruth.Alsoasforbeingacontemporaryphysician,hewasn’tinvolvedinthosemathduels inthefirstplace,sohehadnoreasontomaintainthe secrecy.Thatbeingsaid,theonlythingthatstopped himfromdoingsowastheoathhesworetoTartaglia. SincehesworenottopublishTartaglia’smethod,he wasobligatedtokeepthesecret,whichonlyhappened until1543.In1543,CardanotravelledtoBolognawith hisstudent,Ferrariandmetanothermathematician, HannibalNave.

(14)

HannibalNavewasason-in-lawofDelFerroand healsotookoverDelFerro’sstatusintheuniversity ofBologna.HehadthenotebookpassedbyDelFerro whichDelFerrokepttherecordsofhisdiscoverywhile hewasalive.HestillkeptthenotebookbythetimeCardanotraveledtoBolognatoseehimandCardanogotto knowthatTartaglia’smethodwasalreadypredatedby anothermathematicianashesawDelFerro’snotebook. Inthisway,theoathCardanosworetoTartaglianotto publishthemethodwasnolongervalidasTartagliawas notthefirsttosolvethedepressedcubic.Thatbeing so,in1545,Cardanopublishedabook,ArsMagna,in whichhecoveredabunchoftopicsandconceptsinalgebra-whichhappenedtobemainlyaboutthedepressed cubic-.ThefollowingisfromtheArsMagna:

Inthisway,theformoftheequationcanbefurther solvedbyTartaglia’sformulaandthenwecouldsimply

ScipioneFerroofBologna,almostthirtyyearsago, discoveredthesolutionofthecubeandofthingsequalin number[thatis,theequationx3+px=q,wherepand qarepositivenumbers],areallybeautifulandadmirable accomplishment.Indistinctionthisdiscoverysurpassess allmortalingenuityandallhumansubtlety.Itistruly agiftfromheaven,althoughatthesametimeaproofof thepowerofreason,andsoillustriousthatwhoeverattainsitmaybelievehimselfcapableofsolvinganyprob-

Fig.7:GerolamoCardano

Fig.8:MilantoBologna

lem.InemulationofhimNiccolòTartagliaofBrescia, afriendofours,inordernottobeconqueredwhenhe enteredintocompetitionwithadiscipleofFerro,AntinioMariaFiore,cameuponthesamesolution,and revealedittomebecauseofmymanyentreatiestohim.

3ComplexNumbers

3.1Whyiscomplexnumberexpressedas a + bi?

However,therewasonlyoneproblemCardanofaced withDelFerro’smethod.

Considerthecase:

x 3 =15x +4 (15)

Thisproblemcouldbesolvedeasilyusinghighschoolmathematics,wheretheansweris x =4.However,aswepluginthevaluesintoDelFerro’sformula, somethingverystrangehappens.

Magna,thenheaddedthequotethatitis“subtleasit isuseless”.Todiscussthisobscurity,wehavetoclarify thethingswehaveobservedsofar:

1. Thegivenequation,x3=15x+4hasx=4asthe solution.

2. However,thedepressedcubicformulaseemstobe brokenaswepluginthevaluesfromtheequation.

3. Thus,wecansuspectthatalthoughtheequation seemstonotmakeanysense,somehow,thereisthe correlation.

Totellthetruth,thishashappenedneitherbecause Cardano’sformulawaswronginthefirstplace,norbecausetherewasamistakewhilederivingtheequation, butinfact,theequationwejustderivedactuallydenotesthevalueof4.

Toprovethis:

First,wehavetosubstitutevaluesassymbolsfrom theequationsothatthingscouldbemoremanageable.

Theformulahasslightlychangedascbecamenegative.

4= a + bi + a bi andastheterm bi wouldbecancelledout,weareleftwiththevalueof 4=2a,a =2

Asanidenticalequation,thevalueof a3 3ab2 shoulddenote2whichisatermthatdoesn’tinvolve anegativeterminasquareroot,andbythesamelogic, thevalueof 3a2 b b3 shoulddenote11.(Bothremoved negativeroots).

Aswealreadyknowthata=2, a3 3ab2 =2 could bewrittenas 83 6b2 =2,whichmakes b = ±1

Linkingthistoourfirstexpression,

Andinthisway,thislastwithoursupposedanswer 4.

Asshown,anequationwitharealnumberasthe solutionhasbrokendownintoanexpressionwithcomplexnumbers.Infact,CardanoavoidedthecaseinArs

I’mawaresomeindividualsarefacingdifficultiesat presentbutthissharesthesamelogicthatmakesyou abletowriteacomplexnumberinaform:a+bi.Apparently,thisisonlyasimplesubstitutionofnumbers whereweseparatedtermsthatinvolvenegativeroots fromthosewhichdon’t.Justlikewecanrewritethe expression: x +3x +4+1 5 into 4x,wejusthavesimplifiedtheexpression.

Fig.9:ArsMagna-Thegreatart

3.2ComplexPlanes

However,somethingreallymakesthisform, a + bi, verydifferentfromanyothersimplificationofexpressionisthatthisentireformitselfdenotesavaluefrom amoresophisticateddimension.Unlikeanyrealnumberwhichitsvaluecanbemerelyexpressedasasingle numberwithoutusinganycalculationsignssuchas+or -,acomplexnumbershouldinvolveatleasttwoterms, thefirsttermwhichdenotesthevalueoftherealnumber,thesecondtermwhichdenotesthevalueofthe imaginarynumber.Inthisway,thesetwoincompatible terms,realnumberandimaginarynumbersrequiretwo dimensionsinordertorepresentitstruevalue.

Whenthinkingofacomplexplane,itiseasytostart withanimageofacartesianplane.However,thenoticeabledifferencebetweenanormalgridandacomplex planeisthatonecomplexplanedenotesthevalueofone complexnumberwhereasapointplottedinanormal griddenotesacoordinate,apairoftworealnumbers todetermineaposition.Thegeneralideatoseparate imaginarynumbersfromtherealnumberswasoriginally devisedbyanItalianmathematician,RafaelBombelli.

RafaelBombelliwasacontemporarymathematicianwiththoseaforementionedmathematicians,andhe wasalsotheactualfigurewhosolvedthebrokenDel Ferro’sformulawhichCardanostruggledwith.

Basicallytheideabehindthesubstitution:

wasbasedonhisinterpretationthatanegativenumberinsideasquarerootwasanunprecedentedconcept,

therebytherehadtobedistinctionsbetweenrealnumbersandimaginarynumbers,thuslettingatodenote therealvalueand b todenotethevalueofanimaginary number.-Inthiscase,theactualvalueof b isareal number,butithappenstodenotethevalueofimaginary termssinceitismultipliedwith i,intheform:a+bi.In acomplexplane,thevalueof a isplottedinthehorizontalaxiswhichcalledthe“real-axis”,denotingthevalue ofrealnumberinacomplexnumberandthevalueof b isplottedintheverticalaxiswhichiscalled“imaginaryaxis”,denotingthevalueofanimaginarynumber inacomplexnumber.Inthisway,therepresentation ofacomplexnumberonacomplexplaneresemblescoordinatesonacartesianplane.However,asmentioned beforethisshouldnotbeconfusedthatthepointona complexplaneisnotacoordinateofapairofnumbers, butthisrepresentsavalueofasinglecomplexnumber. Alsoasyoucansee,anynumberinvolvinganimaginaryterminitsvaluecan’tbeplottedonthehorizontal plane.Thisisbecauseintheform a + bi,theonlyway acomplexnumbercouldbelieonthehorizontalaxisis whenthevalueofimaginarynumber, b is0.However, asthenumbersinvolveimaginarytermsinitsvalue, b getstohaveavalueother0,thusmakingthenumber tolieuponorbelowtherealaxis.Infact,thecomplex planecouldbesaidasanupgradeversiontotheinitial numbergridwheretheseconddimensionofimaginary numberhasbeenadded,thusbecomingatwodimensionednumberfromapointonastraightline.Inthis way,youcouldalsoconcludethateveryrealnumberis writteninaform a +0i,sothatonlythevalueofadeterminesanumber’spositionontherealaxis.Justlike inacartesiangrid,eachaxisstartsfrom0andincreases by1asitgoesupandby-1asitgoesdowntheaxis. Itcouldbesaidthatthecomplexplaneistheultimate weaponofmathwhichtheplaneitselfcouldrepresent everynumberhumanshaveinventedsofar,andsurpris-

Fig.10:Thecomplexplane

Fig.11:RafaelBombelli

inglycomplexnumbersseemstobetheultimatelyfinalcategoryofnumberswhereanynewkindofnumber can’tbemadeanymore.Supposemultiplying,square rooting,ordividingacomplexnumber.Althoughits valuecouldbeabitmessy,stillthenumberisdefined asacomplexnumber.

3.3Euler’sidentityandthespiral

Consideraunitcircledrawnonacomplexplane whereitsradiusis1,thusaright-angletriangledrawn withinthecirclehasahypotenuseof1.Inthisway,the pointontheunitcirclewhichhappenstobeoneofthe

verticesofthetrianglewouldalsobe1.Inthisway, letting ϕ tobetheanglethatacomplexnumberhas fromtheorigin,thiscouldbewrittenas cos ϕ + i sin ϕ Thisisbecauseinthissituation,thehorizontalheight oftherightangletriangleisdenotingthevalueofthe imaginarytermofacomplexnumber.Rememberthis istechnicallynotagraph-thefollowingcanbealso writtenas: ϕ asanabbreviation.Inthisway,thiscould beconsideredasafunctionwheretheformis: f (ϕ)= cos ϕ + i sin ϕ anditshouldbealsonoticedthatthisform isaformfurtheredfromtheinitialform: a + bi.By consideringthefollowingasafunction,wecananalyze thefeatureofthisfunction:

Firstly,bymultiplyingthefunction f (a) and f (b):

f (a) f (b)=(cos a + i sin a)(cos b + i sin b) (22)

Whichbysimplificationturnsouttobe:

cos a(cos b + i sin b)+ i sin a(cos b + i sin b)

=cos a cos b + i cos a sin b + i sin a cos b sin a sin b (23)

Howeverasthetrigonometricidentitysuggest,

cos a cos b sin a sin b =cos(a + b) (24)

Andalso i cos a sin b + i sin a cos b canbewrittenas i(cos a sin b +sin a cos b).Andinthesameway,thiscould bewrittenas i(sin a + b)

Bytrigonometryidentitiesandthusthewholeexpressioncanbewrittenas:

cos(a + b) i sin(a + b) (25)

Thisalsocorrespondstothevalueof f (a + b).In thiswaythefirstfeatureofthisfunctionisthat:

f (a + b)= f (a) f (b) (26)

Tofindthesecondfeatureofthisfunction,wesimplysquarethefunctionandget:

f (x)2 =(cos x + i sin x)2

=cos 2 x +2i cos x sin x sin2 x

=cos(2x)+2i cos x sin x

=cos(2x)+ i sin(2x) (27)

Thusthesecondfeatureisthat:

f (2x)= f (x)2 (28)

Fig.12:ComplexPlane2

Fig.13:TheComplexPlane3

Thethirdfeatureofthefunctioncouldbefoundby takingthereciprocal:

1 f (x) = 1 f (x) f ( x) f ( x) = f ( x) f (x) f ( x) (29)

Byinvokingthefirstfeatureofthisformulawhere f (a + b)= f (a) f (b).Thiscanwrittenas f ( x) f (x x) .However,sincethevalueof f (x x)=1 -as cos0+ i sin0=1, where i sin0=1, 1 f (x) = f ( x) becomestrue.

Lastly,wewoulddifferentiatethefunction:

f ′ (x)=( sin x + i cos x) (30)

Andinthisway, f ′ (x)= if (x).Thatsaid,amathematiciannamedEulerfoundaparticularsingularity whichthewholefeatureofthefunctionisveryanalogouswiththerulesofhisconstant, e,wherethefunctionhadthealmostsamefeaturewithanotherfunction, f (x)= ex Tocomparethis,thefunctionwediscovered hadarule:

f (a + b)= f (a) f (b)

f (2x)= f (x)2

1 f (x) = f ( x)

f ′ (x)= if (x)

Whilethefunction f (x)= ex hassimilaridentities: e( a + b)= ea e b

Alsotoreiterate, ϕ standsfortheangleofacomplex numberfromtheoriginandrstandsforthedistancea complexnumberisfurtheredfromtheorigin.

Inthisway, ex and (x) showedverysimilartrait wheretheonlydifferencewasthatwhendifferentiated, (ex )′ didn’tcorrespondtothevalue.ThuswhatEuler divisionwasthatbyslightlychangingtheformto ei x thefunctiontotallyworkedthesameas (x) wherestill otherrulesworkedthesame.Thisisbecause (ei x)′ = iex ,thusmakingthefunctionfollowthefeatureof (x) perfectly.

Theultimatebottomlineisthatwhenthecomplex number’sdistancefromtheoriginequalsto1,itsvalue canberepresentedas cos ϕ + i sin ϕ orinanotherway around: eiϕ .However,sincenoteverycomplexnumberhasadistancefromtheoriginas1,wemultiply thedistancebetweentheorigin, r infrontoftheboth functions,thusthefinalderivationis:

z = reiϕ orr cos ϕ + ri sin ϕ (33)

Byusingtheform z = reiϕ ,thisallowstheperformanceofmuchmoresophisticatedcalculationssuchas thevalueof 2i ,ii oreven i √i.First,weshouldbasically knowtherulesofthecomplexplane.Althoughitseems likeacomplexnumberisplacedatsomekindofrandompositionontheplaneaswemultiplyordividesome numbers,thatactuallydoesn’tturnouttobethecase. Firstofall,forthemultiplicationrules,anyrealnumberdoesn’tgetawayfromtherealaxis,anyimaginary ismultipliedandviceversa,anypureimaginarynumberdoesn’tmoveawayfromtheimaginaryaxisunlessa complexnumberinvolvingarealtermisn’tmultiplied.

Justlikethediagramabove,arealnumberlieson therealaxis,thusitcan’tincludeanyimaginaryterm unlessitismultipliedbyanothercomplexnumber.In thisway,therealnumberrotatesexactly180degrees clockwise(whennegativenumbersaremultiplied)or keepsinthesameanglewhenthesametypeofnumberismultiplied.

Inthisway,theruleofmultiplicationofcomplex numbersisthatwhentwocomplexnumbersaremulti-

Fig.14:ComplexPlane4

Fig.15:ComplexPlane5

plied:

1. Theanglesoftwocomplexnumbersfromtheorigin areaddedtogether.

2. Theeachdistanceoftwocomplexnumbersfromthe originaremultiplied.

Forexample,consideracase 5 (1+2i).Inthis case,5hasadistance r ,and r =5.Also,theangleof 5fromtheoriginequals0or2sincethenumberstays onthepositivesideoftherealaxis.Thus,thiscouldbe writtenas reiθ =5 ei 0 =5,wherewearebacktothe initialvalue,provingthatthisisnotthewrongmethod todenoteacomplexnumber.

Ontheotherhand,thedistanceof(1+2i)from theorigincanbecalculatedeasilyusingthePythagoras theorem, adjacent2 + height2 = hypotenuse2 ,wheredistance, d = √12 +22 = √5 andthe angleθ =arctan( 2 1 )= 1 1rad

Thevalue 5 · (1+2i)=5+10i hasadistanceof √52 +102 = √1025=11 18.Andhasanangleof arctan( 2 1 )=1 1rad.Ascouldbeseen,thedistance, 11.18isactuallythevaluewhichthedistanceof5from theorigin,5multipliedby √5,thedistanceof 1+2i fromtheoriginandtheangle,11.18radisaresultof theadditionofanglesoftwocomplexnumbers,0and

11.18.Inthisway,whenacomplexnumber, z1 = reiθ is multipliedbyanothercomplexnumber, z2 = veiϕ ,the valuecanbewrittenas: rv (ei (θ + ϕ)).Thisisthesame reasonthatmakesthecycleofmultiplicationof i which:

Althoughthisisacommonsensethatnoone doubts,byapplyingtheterminologywehaveacquired, thiscouldbeexplainedbecauseihasadistanceof1from theoriginanditsangleis90degreesfromtheorigin.In thisway,thedistancegetsmultipliedbutsinceitis1, itremainsthesamebut,thedegreesgetsincreasedby 90degreesasitkeepsgettingmultiplied,thusthefirst 90degreesrotationlastwith i,thesecondwith-1,the thirdwith i andreturningbacktotherealaxisandso on.Inthisway,letusputourselvesintosomeridiculous casessuchas 2i .Intuitively,thismeanswearemultiplying2,itimes,howeverthisdefinitioniswaytoosubtle togothroughourcalculation.Instead,letusturnthis intoafrom:reiϕ .Firstly,astheruleoflogarithms,2 canbewrittenas eln2 .Sincewearefindingthevalue of 2i ,thisequalstothevalueof (2ln2 )i .Thus,weare leftwiththesolution, ei ln2 ,valuewhichitsdistanceis 1furtheredfromtheoriginandtheangle, ln2 fromthe origin.

Thenhowabout ii ?Toquicklygothrough:

Inthisway,wegettoknowthatanimaginarynumbertothepowerofanimaginarynumberissurprisinglycorrespondingtoarealnumber,furtherproving thatimaginarynumbersandrealnumbersarenotentirelyseparatetermsbuttheyshowcorrelationunder thealgebrawhichcalculationsinvolvingcomplexnumberscanalsoresultwithrealnumbersasthesolution. Inaddition,thereasonwhythesolutionderivedfrom theequationinvolvingimaginarynumbersisvalidis becausealthoughimaginaryandcomplexnumbersare subtletermsinnature,theystillfollowthewholerules ofmathjustlikerealnumbersdo,thustheirvaluesare consideredequallyinthealgebra.

Lastly,solvingfor i √i:

Fig.16:ComplexPlane6

Fig.17:ComplexPlane7

Alsointhiscase,theequationmerelyinvolvingacomplexnumberhasarealnumberasitssolution.

3.4TheFundamentalTheoremofalgebra& Themissingpuzzlepieces

Thebottomlineofthisjournalshouldbespoken withoneparticularterminvoked,thefundamentalTheoremofAlgebra.Thegeneralideaofthistheorem,conceivedbyCarlFriedrichGauss(1799),statesthatevery polynomialequationofdegree n,mustcomeupwith nsolutions/rootsintotal.Thefundamentaltheorem ofalgebrastandsasthebasicfoundationwhichmost studyofAlgebraisbuilton.Anyequations,polynomials,functionsarefollowingthetheoremandthisisthe theoremthatstatesalinearequationhasasingleroot andthequadraticequationhastwo.Supposeacase y = x2 +1.Thisisakindoffunctioninwhichitsvertex liesuponthex-axiswithaconcaveup,thushavingno intersectionstothex-axis.

Thisisbecauseinorderforafunctiontohavea x-intercept,thevalueofymustbe0,thusmakingthe equationtobewrittenas x2 +1=0.Asexpected,there isn’tanypossiblerealnumberthatcanbethesolution ofthisequationsinceeveryrealnumbermusthavea positivevaluewhenit’ssquared.Thatsaid,thisisthe primarypartinwhichusingimaginarynumbersiscrucial.Sincetherootoftheequationisiand-i,imaginary termsarenecessaryinordertoconservethefundamentaltheoremofalgebra,sincewecan’tonlyallowthose equationswithsolutionsofrealnumberstofollowthe rule.

3.5Howtofindeverycomplexrootsofanequation

Asthefundamentaltheoremofalgebrastates,any polynomialshouldhavethesamenumberofrootswith itsdegree;however,Isuspecttheschoolhasonlytaught thewaystofindtherealsolutionsanddon’tconsider thoselateralcomplexroots.However,thiscouldbea hugemistakeinthegreatmeaningofalgebrasinceas wesawpreviously,complexnumbersworkandfollowthe

sameruleinmathandtheyarenotnecessarilyalways remainingcomplexnumberssincetheycanbecomereal numbersthroughadditionalcalculationsandmostimportantly,thefundamentaltheoremofalgebrashouldbe conserved.Usingthis,amethodexistsinwhichdervies all n rootsfromanequationapplyingtheconceptswe havelearned.Firstly,considerasituationwhere x4 = i Sofar,thewaythattheequationabovecanbeexpress intermsofxis x = 4 √i.

However,althoughthisistrue,sincetheequation hasadegreeof4,thereshouldbe3moresolutionsto thisandalsowedon’tknowtheactualvalueof 4 √i, expressedinaform:eiϕ

Tostartwiththegeneralpremise,weknowthat thereare4possiblesolutions,thusthereshouldbe4 complexsolutionsintotalthatcouldbelyingonthe complexplane.Inthisway,wecanconcludethateach solutionhasadifferenceof24inanglewiththeprevious root.ThiscouldbeprovenbyfurtheringDeMoivre’s rule,firstlettingntobethedegreeoftheequation, andtobetheangleofthefirstrootoftheequation,it couldbesaidthat n(θ )= ϕ sinceasthedegreeofthe equationis n.Thedistanceissquaredntimesandthe sameangleisaddedntimes.Thisisthesamewith n(θ + 360 n + + 360 n )= ϕ

Sinceanangleaddedwith360degreesequalsitself. Inthisway,theanglewillrotatebythevalue 360 n as wegodownthenumbersthatcanbetherootandthe distancewouldremainallthesameforthoserootswhich wouldbe n √d,where d isthedistanceofthenumber whichhastheseroots.Therotationgoesaroundthe wholecomplexplaneanditwillcomebacktoitsoriginal startingpoint,whichisthefirstroot.Thismeansthat therootshaveadifferenceof 360 n intheiranglesand allofthembecomewhenaddedntimes.Thiscould happenforinfinitybuttheresulthasonlynanglesin totalthattheywouldlastwithwhichisthesamereason whythecycleof i repeats.

Toconsiderthecurrentsituation n(θ )= ϕ where ϕ =90 and n =4.Inthisway,thismeansthat 4(θ )=90, andthisconcludesthatthefirstrootoftheequationis x = ei π 8 .Subsequently,weknowthat 4( π 8 + 360 4 + + 360 4 )=90,theangleofthenextrootwouldbe π 8 + 360 4 whichis112.5degrees,thenextwouldbe202.5andthe nextwouldbe292.5andthenextis382.5whichisthe sameanglewiththefirstangle,22.5( π 8 inrad).Thus the4solutionswouldbe x = e0 392i or e196 3i or e3 53i or e5 1i

Onceyouhavefoundtheangle,youcaneasilylook attheunitcirclediagramandguessthevalueofthe rootintheform: a + bi aswealreadyknowthevalueof thoserootsintheform: eiϕ

3.6RootsofUnity

Forthelastbutnotleasttopictocover,thereare rootsofunity.Byapplyingtheconceptswe’velearned we’llbeabletoprovetwomorerulesthatcouldbe

Fig.18:FunctionwithNoRoots

foundincomplexnumbers.First,wecanknowhow togetthemultipleofeveryrootofanumberbygoingthroughasimpleproof.First,considertheprevious casewhere x4 = i.Ultimately,therootsoftheequationwere x = e0 392i or e196 3i or e3 53i or e5 1i andthe sumoftheangleswere 22 5+112 5+202 5+292 5=630, whichequatesto270degrees.Inthisway,thevalueof themultipleofeveryrootis-isinceacomplexnumber with270degreesrotatedand1furtheredfromtheoriginis i whereasthenumberwhichhastheserootsis i.Wecanprovethisbyunderstandinghowthecalculationisdonewhilewearemultiplyingtheroots.First, lettingthefirstrootofacomplexnumberis eiθ where itis ei π 8 inthiscase.Assaidbefore,eachrootshasa differenceof 1 2 π ,sothatthenextrootcanbeusedthat itis ei π 8 · ei π 2 = ei( π 8 + 2π 4 ) .Thiswouldbe e πi 8 · e 2πi 4 · e 2πi 4 andsoon.Thatsaid,thechangeinanglescanbeconsideredasageometricprogressionandtheprogression ofrootis w,w e 2π n ,w e 2π n e 2π n .Wherewisthefirst rootofthecomplexnumber e ϕi n .Intermsoftheangles ontheotherhand,itisanarithmeticsequencesinceas e 2πi n ismultipliedeachtimetoacomplexnumber,the anglesareadded.Thusitis θ,θ + 2π n ,θ + 2π n + 2π n

Weknowthatsinceallthedistancesoftheseroots arethesametheyhavethesamenumbertheyarelasting withwhentheyaremultipliedntimes.Inthisway, themultipleoftheirdistancewouldcorrespondtothe complexnumberthathastheserootsbuttheonlything thatmightchangeistheangleitisawayfromtheorigin. Byturningthisintoasumofanarithmeticprogression formula,Wecansaythat:

2πi n .Thisisbecausethedistanceof1fromtheorigin is1,thusthedistanceofrootsallequalto1sincethey shouldremainthesameeventhoughtheyaresquared ntimesandtheangleis0,butthisalsomeans360. Youcanalsostarttheprogressionwith e θi n ,ifyoulike butthismaymakethecasemorecomplicatedsince youarestartingthesequencefromthenthterm,since w n =1.However,unlikeothergeometricsequences,this sequencerevolvesforeversincetheprogressionwould eventuallycomebacktotheinitialtermasitwillrotate thewholecomplexplane.Inthisway,youmaystart with1,butforthesakeofsimpleexplanation,Istarted theprogressionwith e 2πi n .Butstill,thisworksthesame regardlessofthetermoftheprogressionyouarestarting with.

Thisway, w = e 2πi n ,theprogressionofrootsis w,w e 2π n ,w e 2π n e 2π n andrepeat.Inthisway, w = e 2πi n ,sincetherearenrootsintotalandtheirangles arehavingadifferenceof 2π n andtheprogressioncanbe changedas w,w 2 ,w 3 , ,w n 1 ,w n

Byusingthesumofgeometricsequenceformula, sum = w (w n 1) w 1 .Inconclusion,nomatterwhatvaluew takes,thewholeexpressionbecomes0,thusthesumof rootsof w ,where w n =1 is0.

4Conclusion

Overall,wehavegonethroughthehistoryofimaginarynumbersandthebasicrulesofthecomplexplane. Apparently,thefundamentaltheoremofalgebramakes theusageofimaginarynumbersessentialeventhough theirexistencemaynotmakesenseinreality.Imaginarynumbersarenecessarytomakeanyequationto haveasolutionwhichcan’tbedonebyusingonlyreal numbers.

Asaresult,wenowknowthatthemultipleofroots ofacomplexnumberwillbecomethesamewiththat complexnumberwhenthedegreeisoddnumbersince whenn=oddnumber,thewholevalueoftheexpressionwouldbe ϕ + π ∗ evennumber ,makingthenumber tobethesameas360degreesrotationequalstothe samepoint.Ontheotherhand,whenthedegreen= evennumber,thevalueofmultiplesofrootswouldbe theconjugateofthecomplexnumbersinceitmakesthe pointtoberotated180,makingittheconjugate.This isthereasonwhyweareleftwith-iasthemultipleof rootsofxwhere x4 = i

Lastly,wewillproveasimpleruleofacomplex numberwithrootsofunity.Consideracomplexnumber w and w n =1.Surprisingly,thesumofrootsofwequals 0.Toprovewhythisishappening.

First,considertheprogressionoftheanglesofroots likewehavedonebefore.Theprogressionstartswith

Alsousingimaginarynumbersisnotakindofwordplay,butsometimesevenanequationwithasolution whichhappenstobearealnumberinvolvesimaginary numbersinitssolvingjustlikethespecialcaseinDel Ferro’sformula.Eventhoughtheexpressionseemed virtuallymeaningless,wecouldderivethesolutionby goingthroughthecalculationwhileseparatingimaginarynumbersfromrealnumbers,andinthisway,it couldbesaidthatusingimaginarynumberswasacrucialpartinthesituation.Imaginarynumbersareconsideredsignificantbecauseeventhoughthevaluethey denotedoesn’tmakesense,theirexistenceistheonly waytoconservethelawofmath,-literallyanybasiclaw suchasthemultiplication,subtractionandespecially thefundamentaltheoremofalgebra.Imaginarynumbersstillfollowtherulesofmathandinthisway,the ruleofmathdoesn’thaveanyparadoxorerror.Thus makingmathanunerringsubject.-Inaddition,calculationsinvolvingimaginarynumbersdon’talwaysend upinasubtleandmeaninglessnumberbutimaginary numbersaresometimesconsideredcrucialintermsof derivingareal-numbersolution.

1. NegativeNumbers. (n.d.). https://nrich.math.org/5747

2. OpenCourseWare,M.(2017,July 5).Necessityofcomplexnumbers [Video].YouTube.https://www.youtube.com/ watch?v=f079K1f2WQkfeature=youtu.be

3. 3Blue1Brown.(2017,May2).What’sso specialaboutEuler’snumbere?|Chapter5,Essenceofcalculus.YouTube. https://www.youtube.com/watch?v=m2MIpDrF7Es

4. Veritasium.(2021,November1).HowImaginaryNumbersWereInvented.YouTube. https://www.youtube.com/watch?v=cUzklzVXJwo

5. WelchLabs.(2015b,August28).ImaginaryNumbersAreReal[Part1:Introduction].YouTube. https://www.youtube.com/watch?v=T647CGsuOVU

6. WelchLabs.(2015b,September4).

ImaginaryNumbersAreReal[Part 2:ALittleHistory].YouTube. https://www.youtube.com/watch?v=2HrSG0fdxLY

7. WelchLabs.(2015b,September11).

ImaginaryNumbersAreReal[Part 3:Cardan’sProblem].YouTube.

https://www.youtube.com/watch?v=N9QOLr fcKNcfeature=youtu.be

8. WelchLabs.(2016,September3).

ImaginaryNumbersAreReal[Part 13:RiemannSurfaces].YouTube. https://www.youtube.com/watch?v=4MmSZrAlq Kcfeature=youtu.be

9. WelchLabs.(2015c,September18).

ImaginaryNumbersAreReal[Part 4:Bombelli’sSolution].YouTube. https://www.youtube.com/watch?v=DThAoT3 q2V4feature=youtu.be

10. WelchLabs.(2015d,September25).

ImaginaryNumbersAreReal[Part 5:NumbersareTwoDimensional]. YouTube.https://www.youtube.com/watch?v =65wYmy8Pf-Yfeature=youtu.be

11. WelchLabs.(2015d,October2).

ImaginaryNumbersAreReal[Part 6:TheComplexPlane].YouTube. https://www.youtube.com/watch?v=z5IG_6_zP Dofeature=youtu.be

12. WelchLabs.(2015f,October9).ImaginaryNumbersAreReal[Part7: ComplexMultiplication]. YouTube. https://www.youtube.com/watch?v=YHv R8siIiD0feature=youtu.be

13. Dunham,W.(1990).Journeythroughgenius:The greattheoremsofmathematics.NewYork.–https://ve42.co/Dunham90

14. Merino,O.(2006).Ashorthistoryofcomplexnumbers.UniversityofRhodeIsland.–https://ve42.co/Merino2006

15. ScipionedelFerro.(n.d.).math

ThePainter’sParadox

AlbertJinyongShin Year10Noro

MemberofMathematicsSociety

Email:jyshin26@pupils.nlcsjeju.kr

Editor

JooHyunKim∗

RecommendedYearLevel:KS4andabove

Keywords:Integration,Volume,SurfaceArea,Limit, Infinity,Paradox

Gabriel’sHornParadox,orthePainter’sParadox isnamedafterGabriel,anangelintheChristianBible, whoblowsthehorn.Inmathematics,itreferstothe 3-dimensionalshapeofahornmadefromafull 360◦ revolutionofthegraphof y = 1 x ,where x> 1.The actualparadoxisthattheinnersurface,orthesurface area,isinfinite,andhenceaninfiniteamountofpaint shouldbeneededtopaintalloftheinnersurfaces., butthevolumeofthehornisactuallyfinite,sothe horncouldbepaintedbypouringpaintthenemptying it-aninfiniteamountofpaintisneededtopaintthe sameareathatcouldbepaintedwithafiniteamount ofpaint.Simply,itsaysthatthesurfaceareaisinfinite whilethevolumeisfinite.

3SurfaceArea

Theareaofthetotalsurfaceofanobject.

4Limit

Avaluetowardwhichanexpressionconvergesas oneormorevariablesapproachacertainvalue.

5Infinity

Aconcepttodescribesomethingbiggerthanany naturalorfinitenumber-somethingwithoutaspecific valueorlimit.

6Paradox

Anideawheretwoviewsmakesenseseparatelybut don’tmakesenseatthesametime.

Tounderstandthis,therearefewterminologiesto knowandunderstand:

1Integration

Awayofgettingareaunderagraphandbasedon splittingtheareaunderthegraphintoinfinitelysmall rectangles.

2Volume

Theamountofspacethereisina3-dimensional shape.

∗ MemberofJPAMEditingTeam

Firstly,howdoweknowthesurfaceareaisinfinite, whilethevolumeisfinite?Ifwethinkaboutit,integrationistheprocessofsplittingthegraphintoinfinitely smallrectangles.Inthiscase,asitisa3-dimensional horn,itissplitintoinfinitelysmallcylinders.Theheight ofthecylinderis dx.Lettheradiusbe r .Thenthearea ofthecylinderbecomes:

A = πr 2 dx (1)

Thistendstoinfinity.Hence,wegettheintegral: ∞ 1 πr 2 dx (2)

Thehornisalsosymmetricalwhenviewinga2dimensionalview,wherethetoppartis:

y = 1 x (3)

Andthebottompartis: y = 1 x (4)

Thismakestheshapesymmetricalalongthe x-axis; hence,thedistancefromthe x-axistothegraphis y , or 1 x ,whichisequaltotheradiusofthehorn.Ifwe substitutethisvalueintotheintegral,weget: ∞

π ( 1 x )2 dx (5)

Wecouldgetthevolumeusingthis. π ,sinceit isaconstant,couldbepulledforwardsinfrontofthe integral.Moreover,itisquitecomplicatedtoworkwith aninfiniteintegral,sowelet ∞ be c.Thenwegetthe expression:

π c 1 ( 1 x )2 dx (6)

Ifwesolvethisintegral,weget:

π 1 x

c 1 (7)

Butcisinfinite,soweshouldputalimitaroundit toget:

However,wemustassumethatthecylinderisnot perfectlyround,andisslightlyslanted.Ifwelayit out,itlookslikeapartofacirclesegment,likeapizza crust,orFigure1(Fig.1).

Let’ssaythattheinnercurveis A,andthestraight edges B .Wecouldmergethetwoendswithlengths B togethertoformapartofaslantedcone.Whenwe shrink B ,wecouldmakeitintoaderivative;let’ssay thatthisis ds.Weshouldn’tuse x becausethistime, B isnothorizontal;itisslanted. A wouldnowbecome thecircumference,asnowthetwo B sidesarejoined. Togetthecircumference,wesaythattheformulais 2πr ,butforthereasonsexplainedearlier,theradius isagain y ,or 1 x ,Togetthesurfacearea,weagainuse integrationtoget: ∞ 1 2π 1 x ds (9)

(8)

Weshouldnowconvert ds into dx.Thiscanbe expressedasaright-angletriangle;thehypotenuseis theslanted ds,thehorizontalsideis dx,andthevertical sideis dy .UsingPythagoras’Theorem,weget:

Hence,thevolumeisequalto π ,whichisaconstant andthereforefinite.

Wecouldalsofindtheinnersurfaceareaina similarway.Thesurfaceareaalsoworksbyslicing thehornintoverysmallcylinderwithheightof dx

ds2 = dx2 + dy 2 ds = dx2 + dy 2 = 1+( dy dx )2 (10)

Weknowthat:

Fig.1:Source:Youtube-NumberPhile

Althoughthismightseemcomplex,itisquiteeasy tounderstand.Weknowthat x ∈ [1, ∞].Ifweplugin anyvalue, 1+ 1 x4 willalwaysbebiggerthan1,as x4 alwayshasapositivevalue(evenexponent),and 1+ a, where a ∈ R+ ,willalwaysbebiggerthan1; √b,where b> 1,isalwaysgreaterthan1.Therefore, 1+ 1 x4 is alwaysgreaterthan1forallvaluesof x greaterthan1. Wecannowsolvethis.Thesurfaceareais:

Thenwecansolvethistoget:

Ifwegobacktotheparadoxitself,nowwecan saythat ∞ unitsofpaintisneededtopaintthesame areawhere π unitsofpaintcouldalsopaint;otherwise, itissaying π = ∞,whichisobviouslynottrue,which istheparadox.

ThePowerTowerFunction

JamesJiminLim Year10Mulchat

MemberoftheMathematicsSociety

Email:jmlim26@pupils.nlcsjeju.kr

Editor JooHyunKim∗

RecommendedYearLevel:KS5

Keywords:LambertWFunction,Derivative,Integral, Tetration,ConvergenceandDivergence

1Introduction

Thepowertowerfunction,alsoknownastheinfinitetowerfunction,isconsideredoneofthemostcomplexfunctions.ThemathematicianLeonardEuler,the fatherofmanydifferentmathematicaldiscoveries,also statedinhisownpaperthat"nomanclearlyknows aboutthepowertower,"denotingthecomplexityofthe function.

2Representation

2.1Generalexpression

Thepowertowerfunctionisusuallyexpressedas:

f (x)= xx (1)

However,toexpressitmorealgebraically,itcanbe furthersimplified.Sincethereareinfinitelymanypowersof x,wecanexpressthisas:

y = xy (2)

Thismethodismorepreferredwhensolvingmathematicalquestionsandconjectures,howeverisnotthe mostelegantmethodtoexpressthefunctionitself.

2.2AlternateMethod:LambertWFunction

2.2.1LambertWfunction

TheLambertWfunctionisoneofthespecialfunctionsandisalsoknownastheomegafunctionorthe productlogarithm.Itwasdiscoveredandnamedafter aFrenchmathematiciannamedJohannHeinrichLambertandwasfirstintroducedinhisarticle"Lambert’s TranscendentalEquation"in1758.Thisspecialfunctionisoftenusedwhensolvingequationsinwhichthe unknownvariableisbothinsideandoutsideofanexponentialoralogarithmicfunction.TheLambertW functioncanbenotedas:

TheLambertWfunctionalsohasauniqueproperty where:

Whichcanbefurtherappliedindifferentbranches ofalgebraandcalculus.

2.2.2UsingtheLambertWFunction

Fromthegeneralexpressionofthepowertower functionabove,thenaturallogofbothsideswilllook as:

) (5)

Throughmultiplesteps,theequationcanbeturned intothefollowing: 32

1 y ln(y )= ln(x)

1 y ln( 1 y )= ln(x)

ln( 1 y )eln( 1 y ) = ln(x)

(6)

usedinsimpleproblemsolvingorfindingspeed(in kinematics),butarealsocreativelyusedinotherparts ofmathematicssuchaspredictingfluctuationsinstock marketsetc.

Thederivativeofthepowertowerfunctioncanbe rathereasilycalculated.Fromthegeneralexpression ofthepowertowerfunction,puttingthenaturallogon bothsidesleadto:

Fromthisstage,theimplementationoftheLambert Wfunctionwillsimplifythisequationinto:

ln( 1 y )= W ( ln(x))

1 y = e W ( ln(x)) (7)

Beforeanymoremanipulationofthecurrentequation,thepropertyoftheLambertWfunctioncanbe slightlymodifiedinto:

W (x)e W (x)= x

e W (x) = x W (x) (8)

Then,theequationfinallyresultsinto:

1 y = ln(x)

W ( ln(x))

y = W ( ln(x)) ln(x) (9)

AlthoughtheutilisationoftheLambertWfunction makesthefunctionmoreelegant,therearelimitations ofthisformula.Thebiggestlimitationisthatthisformulaforcesthepowertowerfunctiontoconvergewith anycomplexvaluesof x,whichisnottrue.Thevalues of y onlyconvergesintheinterval[e e , e 1 e ]intheinitial powertowerfunctionandoutsidethisdomain,thefunctiondiverges.Therefore,inordertousethemodified formulawiththeLambertWfunction,itisnecessaryto maketheassumptionthattheinfinitetetration(power) of x convergestoavalue,orthevalueof x isinthe interval[e e , e 1 e ].

3Calculus

3.1Derivative

Thedefinitionofderivativevariesindifferent branchesofmathematics,butincalculusthederivative ofafunctionistherateofchangeofanunknown variablewithrespecttoanotherunknownvariable. Inpuremathematics,derivativesoffunctionsare

Usingimplicitdifferentiation,amethodofdifferentiationwhenitisnecessarytodifferentiatebothsides treatingeachvariableasawholefunction,theequation becomes:

3.2Integral

3.2.1Integration

Comparedtothederivative,theintegralofthe powertowerfunctionisnearlyimpossibletobemanuallycalculatedbyhandduetothedifficultyinthe integration.However,itcanbepartlyderivedusingthe LambertWFunction,thoughitstilltakesnumerous stepsbeforehand.

3.2.2Integraloftheinverse

Theintegraloftheinverseofafunctionexistsand thewayitisderivedisasthefollowing:

Substitutingthisintotheoriginalequationbecomes:

Usingintegrationbypartsthisisexpandedas:

uf (u) f (u) du = xf 1 (x) F (u) = xf 1 (x) F (f 1 (x))+ C

3.2.3IntegralofLambertWFunction

(15)

Usingtheproductruleandletting u = 1 ln(x) and dv = W ( ln(x)),theintegralofthepowertowerfunctionis:

Theformulaoftheintegraloftheinversecanbe usedtocalculatetheintegraloftheLambertWFunction.However,astheinitialgoalistocalculatethe integralofthepowertowerfunction,itisrequiredto calculatetheintegralof W ( ln(x)) ratherthan W (x) Thecalculationisasthefollowing:

f 1 ( ln(x))= W ( ln(x))

f ( ln(x))= ln(x)e ln(x)

AstheintegraloftheLambertWFunctionisalreadycalculated,iftheexpressionissubstituted,itbecomes:

f ( ln(x))= ln(x) x

(16) Let u = ln(x) du = 1 x dx (17)

Goingbacktotheintegralof ln(x) x

Substitutingthisbackintotheoriginalequationwill finallygivetheintegral:

Usingtheformulafortheintegraloftheinverse,the integraloftheLambertWFunctionis:

xW ( ln(x))+ ln2 (W ( ln(x)) 2 (19)

3.2.4CalculatingtheIntegral

Tocalculatetheintegralofthepowertowerfunction,theproductrulemustbeused.Theproductrule isasthefollowing:

= uv vdv (20)

Asmentionedabove,sincethisintegralusesthe LambertWFunction,thisintegralisonlyvalidinthe domain[e e ,e 1 e ]

4PowerTowerFunctionandtheComplex Plane

4.1ComplexPlane

Thecomplexplane,alsoknownastheArgand planeoftheGaussplane,isaplanetorepresent plottingofcomplexnumbers.Thecomplexnumbers thatareplottedusuallycomeintheformof z = x + yi, where x and y arethehorizontalandverticalaxis respectively.However,dependingoncircumstances, thepointscouldbeplottedas z = reiθ ,where θ isthe anglebetweenthelinethatconnectsthepointandthe originandthehorizontalaxisand r isthelengthofthe line.

Themaindifferencebetweenthecomplexplane andthecartesianplaneisthatinthecomplexplane, the y axisisimaginaryinsteadofreal,whichenables ustoplotcomplexnumbers.

Themainreasonfortheuseofthecomplexplane istogeometricallyrepresenttheconvergenceandthe divergenceofafunction.Inthecomplexplane,xcoordinateswithyvaluesthatconvergeinthecartesian planeareplottedasablackandvaluesthatdivergeare plottedaswhitedots.Thisway,differentfunctionsshow differentinterestingshapesinthecomplexplane.

4.2PowerTowerFunctioninthecomplex planeandthefractalshape

Thepowertowerfunctioninthecomplexplanehas averycomplexandaninfinitefractalshape.

Fractalisdefinedasamathematicalshapewhere thepatterninfinitelycontinuesforever.Nomathematicianhasfoundoutthereasonbehindthefractalshape ofthepowertowerfunctioninthecomplexplane, howeveritisassumedtoberelatedtotheinfinite tetrationof x.

5Bibliography

1. Peter,L.(2013).ThePowerTowerFunction[online]https://thatsmaths.files.wordpress.com/2013/ 01/powertowerlambert.pdf

2. Steve,C.(2018). TheMost FunDerivativeofxˆxˆxˆx... https://www.youtube.com/watch?v=i_l1lz26C2M

3. Mathoma.(2016).CoolMath:TheLambertWFunctionAndInfiniteTetration https://www.youtube.com/watch?v=TNeJyVzDU20

4. Steve,C.(2021).LambertWFunction https://www.youtube.com/watch?v=Qb7JITsbyKs

5. CueMath.ArgandPlane-Definition, Formulas,Properties,PolarForm... https://www.cuemath.com/geometry/argandplane/

6. PeterL.TheFractalBoundaryofthePower TowerFunctionhttps://maths.ucd.ie/plynch/Publications/PowerTower.pdf

Fig.1

AsseeninFigure1(Fig.1),thecomplexnumbers areplottedasdotsandthevectorcanberepresented asarrows.

Fig.2

ConicSectionasLociofPoints

SophieJisuLee

Year11Mulchat

MemberoftheMathematicsSociety

Email:jisulee25@pupils.nlcsjeju.kr

Editor EmmaChaeeunChung ∗

RecommendedYearLevel:KS4andabove

Keywords:ConicSection,Eccentricity,Ellipse, Parabola,Hyperbola

1Introduction

Conicsectionisoneoftheconceptsofgeometry.It demonstrateshowdifferentfiguresareformedbyonly usingtwoobjects,aconeandaflatplane.Itseemslikea simplemethod,butitisacomplexideatounderstand theprocess.Howiseachfiguredefinedasaspecific shape?

2Cone

Coneisathree-dimensionalshapewithaflatcircularbasecalleddirectrixandapointededgecalledthe vertex.

foundbytheformula θ 2 whilethethetaisequaltothe givenangle.

3ConicSection

Conicsectionisafigureformedbytheintersection ofaflatplaneandacone.Thereisacircle,ellipse, parabolaandhyperbolaforthetypesofaconicsections. Theformedshapesdependontheangleoftheflatplane intersectingthecone.

4DandelinSphere

ADandelinsphereisasphereinsidethatistangent toacone.Itislocatedperpendiculartotheconeaxis, andtheaxisdividesthethetaintotwo.Thissphere isalsocalleda‘focalsphere’becauseofthename‘focus’ofitspointsthattouchtheplane.Insomecases, twoDandelinspheresareneededinoneconetoforma particularfigure,ahyperbola.Thisideacanprovehow theshapesareformedinaconethroughclearimages showingthelocusofpoints,furtherexplainedinlater sections.

5Eccentricity

Inafocus-directrixproperty,thetheoremproven byDandelinspheresstatesabouttheconstantineach conicsectioniscalledeccentricity.

Fig.1:

Thediagramaboveshowsthefigureoftwocones reverselyattachedattheirvertex.Thegeneratorsare thelinespassingthroughthevertexatavertexangle fromacone.Thevertexangleistheanglebetweenthe verticalconeaxisandthegeneratorofacone.Itcanbe

Eccentricityisanon-negativerealnumberedvalue inaconicsection.Itistheratioofthedistancefrom anypointtothefocusdividedbythedistancefromthat pointtothevertex,expressedin [e = c a ].

Aseccentricityisaconstant,itcandistinguishthe shapeofaconicsection.Itisduetothedifferentdistancesandanglesthateachshaperequirestobeformed. Thiscanbeinferredfromtheequation e = c a above,as thevalueofewillvarybyvaluecanda.Thus,thereis arangeofeccentricityforeachconicsectionthatdefines

itsshape.Thiswillbeintroducedinthesectionswhere eachconicsectionisexplained.

6Ellipse

Anellipseisformedundertheconditionwhere 0<e<1.Thefigurecanbeprovedthatitisanellipse throughitsproperty.PointP.isthelocusofthepoint. F1.andF2indicatesthefocalpointsoftheintersectioncurveC.ThesefocilieonthecurveCandarerespectivelylocatedonthefirstandtheseconddandelion sphere.

Inordertoprovetheconicsectionasacircle,only onefocuspointisrequired.Similartotheellipse,assumethatthisfocuspointismovingaroundthecircumferenceoftheconicsection.Asthepointwherethe verticalaxisandtheintersectionpointmeetsisthecentreoftheconicsection,thedistancebetweenthefocus pointandthecentrewillremainconstantasthefocus moves.Thisisthepropertyofacirclethatanypointon acircumferenceisalwaysequidistantfromthecentre,as mentionedabove.

8Parabola

Aparabolaisasymmetricalcurvewhereanypoint isatanequaldistancefromafixedpoint,focus.Itis formedwhentheplaneintersectstheconeinparallelto thegeneratorline.

Theeccentricityshouldbe1.Aparabolahasthis specificvaluefortheeccentricityratherthanarange becauseitcanbeformedintheonlycasewhereitsgradientisequaltotheslantofthegeneratorline.This meansthatthedistancefromanypointtothefocus andthedistancefromthatpointtothevertexmustbe thesame.

AssumethatthepointPmovesaroundthealong thecurveC.ThedistancesbetweenthepointPand eachF1,F2varydependingonthelocationofP.The importantfactorofthiscaseisthesumofthedistance fromPtoF1andPtoF2,whichcanbewritteninthe expression:

S = d(P,F 1)+ d(P,F 2) (1)

Thesumwillremainthesame.Thisistheproperty andalsothedefinitionofanellipse,wherethesumof thelocusofapointfromthetwofocalpointsisthe constant.

7Circle

Acircleisafigureofallpointsequidistantfromone point.Acurveneedstointersecttheconeinparallelto thebase,perpendiculartothedirectrix.

Theeccentricityis0foracircleduetothesame locationofthecentreofacircleanditsfocus.Using theexpression e = c a ,eresultsin0regardlessofthe valueofaandthevalueofcis0.

Toprovethattheintersectionisaparabola,aflat planeperpendiculartotheconeabovethesphereandan additionaltwocirclesparalleltothisplaneareneeded. TheplaneEwillbetheplaneinwhichitsgradientis equaltotheslantofthecone.

ThegeneratorlinegistangenttopointMonthe sphereandstartstorotatearoundthecircle.Theplane K,paralleltothebaseofthecone,intersectsthesphere throughthecirclethatpointsMandAlieon.

WhenlinegmeetstheintersectionplaneE,pointP istheintersectionpointofplaneE,linegandthecircle closesttothecone.AsthepointsFandMareboth tangentstothesphere,PFisequaltoPMaccordingto two-tangenttheorem.PointNliesonplaneEandplane K,formingaperpendicularlinetopointP.Thisinforms thatNPandABisthesamelengthastheyareinthe

Fig.2:

Fig.3:

samegradientwithpointNandA,andpointPandB eachlieatthesameheight.ABequalsMP.Thus,it isNP==PM=BA.Thisisinterpretedthatthedistance fromthemovingpointPtothefixedpointFandthe linedwillalwaysbethesame,matchingthedefinition ofaparabola.

9Hyperbola

Ahyperbolaisatwo-branchedopeninfinitecurve symmetricalongitsconjugateaxis.Itisformedwhen theplaneintersectsbothnappesofadoublecone.

Itcanalsobedefinedasthelocusofapointmoving inwhichtheratioofthedistancefromthefocustothe distancefromthedirectrixisaconstantgreaterthan one.Thismeansitseccentricityshouldbegreaterthan 1,expressedine>1.

tothepointP.Thentwoequationscanbewritten:

PM = PF 1

PN = PF 2

ThepointP,MandNlieonthegeneratorlineg, whichmeansthatMN=PM-PN.Also,asonlythepoint PismovingwhileF1andF2areatthefixedpositions,it isdeducedthatPF1-PF2isconstant.Thelocusofpoint PontheintersectioncurvehfromthefixedpointsF1 andF2isconstant,thereforethecurveisahyperbola.

10Bibliography

1. https://mathimages.swarthmore.edu/index.php/ConicSectionCircles

2. https://mathimages.swarthmore.edu/index.php/DandelinSpheres-TheoryTwo-Theorems

3. https://www.ucl.ac.uk/rmhajc0/cone.pdf

4. https://mathimages.swarthmore.edu/index.php/ConicSectionCircles

5. https://byjus.com/maths/eccentricity/

6. https://mathimages.swarthmore.edu/index.php/DandelinSpheres-TheoryTwo-Theorems

7. http://www.nabla.hr/PC-ConicsProperties2.htm

Todemonstrateahyperbola,adoubleconeandtwo spheresineachnappearerequired.TheflatplaneE verticallyintersectsatasmalleranglethantheangleof thegeneratorlinetogoacrossbothnappesofthecone. Thehyperbolahisthelineofintersectionoftheplane Eandthecone.

IttouchesthetwospheresatthepointsF1and F2.Insideeachtwosphere,therearethecirclesk1 andk2touchingthecone.ThepointsMandNlieson eachcirclek1andk2,andthegeneratorlinegintersects acrossthesepoints.

AsthelinegrotatesaroundtheconeatthevertexV,thePmovesalongthehyperbolah.Theline segmentsPF1andPMarethetangentsfromtheupper sphere,PF2andPNthetangentsfromthelowersphere

Fig.4:

https://mathimages.swarthmore.edu/index.php/Dandelinhttps://mathimages.swarthmore.edu/index.php/Dandelin-

InscribedRectangleProblem:Topology

DerekYejunYoo Year10Noro

MemberoftheMathematicsSociety

Email:yjyoo26@pupils.nlcsjeju.kr

Editor EmmaChaeeunChung ∗

RecommendedYearLevel:KS4andabove

Keywords:Topology,InscribedRectangle,Mobius Strip

1Introduction

Oneofthemostcommonstatementsrelatedto Topology;topologistsviewacoffeemugandadoughnut tobethesame.Butthesestatementsalwaysleavea question:howisthisrelatedtomaths?Asanexample oftopologybeingusedtosolveamathsproblem,The “inscribedrectangleproblem”iselegantlysolvedby understandingthepropertiesoftheconcept.

fourverticesthatwillcreateanyformofasquare?This probleminvolvesunderstandingtopologyatahigher level,sinceitisaquestionthathasbeenyetanswered.

2Theinscribedrectangletask Toshowtheessenceoftopologyinvolvedinthis question,asolutionwouldbefurtherexploredfora weakerversionoftheproblem.Asaninitialapproach totheproof,wehavetounderstandthefeaturesofa rectangle;thetwodiagonalsegmentswouldalwaysbe equalinlengthpassingthrougheachother’smidpoint.

Theinscribedrectangleproblemoriginatedfrom along-lastingproblem;Theinscribedsquareproblem, alsoknownastheToeplitz’conjecture:WouldaJordan curve(figure.1)-planesimpleclosedcurve-alwayshave

Beingthekeytotheproblem,Infigure.2,ifwecan guaranteethatsegments(A,C)and(B,D)sharetheir midpointFanddistance,thequadrilateralwouldbe provedarectangle.Hence,atthispoint,arectangle canbeevaluatedbythreepoints:twodescribingthe distancebetweenthepoints,andonedescribingthemidpoint.

3Dataintothe3Dplane

Theinformationcanbesimplifiedwiththefunction f(A,B)=x,y,z.Pointsxandywouldcoordinatethe segmentontheinitialcurve,butthezvaluewouldhave

Fig.1:

Fig.2:

xandyvalueatthemidpointandazvalueofthelength ofthesegment.Then,thismeansiftwosetsofvalues haveasharedzvalue,itshouldsimultaneouslysharethe midpointandlength:ultimately,aprovenrectangle.

4OrderedorUnorderedpairs?

Ifweassumethatwearecuttingthecurveatone point,andstraighteningit.Thetwopointsmakingup asegmentwouldlieonastraightlinerepresentingthe curve.Figure.3wouldbeshownasfigure.4.

thiscasethereciprocalrepresentationjustshowsthe samesegment,wehavetorepresentthesepointsasan unorderedpaironfigure.5.

Then,extendingthisrepresentationintoa2dimensionalplane,eachpointwouldrepresentingavalueof anaxis(figure.5)

Inthisrepresentation,wecanfindalineofsymmetrythatjustreflectstwopointsprovidingthesame information.So,thefollowingisrequired.

However,nowthereisadistinctionbetweenanorderedpair,andanunorderedpair.Anorderedpairdescribes(a,b)and(b,a)tobeseparatepoints;whereas, unorderedpairssaytheopposite.Consideringthatin

Theplaneisfoldedinhalf,dissolvingtwosame pointstoone.However,toproperlyrepresentthe points,thebluearrowshavetobeconnectedtogether, seemingimpossible.

Fig.3:

Fig.4:

Fig.5:

Fig.6:

Fig.7:

Fig.8:

Alternatively,theplaneiscuttedinhalf,imposing arathereasierway:yellowarrowsshouldbetogether inthesamedirection,andsameforthebluearrows.

tothe3D-plane,itmeansthatatleasttwounordered pairsofpointscorrespondstothesameoutputofthe zvalue:provingourpremisethatiftherearetwodistinctsegmentsoutputtingthesamezvalue,itcreatesa rectangle.

6RequiredImprovements

Theprocesswascorrect,howeverthefinalstepof consolidatingthestripintothe3D-planehasbeenmore “seemingtrue”thanproved.Furthermore,inadifferent circumstanceknownasthe“SudaneseMobiusBand”, Dr.BlainLawsonhasproventhatitispossibletoarrangetheMobiusstripinawaythattheedgesdon’t passeachother.However,theprocessexplainedinthis writingstillhasspacetoremainsolid.Thepractical proofthatDr.Lawsonprovidedthesurfaceofthestrip togobothunderandabovethecurveformedbythe edge,whichisnotsoinourcase.

Theorientationoftheplanenowallowstheblue arrowstounify,followedbyflippingoneoftheedgesto connectwiththeotheryellowarrow.

4.1TheMöbiusStrip

Surprisingly,theshapeappearstoshowdirectcongruencewiththeMobiusstrip(!!).Evidently,theplane thatrepresentsallpointsonacurvehappenstobea Mobiusstrip.

5Proof

Now,bringingbackthe3Dplanepreviouslyexplained,thereisahighcorrelationbetweentheMobius stripandthegraph.Sincethestripisadirectrepresentationofthecurveonthexy-plane,theMobius stripiscapableofreplacingtheprimitivecurve;Inthe Mobiusstrip,theedgesarerepresentingasinglepoint, notasegment.Sodoesthexy-plane.Hence,wewould havetomanipulatethesizeandshapeofthestrip(In thisprocess,thevaluesareneverdistortedsinceitis justchangingthescalefactorsofthevalues).Inthis process,thenatureoftheMobiusstripdoesnotallow thisharmonisingsteptohappenwithouttheedgespassingeachother.SincetheMobiusstripvisualisespoints onthecurve,ifthestiphastointersectitselftofitin

Fig.9:

Fig.10:

FindingtheOptimalPolicytoExploretheGrid WorldusingDynamicProgrammingand Q-learninginReinforcementLearning

AarenJoonseokKang

Year12Halla

ChairoftheMathematicsSociety

LeaderoftheJPAMEditingTeam

Email:jskang24@pupils.nlcsjeju.kr

Editor

AarenJoonseokKang∗

RecommendedYearLevel:KS5andAbove

Keywords:ReinforcementLearning,DynamicProgramming,Q-learning,Grid-World,OptimalValue Function

1Introduction

1.1MachineLearning

Therearenumeroustasksthatcomputersarecapableofandarefasterinthanhumans.Thisisthe mainreasonforcreatingnumerousmachinesandsystemswithartificialintelligence(AI)embedded.Althoughanartificialintelligencethatperformssimple tasksseemseasytocreate,thereareheavybehind-thescenescalculationsconcurrentlyprocessed.

Themachineswithartificialintelligencethatare mostseeninoureverydaylivesarebasedonatrainingprocessnamedmachinelearning.Machinelearning canbefurtherbranchedintothreedifferentparadigms oflearning:supervisedlearning,unsupervisedlearning, andreinforcementlearning.Supervisedlearningand unsupervisedlearningarebasedonpre-givendata,while reinforcementlearningistraininganAIbasedontrial anderror.Thispaperwillparticularlyfocusonreinforcementlearningonagridworldenvironmentthatis designedincustom.

1.2ReinforcementLearning

Onekeyconcepttoreinforcementlearningisthat theagent,theAIitself,willlearnhowtoproducethe bestoutputinthegivenenvironmentbyexploringthe

∗ ChairofMathematicsSociety,LeaderofJPAMEditingTeam

environmentthroughtrialanderror.Itwillexploreaimingtomaximisethenumericalrewardthatisgivento theagentineachaction.Theultimategoaloftheagent istomaximisetheaccumulatednumericalrewardatthe endofoneepisodeoronetrial.

Thispaperaimstoimplementbothsimulationsof dynamicprogrammingandQ-learninginacustomised basicgridworldenvironmenttocomparetheprocessof training.

2Keywords&BackgroundKnowledge

2.1ReinforcementLearningKeywords

Itisessentialtounderstandthebasicconceptsand keywordsofreinforcementlearningasitissignificantin understandingthelargerconceptsthataregoingtobe mentionedinthispaper:

1. Agent -Agentisthecomputerwhichinteractswith theenvironmentthroughobservingandtakingactions.Itsultimategoalistomaximisethereward inasingleepisodebyfindingaseriesofoptimal actions.

2. Environment -Theagentinteractswiththeenvironmentthroughaseriesofactions,statechanges, andtherewarditreceives.Theinteractionisdisplayedinthefigure1:Thekeywordsstate,reward, andactionsdisplayedinthediagramarefurtherdefinedbelow.

3. State -Stateisasetofinformationwhichdefines thecurrentsituationoftheagent.Theinformationinthestatecanbetransformedwhentheagent takesanactionintheenvironment.Thestatein timestep t isdefinedas St

4. EndCondition -Theendconditionisthecon42

Fig.1:Agent-EnvironmentInteraction ditioninwhichtheepisodewillterminate.The agent’saimistoreachthisstatewiththemaximised reward.

5. Action -Actionisthedecisionmadebytheagent dependingontheenvironment.Eachactionwill changethestateoftheenvironmentandtheaction takenattimestep t isdefinedas At

6. Reward -Rewardisanumericalvalueinwhichthe agentaimstomaximiseattheendoftheepisode. Rewardisawardedtotheagentforeveryaction thattheagenttakes.Similartostatesandactions, therewardattimestep t isdefinedtobe Rt .The rewardfunctioncanbemodelledintermsofthe currentstateandactionusingtheconditionalprobability:

2.2GridWorld

Itisalsoakeysteptodefinetheenvironmentin whichtheagentwillinteractthroughoutthispaper. Therearegoingtobetwodifferentgridenvironments whichtheagentisgoingtobeinteractingwith.

Agridenvironmentingeneralisasimpleenvironmentwhichisformedofatwodimensionalplane.Each gridisnumberedwiththetuple (x,y ) andtheagentcan movealongthegridinthefourdifferentcompassdirectionsineachtimestep.Unlessthereareotherspecific rules,theagentwillbeabletomovealongasinglegrid inasingletimestep.

Thefirstgridinwhichtheagentwillreactwithisa fourbyfourgridinwhichtheendconditionisdefinedto beatthecoordinates (0, 0) and (3, 3) whicharethetop leftandthebottomrightcorner.Everyothercoordinate ofthegridwillreturnthenumericalrewardof 1.The gridwilllooklikethefollowing:

7. DiscountFactor -Discountfactorisusedtocalculatetheprojectedrewardaswellasthereward receivedineachaction.Asthesignificanceofthe actionwilldecreaseasitisdoneinthefurtherpast orfuture,thediscountfactorisusedtoweigheach action’sreward.Thediscountfactorisdefinedas γ whichisintheinterval[0,1]asitismultipliedtothe actionsthataredoneinthepastinanexponential sense.Forexample,theweightedrewardforthe actiontakenntimestepsagowouldbe γ n Rt n where n ≤ t

8. ExpectedReturn-Expectedreturnisthesumof theprojectedrewardbasedontheactionsthatthe agentchoosesinthecurrentstate.Itissignificant whentheagentisdecidingforanactionasitsgoal istomaximisetherewardatthefinalstate.The expectedreturnGtattimestep t ismodelledas:

Thesecondgridworldwouldbearectangulargrid withthesizethreebyfour.Theendstatewillbeatthe twocoordinates (0, 3) and (1, 3),yetthetwocoordinates willhavedifferentrewardsof +1 and 1 respectively. Thestartingcoordinatewillbe (2, 0) asseeninthediagram,aswellashavingagridwhichtheagentcannot enterat (1, 1)

9. Policy-Policyistheactionthattheagentformulatestofollowthroughtheepisode.Inorderforthe agenttomaximisetherewarditreceives,itwould havetofindanoptimalpolicywhichisdefinedto havetherewardthatisequalorlargertoallthe otherpolicies.Thepolicyisnotatedwith π andis modelledasaconditionalprobability:

MarkovDecisionProcessisanessentialconceptunderlyingreinforcementlearningasitisalargepartof definingtheenvironment.TheMarkovdecisionprocess willonlybeimpliedwhentheenvironmentisfullyobservablewithanadditionalpropertythat“thefutureis independentofthepastgiventhepresent”.Thisspecific propertycanbemodelledasanequationofaspecific

Fig.2:GridWorld1

Fig.3:GridWorld2

2.3MarkovDecisionProcess(MDP)

inwhichtheexpressionimpliesthatthecurrentstate containsalltheinformationofthecurrentstatestraversedbytheagent.

Themarkovdecisionprocessinreinforcementlearningisdefinedasthetuple (S,A,P,R,γ ) where S isthe finitesetofstates, A isthefinitesetofactions, P beingthestatetransitionprobabilitymatrix, R beingthe rewardfunction,anddefinedasthediscountfactor.

2.4ValueFunctionsandBellmanExpectation Equations

DerivedfromtheMDP,therearetwovaluefunctionsdefinedinreinforcementlearningfortheagentto basethelearningfrombothvaluefunctionsreturnsthe expectedreturn.

Thestate-valuefunctionistheexpectedreturn whengiventhestatethattimestep t whichis St followingpolicy π .Thefunctionismodelledasafunction v whereittakestheparameter s:

Thus,byutilisingthetreediagrambelow,thevalue function vπ canbeexpressedintermsof qπ (s,a) tofind theBellmanExpectationEquationfor vπ :

Thesecondvaluefunctionistheaction-valuefunctioninwhichisexpressedastheformoftheexpected returnfromstate s takingaction a,followingpolicy π Thisismodelledasthefunction q whereittakes s and a asthetwoparameters:

Usingasimilartreediagram,buttransformedin orderfortheaction-valuefunction qπ tobetheroot nodeofthetree(Figure4),theBellmanExpectation Equationfor qπ canbeexpressedas:

BothvaluefunctionscanbeexpandedtobeexpressedastheBellmanExpectationEquationwhich showstherelationshipbetweenthestate-valuefunction andtheaction-valuefunction.

Inordertogettotheequationswhichareexpressed intermsof vπ and qπ ,thetwofunctions vπ and qπ have tobedecomposedtopossessrecursiveproperties:

Thesevaluefunctionsaresignificantintheprocess oftrainingtheagentusingQ-learningasitisQ-learning isbasedontheactionvaluefunctioninordertofindthe bestactionsateachstate.

3ExploringusingDynamicProgramming

Inthissection,thegridworldwillbeexploredusing dynamicprogramming(DP).DynamicProgrammingis ageneraltechniqueofsolvingtheproblemusingtwo differentcharacteristics:optimalsubstructureandoverlappingsub-problems.

Thesimilarprocesscanbedonefor qπ whichderives thenew qπ (s,a) topossessarecursiveproperty: q

3.1DPSimulation

SimulatingDPinreinforcementlearninginvolves twomainsteps:iterativepolicyevaluationandpolicyimprovement.DynamicProgrammingisoneofthe

Fig.4:TreeDiagramfrom vπ

Fig.5:TreeDiagramfrom qπ

trainingprocessestofind*whichistheoptimalpolicy inthegivenenvironment.

3.1.1PolicyEvaluation

Iterativepolicyevaluationistheprocessoffinding vπ byiterativelyapplyingtheBellmanExpectationbackup.WhenapplyingtheBellmanExpectation backupfornumerousiterations,thevaluefunctionwill convergetoacertainvaluefunction vπ

Theprocessmainlyusessynchronousbackupin whichtheprogramupdatesthevaluefunction vk +1 from thepreviousvaluefunction vk .Thisvaluefunctionis calculatedforallpossiblestates s′ ∈ S foreveryiteration.Thevaluefunctionforstate s′ - vk +1 (s) -will beupdatedfromthevaluefunctionofstate s′ fromthe previousiteration- vk (s′ )

Usingthediagram,theprocessoftheevaluation willlooklikethefollowingtreediagram:

considered.Then,wecanimprovethepolicygreedily usingthe argmax functioninwhichactsforallactions which a ∈ A:

Inanequationtoexpressthenewvaluefunction v

3.1.2PolicyIteration&Improvement

Toimprovethepolicy π ,thereneedstwomainsteps init:evaluatingthepolicy π andimprovingthepolicy π greedily.While vπ (s) canbefoundin3.1.1usingpolicy evaluation,thisvaluefunctionofthepolicyneedstobe evaluated,whichultimatelyevaluatespolicy π :

Then,thisfunctionwillimprovethevaluefromall state s ∈ S overthenewpolicy π ′ whichcanbeexpressed astheequation:

Then,actinggreedily,anewpolicy π ′ canbecreatedinreturnofthevaluefunction vπ .Repeatingthis stepagainnumerously,itisguaranteedtoconvergeto π ∗,whichistheoptimalpolicy.Thisiterationcanbe describedusingfigure7:

Toprovethattheiteratingthestepsofevaluating thepolicyandactinggreedilytoimprovethepolicyconvergesthe π ∗,adeterministicpolicy a = π (s) hastobe

improvingthenewvaluefunction v

π (s) fromtheold valuefunction vπ (s).Thiscanbeexplainedthrough theexpansionoftheequation q (s,π ′ (s)),inwhichis guaranteedtohavevalueswhicharegreaterorequalto thevaluefunction vπ (s):

s]

≤ Eπ ′ [Rt+1 + γRt+2 + γ 2 qπ (St+2 ,π ′ (St+2 ))|St = s]

≤ Eπ ′ [Rt+1 + γRt+2 + )|St = s]= vπ ′ (s) (15)

Then,whentheimprovementstops,theactionvaluefunctionofthecurrentdeterministicpolicy π will be:

qπ (s,π ′ (s))=max a∈A qπ (s,a)

= qπ (s,π (s))

= vπ (s) (16)

Thus,theBellmanoptimalityequation vπ (s)= max a∈A qπ (s,a) issatisfiedwhichmakestheequation vπ (s)= v∗ (s) true.

Fig.6:PolicyEvaluationof vk +1 (s)

Fig.7:PolicyImprovementof vπ

3.2DynamicProgrammingSimulationonGrid World

Inthissection,applyingtheprinciplesofDynamic Programming,willbeperformedongrid1mentioned previouslyinsection2.2willbeused:

Asstatedinthefigureabove,therearefouractions -north,south,east,andwest,andeachtransitionhas thereward Rt of 1

Additionally,eachofthefouractionswillhavea uniformrandompolicy:

Then,afteroneiteration- v1 (s) -thevalueofeach gridwillchangeto-1exceptthetwoendstatesasthe valuefunctionwillmodelthevaluefunctionafterone moveofthemodelfromthespecificgrid.Thus,each gridwillbecalculatedwiththevaluesofthefoursurroundinggridsweightedequallyasthegridhasauniformrandompolicy.Thefigures11and12displaythe changedvaluefunctionandthepolicy.

Thesimulationwillendintwoendstatesatthe squares (0, 0) and (3, 3) inthefourbyfourgrid.

ThesimulationhasbeencreatedusingPythonand thefullcodeisattachedinAppendixA.

Toexplainthecodebriefly,thefunction“policyEvaluation”withtheparameter“grid”willcalculatethe newpolicy π ′ (s) fromthepreviousvaluefunction vπ (s) Additionally,thenumberofiterationsisdeterminedby thevariable“ITERATIONNUM”statedatthetopof thecodeinthe“Variables”section.

3.3Results

Theoutputsofthecodeisprogrammedtohavetwo partsforeveryiteration:thevaluefunction vk forthe randompolicyandthegreedypolicywithrespectto vk

Fortrial0,thestartingstate,thevaluefunction v0 hasbeeninitialisedtoall0asitisthestartingstate theandthegreedypolicy π (s) hasbeeninitialisedto randompolicyasthevaluesofallgridareequal.

Figure9and10representsthevaluefunction v0 and greedypolicy π (s)

Thesameprocesswillbeiteratedinfinitelyconverg-

Fig.8:GridWorldforDynamicProgramming

Fig.9: v0 (s) ofthegridworld

Fig.10: π (s) accordingto v0 (s)

Fig.11: v1 (s) ofthegridworld

Fig.12: π (s) accordingto v1 (s)

ingto vπ (s) whichisdisplayedinfigure13and14.

Fig.13: vπ (s) ofthegridworld

Fig.14: π (s) accordingto vπ (s)

Thispolicy π (s) startstoconvergetofromwhen k =3 inthisparticulargridworldwhichisveryearly. However,thismayvaryindifferentgridsandenvironmentsdependingonthecomplexity.

Additionally,thisgridworlddidnotneedaseparate stepofpolicyimprovementanditerationasthepolicy v( s) convergedto vπ∗ (s) automaticallyafterthesteps ofpolicyevaluation.

4ExploringusingQ-Learning

4.1Q-Learning

4.1.1BellmanEquation

Q-learningisthemostbasicalgorithmintherealm ofreinforcementlearningasitisanapplicationofa simplegreedyalgorithm-analgorithmwhichchooses thebestactionineverystate.However,itmightnotbe abletoeasilyidentifyhowtoapplythegreedyaspectin a2-dimensional,orevena3-dimensionalenvironment.

Thisisthroughusingaq-valuefunction-action valuefunction q (s,a) -andthebellmanequation.

TheQ-valuefunctionisupdatedusingtheBellman equationlikethefollowing:

functiondoesnothavetohaveadefinitivevalueasthe recursivepropertywillmakethefunctionconvergetoa specificvalue.

4.1.2 ϵ-GreedyAlgorithm

Sofar,Q-learningwasconsideredtobeagreedyalgorithmwhichalwaystakestheactionsbyprioritising theexpectedrewardusingtherecursivecharacteristicof theequation.However,thissometimesdoesnotguaranteeconvergencetoanoptimalpolicybecausetherecan becasesinwhichtheactionwhichisinitiallychosenby theagentisnotthebestaction.Thus,thiswillresult inanexploitationratherthananexplorationinwhich theagentdoesnotknowtheexactrewardsoftheother actions.

Tosolvethisproblem,thereisatechniquecalled “ϵ-greedy”whichencouragesexplorationoftheagent byacertainprobability.Theepsilonvalue(ϵ)isavalue intherange [0, 1] whichtakesagreedyactionwitha probability p =1 ϵ andexploresrandomlyusingthe probability p = ϵ.Thus,whenassumingthattheagent explorestheenvironmentnumerouslymanytimes,the agentwouldconvergetotheoptimalpolicy π∗ bya higherchancethanonlytakingthegreedyactions.

4.2Q-LearningSimulation&Results

ToimplementtheQ-learningalgorithmingrid world2inFigure3(Section2.2),theBellmanupdate hastobeimplementedintheprogram.Thespecificline ofprogramusedisthefollowing:

reward = curQVal + self.learningRate

∗ (self.discFactor ∗ reward curQVal ) (19)

Thefunctioniscomposedoftwodifferentparts:the immediaterewardandthemaximumexpectedreward. r (s,a) istheawardinwhichisgiventotheagentwhen theagentexecutestheactionaatstates.Ontheother hand, q (s′ ,a) istherewardthatisexpectedfromthe nextstatewhichis s′ .Applyingthe max a function, whichtakesaction a astheparametertofindthebest actioninwhichreturnsthemostreward.Thediscount factor γ ismultipliedtotheresultofthe max a function astheexpectedrewardhastobeweightedinorderto balancewiththeimmediaterewardbeingreceived.

ThisBellmanequationisveryusefulintwodifferent ways.ThefirstwayisiftheMarkovianenvironmentis applicabletothereinforcementlearningenvironment, theexpectedrewardfromthefarfuturewillbeableto betakenintoaccountusingtherecursivecharacteristic oftheequation.ThesecondwayisthattheQ-value

Toexplainthegridenvironmentimplementedinthe programfurther,whiletheenvironment’sstateshave beenconsistentwiththediagram,therehasbeenan stochastic,randomisedvariable,aspectaddedtotheenvironmentinwhichtheagentmaybeforcedtomoveto anotherdirectionby20%chanceeachmoveotherthan theintendedaction.

Afterrunning100episodes(simulations)ofthisgrid environment,theQ-valuesateachstatewascalculated tobethefollowing:

Fig.15:Q-LearningResults

Thisshowsthattheagentwillconvergetotheoptimalrouteofthisgridworld.Thus,thesimulation displayshowQ-learningoperatesandshowshowtheQtableofvaluesofeachgridiscalculated.Forthespecific code,refertoAppendixB-Q-learningAlgorithmCode.

AnextensiontothebasicQ-learningalgorithmis theDeepQ-learningalgorithm,whichaddsaDeep learningaspectintothealgorithm.Q-learninghasa highlimitationontheproblemsitcanbeappliedtobecauseofthealgorithmdependsonthecomplexityofthe problem.Iftherearenumerouscasestobeconsidered, eachstateoftheenvironmentwouldnotbeabletocalculatefortheQ-value,whichispivotalintheQ-learning algorithm.

Thus,tosolvethisproblem,atechniquefromdeep learningisaddedtothealgorithm:theneuralnetwork. Theneuralnetworkenablesthealgorithmtoestimate theQ-valueexponentiallyfasterthantheprogramcalculatingitstepbystep.Thisisbecausetheneural networkestimatesthevaluethroughthehiddenlayers inthemiddlewhichisoptimisedtooperateinanextremelyquickmanner.However,althoughtheapproximationmethodchanges,thenatureofthealgorithmremainsequal,thushavingthename“DeepQ-learning” althoughbeingcalledasavariationwiththenamessuch asapproximatingfunctionorapproximator.

FortheneuralnetworktocalculatetheQ-valuesof theenvironment,itneedsa“CostFunction”.Wehave tousetheapproximatorfunctionwhichisexpressedas thefollowingadding θ whichistheweightofthenode intheneuralnetwork:

q (s,a; θ ) (20)

ApplyingthistocalculatetheCostfunction,the costfunctionwouldbemodelledasthefollowing:

5Conclusion&FurtherStudies

Overall,inthispaper,explorationofthetwokey fundamentalaspectsofreinforcementlearninghasbeen reviewedusingthesimplegridworldtofindtheoptimal policyusingbothofthemethods.Itcanbeseenthat eachofthealgorithmsshowsitslimitationwhenthe environmentbecomescomplexasitishardtocalculate allthevaluesofeachstate.

Thesetwofundamentalconceptshavebuiltup thebranchof“Model-freealgorithms”inreinforcement learning.Now,thereareseveralmorecomplexways inwhichanenvironmentcanbeexploredsuchasthe DDPGalgorithmorthePPOalgorithm;bothalgorithmsalsoinvolveadeeplearningaspecttooptimise theprocess.

Theareaof“reinforcementlearning”isdrawing moreandmoreattentioninrecentyearsasthereare successfulproductscreatedusingreinforcementlearning suchasAlphaGoandAIsplayingfamousvideogames. Althoughdivingdeepintothearearequiresmoredeep knowledge,understandingthebasicsofreinforcement learningcanprovideagoodinsightintohowtrial-anderrorprocessisimplementedtoanaivecomputer.

6Bibliography

1. R.SuttonandA.Barto,1998,Reinforcement Learning:AnIntroduction[online]Availableat: http://www.incompleteideas.net /book/RLbook2020.pdf

2. DavidSilver,2015,Lecture2:Markov DecisionProcess[online]Available at: https://www.davidsilver.uk/wpcontent/uploads/2020/03/MDP.pdf

3. DavidSilver,2015,Lecture3:Planning ByDynamicProgramming[online]Availableat:https://www.davidsilver.uk/wpcontent/uploads/2020/03/DP.pdf

4. JeremyZhang,2019,ImplementGrid WorldwithQ-Learning[online]Availableat: https://towardsdatascience.com/implement-gridworld-with-q-learning-51151747b455

5. JeremyZhang,2019,gridWorld_Q.py,Availableat: https://github.com/MJeremy2017/reinforcementlearning-implementation /blob/master/GridWorld/gridWorld_Q.py

ThisisextremelysimilartotheMeanSquaredError (MSE)asQ-learningiscalculatedinasimilarwaytoa calculationoftheMSE. MSE

Usingthischaracteristic,themodifiedQ-valuefunction q (s,a; θ ) iscalledaQ-target.

6. ,2018, (Q-learning,ReinforcementLearning)[online]Availableat: https://bluediary8.tistory.com/18

7. Jeina,2019,DeepQNetworks,10[online]Availableat:https://jeinalog.tistory.com/20

ADynamicProgrammingCode

4.3DeepQ-LearningNetwork(DQN)

BQ-LearningAlgorithmCode

EstimationoftheChangingVelocityDuring RocketLaunchUsingEuler’sMethod

JamesDaewoongKang Year12Halla

Co-ChairoftheMathematicsSociety Email:dwkang24@pupils.nlcsjeju.kr

Editor

AarenJoonseokKang∗ JooHyunKim†

RecommendedYearLevel:KS5andAbove

Keywords:ResultantForce,Euler’sMethod

1Introduction

1.1ModellingthePathofRocketMotion

Inthisinvestigation,rocketmotionduringliftoff willbeanalysedmathematically.Theliftoffofarocket isacrucialphaseinitsjourneytowardsspace,its motiondeterminedbyacomplexinterplayofvarious physicalfactorssuchasthrust,weight,dragand gravitationalforces.

Inordertoaccuratelyunderstandandpredictthe motionofarocketduringliftoff,itisnecessarytoapplymathematicalmodelsandequationstoanalysethe differentcomponentsoftheseforces.Thispaperdelves intothemathematicsbehindrocketmotionduringliftoff andexplorestherelationshipsbetweenthevariousphysicalfactorsthatdetermineitsbehaviourusingcalculus andtheEuler’smethodforestimation.

2Calculations

2.1ForcesActingonaRocket

Onarocket,therearefourforcesthatareinaction: weight,thrust,drag,andlift-asseeninthediagram below.Specificallyduringrocketlaunch,liftforcesare ignored,asthrustandliftforcesarebothindirections paralleltotherocketmotion.(TomBenson, Four forcesonaRocket )

Letthethrust,drag,andweightforcesbe T(t), D(t) and W(t),giventhatthesevalueschangeover time.Similarly,somevariablesthatareinvolvedinthe equationchangeovertime-massandvelocity-and thereforetheyaredenotedas m(t) and v(t).Theequationsfor T(t) and D(t) canbefoundontheNASA website:

Fig.1

Ve standsforexhaustgasvelocity, Cd standsfordrag coefficient, p forpressure, v (t) forvelocity,and A for thecrosssectionalareaofarocketincontactwiththe atmosphere.(TomBenson, Thedragequation )(Nancy Hall, RocketThrustequation )

Thethrustequationisoftensimplified:

3Euler’sMethod

Ifwearegivenvaluesforthefollowing,itispossible toestimatethevelocityusingEuler’smethod.

Assumethat k = dm dt =300kgs 1 , Cd =0 75, A = 10m2 , p =1 0kgm 1 , g =9 81ms 2 , Ve =5000ms 1 , m(0)=2600000kg , v (9177)=0ms 1

Theequationforweight,W(t),isgivenas:

2.2ResultantForce

Theresultantforceactingontherocketiscalculated by T (t) W (t) D (t) inrocketmotion.Inamoregeneralcontext,theresultantforceistherateofchangeof momentum.Therefore,thefollowingdifferentialequationissetasfollows:

Then, dv (t) dt = 3 75 2600000 300t v 2 (t) + 300 2600000 300t v (t) 1500000 2600000 300t 9 81

(10)

Notethatwhen t =9177, v (t)=0,fromtheassumptionsmadeabove.Thisvalueisobtainedbygraphing theequationitself-becausethepointmustbeonthe curve.However,ifrealvaluesareobtainedandused therewillbenoproblemswiththeequation.

Euler’smethodapproximatesthesolutiontoadifferentialequationatdiscretepointsintime,anditworks byusingthetangentlineateachpointtoestimatethe valueofthesolutionatthenexttimestep.Therefore, itisassumingaconstantrateofchangeof v (t),andthe accuracyincreaseswithasmallersteplength.Here,for simplicity,thesteplength, δx,issetas100.

Let xn bethe nth valueof t, yn bethe nth valueof v (t), mn bethe nth valueof dv (t) dt ,and yn bethe nth changein v (t).

Thevelocityoftherocketateachtimeintervalcan betabulatedasfollows.

Substitutethisintotheoriginalequation:

δyn iscalculatedusing mn δx,as mn = δy δx

4Conclusion

Asthegradientvaluesaredecreasing,thevaluesin thetableareanoverestimate.Euler’smethodissimple toimplementandprovidesagoodapproximationofthe solutionforsmallstepsizes,butitmaynotbeveryaccurateforlargerstepsizesorformorecomplexODEs.

Therearemoreaccurateandefficientnumericalmethodsavailable,suchasRunge-Kuttamethods,thatcan beusedtosolveODEs.However,inthesecircumstances wheresecondorderdifferentialequationsareinvolved, estimationmethodssuchasslopefieldsorisoclinesmay beuseful-asdemonstratedinthisstudy.

5Bibliography

1. TomBensonFourforcesona Rocket(nodate).NASA.Available at: https://www.grc.nasa.gov/www/k12/rocket/rktfor.html(Accessed:February11, 2023).

2. TomBensonThedragequation.(nodate).NASA. Availableat:https://www.grc.nasa.gov/www/k12/rocket/drageq.html(Accessed:February11,2023).NancyHallRocketThrust equation(nodate).NASA.Available at: https://www.grc.nasa.gov/www/k12/airplane/rockth.html(Accessed:February 11,2023).

BifurcationDiagram

AlvinSehunJeong

Year12Jeoji

MemberoftheMathematicsSociety

Email:shjeong24@pupils.nlcsjeju.kr

Editor

TaeheeKim ∗

RecommendedYearLevel:KS4,KS5

Keywords:BifurcationDiagram,DrippingFaucet, MandelbrotSet

1Introduction

WrittenbyRobertM.May,thearticle“Simple mathematicalmodelswithverycomplicateddynamics” waspublishedin1976.Thepaperinitiatedadebate abouthowsuchunpredictablechaoscanbecommenced fromasimple,intuitiveequation.Theequationwas designedtoimitatethepopulationofanorganismina controlledenvironment.Itshowsthechangeofthepopulationratio-aportionofthepopulationthataspecies occupiesintheecosystem-whichisintherangebetween0and1.Apopulation(ratio)ofaspeciesinyear nis xn andthenextyearis x[ n +1].Theequationis

xn+1 = λxn (1 xn ) (1)

wherelambdaisaconstantand xn isbetween0and 1. λxn representsthe‘reproduction’ofaspecies.Let (λ -1)astheaveragenumberofoffspringthatananimalproduces,then λ willrepresentthenumberofthe animalmultipliedayearlateronlyifnoneofthemdied. However,everylivingorganismdies.Sotheequation needsa‘death’termwhichdecreasesthepopulationof thespeciesovertime.(1- xn )worksasthedeathterm intheequation.Inaclosedenvironmentofrealnature, limitedresourcesworkasaregulatoroftheoverpopulatedspecies.Forinstance,overpopulation(high xn value)willleadtothelackoffood,consequentlydecreasingthepopulationoftheanimal.

Inthecaseof λ between0and1,theyfound xn convergesto0.Youmaythinkofaninferiorspeciesof rabbitswhicharesexuallydisabled.Intherealecosystem,thespecieswilldisappearbytheprocessofnatural selection.When1< λ <3,thedeathterm,(1- xn ),starts tointervene.Thepopulationinitiallyincreasesgeometricallybythereproductiontermbutasoverpopulation occurs,thepopulationdecreasesrapidlybythedeath term.Afterseveralrepetitionsofincreaseanddecrease ofthepopulation,thepopulationseemstosettledown toastaticvalueasthereproductionrateandthedeath rateattainequilibrium.

Thisistheextentinwhichecologistsareinterested, butwiththeiruncontrollabledesire,mathematicians wantedtofindhowtheresultoftheequationchanges when λ islargerthan3.Theydidwhattheydobestat; theygraphedit.

∗ MemberofJPAMEditingTeam

Fig.1: λ <1

Whattheyfoundisthatthegraphbifurcates-a fancywordforsayingthatthegraphisbranchinginto two-atrandompoints.Thismeansintheregionof valueL1,thepopulationofrabbitsoscillatesbetween twovaluesthan4valuesinregionL2then8inregion L3,andsoon.

Mathematicianscouldnotfindthereasonforthe constantthusthevalueleftasthefundamentalconstant ofmaths,alongwithpi,e,and √2.Graphsthatfollow thesameruleandhavesimilarmathematicalstructures arecalledbifurcationdiagrams.Eventhoughyoumight notbefamiliarwiththeconcept.However,itmight existinyourbathroomrightnowaswell.

Ifafaucetisnotwellclosed,waterdropletsfall withaconstantperiodlikedab,dab,dab,dab.However, ifthewaterflowincreases,theperiodincreasesby2as wellsoitdropslikedadab,dab,dadab,dab.Lateritbecomeschaoticandimpossibletofindtheperiod.dadaddadadb,daddadab,dadab,dadadadb,da....Thisincreaseofperiodcreatesthebifurcationdiagram.

Itwaschaotic,andseemedtohavenorulesat all.However,amathematicianwhosenameisMichael Feigenbaumfoundaverysimpleyetextraordinaryrule.

Twodimensionalversionofamandelbrotsetiswell known.Howeveritsthreedimensionalversionismuch different.

Fig.2:1< λ <3

Fig.3:.

Fig.4:period2

(y-axisisthepopulationofrabbitsandx-axisisthe time)

Fig.5:period4

Fig.6:period8

Fig.7:period18

Theratiobetweenthetwogapsbetweenthreeconsecutivebifurcationsisalways4.669...

2Drippingfaucet

3Mandelbrotset

Thisshowsthatthemandelbrotsetisanotherway torepresentthebifurcationdiagramwherethelessdivergentpartofthegraphisrepresentedasflatareas.

Fig.8:.

Fig.9:.

TicTacToe:WinningStrategyoftheGame

JuneKim

Year11Jeoji

MemberoftheMathematicsSociety

Email:jukim25@pupils.nlcsjeju.kr

Editor EmmaChaeeunChung ∗

RecommendedYearLevel:KS3,KS4

Keywords:TicTacToe,WinningCondition

1Introduction

TicTacToeisaclassicgameoftwoplayerstaking turnsandmarkingthespaceswithXorOinagrid. Despiteitbeingsimple,ithasamathematicalstrategy thatensuresaplayerawinoradraw.Thestrategy canbeexplainedintwosections:whenplayingfirst andwhenplayingsecond.

Players:

1. Player1=X(playingfirst)

2. Player2=O(playingsecond)

TicTacToeGrid:

2Whenplayingfirst(X)

2.1Player1startsfromthecorners

i)Player2doesn’tmarkatthecentrenextturn Startingfromthecornersistheeasiesttogetmany guaranteedwins.Theonlyplacethesecondplayercan winistoputatthecentre.Inalltheothercases,player 1getsthewin.

Forexample,whenplayer1marksXat(1),the secondplayermarksOat(2),andplayer1marksX at(7),thesecondplayerwouldhavetomarkat(4)to preventplayer1fromwinningwiththerow1-4-7.This allowsplayer1tomarkatthecentre,whichguarantees theplayerawinastheplayerwouldbeabletoputat both(3)and(9)tomaketherow3-5-7and1-5-9.

Guaranteedwinswhenplayer1startsfrom thecornersandplayer2doesn’tmarkatthecenter afterplayer1aretotalofseven,includingtheoneabove.

Fig.1:

1. Side:(2),(4),(6),(8)

2. Corner:(1),(3),(7),(9)

3. Centre:(5)

Notethatduetothesymmetryoftheboard,the boardcanberotatedinanydirection,andtheoutcome wouldstillbethesame.

Notethat“awinisguaranteed”whencompleting tworowsispossibleinoneturn.

∗ MemberofJPAMEditingTeam

ii)Player2marksatthecentrenextturn Markingatsymmetryiscriticalinthiscase.Forexample,withXat(1)andOat(5),player1mustmarkat (9),whichispositionedsymmetricalto(1).Forguaranteedwins,player2mustmarkatcorners,(3)or(7),as thiswouldallowplayer1tomarkat(4)or(8)tomake therows1-4-7and7-8-9togetawin.

However,thisdoesn’tmeanthatonlymarkingat thesymmetryguaranteesawinforplayer1inthiscase. Whenplayer1breaksthesymmetryandmarksat(6), withXat(1)andOat(5),player1mustmarkat(3)or (7)forawin,asthisguaranteewinswithpossiblerows 1-2-3and3-6-9or1-4-7and7-8-9.

2.2Startingfromthesides(forexample,(4)) i)Player2marksat(1)

Inthiscase,thepositionofthesecondXthatguaranteesthehighestchanceofwinningisat(3),asitis thepositionofsymmetrytoO.Whenplayer1marks Xat(3),player1willwinwhenplayer2marksat 57

(2),(7)and(9),whichgivesplayer1threechancesof winning.Forexample,withXpositionedat(3)and(4) andOat(1),whenplayer2marksOat(9)player1 wouldmarkXatthecentreinturntopreventplayer2 fromwinning,player1willgetaguaranteedwin,with possiblerowsof3-5-7and4-5-6.

ii)Player2marksat(2)

Inthiscase,player1mustmarkat(1)or(5)for guaranteedwins,asthepositionstouchthefirstX, allowingmorechancesforthefirstplayertomakea row.Forexample,withXat(4)andOat(2),when player1marksat(1),player2wouldhavetomarkat (7)topreventrow1-4-7.Thenplayer1wouldbeable tomakerows1-5-9and4-5-6andwin.

iii)Player2marksat(3)

Inthiscase,player1wouldhavetomarkat(9),a positionofsymmetrytoO,foraguaranteedwin.With Xat(4)and(9),andOat(6),player1willwinwhen player2marksat(2),(5),(6),(7),and(8),givingfive chancesofwinning.Forexample,whenplayer2marks at(5),player1willmarkat(7),topreventplayer 2fromwinningwithrow3-5-7,whichgivesplayer1 possiblewinswithrows1-4-7and7-8-9.

iv)Player2marksat(6)

Inthiscase,player1wouldhavetomarkat(9),which isasymmetricalpositionto(6).Thisgivesfourchances ofwinningforplayer1,asitguaranteeswinsforplayer 1whenplayer2marksat(2),(3),(5),(8).

v)Player2marksatthecentre(5)

Winninggetsharderinthiscase.Twocasesthatguaranteewins:markingat(2)andmarkingat(3).

2.3Startingfromthecentre

(i)Player2markingatoneofthesides

Inthiscase,markingatanyoneofthecornersguaranteesplayer1awin.Forexample,withXatthecentere andOat(2),player2mustmarkatoneofthecorners, suchas(1).Thentheplayer2willmarkat(9)to preventplayer1fromwinningwiththerow1-5-9.The nextturnplayer1markingat(4)guaranteesplayer 1awin,withtwopossiblerowstomake:1-4-7and4-5-6.

(ii)Player2markingatoneofthecorners

Inthiscase,player1shouldbackintothediagonally oppositecorner.Forexample,withXatthecentreand Oat(1),player1mustplaceatthediagonallyopposite corner(9).Thisguaranteesseveralchancesofwinning; whenplayer2inturnmarksat(2),(4),(6),or(8), player1wins.Forexample,withXatthecentreand (9)andOat(1),whenplayer2marksat(2),player1 willmarkat(3)topreventplayer2fromwinningwith therow1-2-3.Thisthenguaranteesplayer1awin,with twopossiblerows:3-5-7and3-6-9.

3Whenplayingsecond(X)

3.1Player1startingfromthecorners

Asmentionedbefore,wheneverOisplacedona side,Xisguaranteedawin,meaningthatforallstarts, thefirstOshouldneverbeplacedonside.Player2 markingatthecentreisthebestwaynottoallowplayer 1fromwinningeasy.

3.2Player1startingfromthesides

Inthiscase,thereisachanceofwinning,butwinningonX’sside-startismoreluckthanlogic.Asanalyzedin1.2,therearetoomanychancesofplayer1 winning,nomatterwhereplayer2marks.

3.3Player1startingfromthecorner

FromX’scentre-starttherearenopossibleOwins, asinbothcaseswhereplayer2marksatanyofthesides andwhereplayer2marksatanyofthecornersplayer 1hasawaytowin.

4Conclusion

Inconclusion,simplysaying,whenstartingfirst, puttingXinanycornerguaranteesplayer1asimplest winaslongasyouropponentdoesn’tputtheirfirst‘O’ inthecentrebox.

Whenstartingsecond,itmaytakemoreworktowin thegame.Ifplayer1takesthecentre,player2should placeOatthecorner.Ifplayer1takesthecorner,player 2shouldplaceOatthecentre.Thiswillmakeadraw inbothcases.Inthiscaseofplayer1markingatthe centeroratthecorner,winningisalmostonlypossible ifamistakeismadebyplayer2.

But,ifplayer1startsonaside,player2canwin. Inthiscase,player2mustmarkOinthecentretowin. Thisistheonlycaseplayer2canwin.Otherwise,player 2maydraworlose.

5Bibliography

1. Aycock,R.(2002).HowtoWinat Tic-Tac-Toe.[online]Availableat: https://www.cs.jhu.edu/jorgev/cs106/ttt.pdf.

2. Cappiello,E.(2020).HowtoWinTicTac-Toe:TheStrategiesYouNeedtoMaster.[online]Reader’sDigest.Available at:https://www.rd.com/article/how-to-win-tictac-toe/.

ProbabilityAppliedtoSocialSciences

JeanKim

Year11Jeoji

MemberofMathematicsSociety

Email:jnkim25@pupils.nlcsjeju.kr

Editor AarenJoonseokKang∗

RecommendedYearLevel:KS3andKS4

Keywords:GameTheory,SocialSciences

1WhatisProbability

Probabilityisdefinedasthelikelihoodofsomething happeningorbeingthecase.

P (x)= NumberoffavourableoutcomestoA Totalnumberofpossibleoutcomes (1)

2GameTheory

Gametheoryisabranchofappliedmathematics thatprovidestoolsforanalysinginterdependentsituations.Accordingtothistheory,eachsideconsidersthe other’spossibledecisionsandusesthemtoformulate strategiesforhimorherself.Thepurposeofthis‘game’ istofindthebestdecisionfortheplayers,andtheresultsmaydifferregardingtheirfindings.

Gametheoryhaswideapplications,especiallywhen players’choicesaffecteachother’spossibleoutcomes. Therearetwokeywordsingametheorythatneedsto bedefined:

1. Game:Asetofcircumstancesinwhichtheresult dependontheactionsofdecision-makers(moreor equalto2)

2. Players:Strategicdecision-makerswithinthecontextofthegame

3CooperativeandNon-CooperativeGames

Therearevariousgametheories,mostcommonly cooperativeandnon-cooperative.Cooperativegame

theoryisabouthowcooperativegroupsactwhenthe payoffisknown,morelikelyagamebetweenagroup ofplayersratherthanbetweenindividuals.Itquestions howgroupsareformed,andpayoffsareallocatedamong players.

Ontheotherhand,thenon-cooperativegametheorydealswithhowindividualsdealwitheachother toachievetheirowngoals.Themostcommonnoncooperativegamehappenstobethestrategicgame, wheretheavailablestrategiesandtheoutcomesresult inalreadyknownresults.Themostwell-knownand simpleexampleofareal-worldnon-cooperativegameis Rock-Paper-Scissorsandprisoner’sdilemma,whichwill appearlaterinthisarticle.

Zero-sumandnon-zero-sumgamesarealsogame theories.Whenthereisaconflictamongmultiple groupswantingthesameoutcome,itisoftenazerosumgame.Thereisalwaysaloserinthiskindofgame, astherearewinners.Alternatively,italsomeansthat thebenefitreceivedequalsthebenefitlost.Mostsports gamescanbeknownaszero-sumgamesasthereare bothwinnersandlosers.

Contrarytozero-sumgames,inanon-zero-sum game,winnerscanbeloserssimultaneously.Abusiness partnershipthatismutuallybeneficialandfostersvalue forbothentities.Insteadofcompetingandattempting towin,inthiscase,bothpartiesbenefit.

5Volunteer’sDilemma

Volunteer’sdilemmaiswhensomeonehastoundertakeachoreorajobforthecommongood.The worstpossibleoutcomehappenswhentherearenovolunteers.Areal-lifeexamplecanbewhentheelectricity ofthewholeapartmentfails.Peoplelivinginthesame apartmentknowthattheelectricitycompanywillfixthe

∗ ChairofMathematicsSociety,LeaderofJPAMEditingTeam

4ZeroSumandNon-zeroSumGames

problemifsomeonecallstonotifythematsomecost. Ifatleastonepersoncallsthecompany,everyonecan benefitfromit.However,ifnoonevolunteers,theworst possibleoutcomeisobtainedforallparticipants.

Forexample,ifIliveinafourstoryapartmentwith twohousesineachstory.Considerineachhousethere isatleastonepersonstayinginthehousewhenthe electricityfails.Theminimumprobabilityofatleastone personotherthanmecallingtheelectricitycompanyis 7 8 .Therefore,inthiscasethereisahigherprobability thatsomeoneelsewillcallthecompanyeventhoughI donothing.Ontheotherhand,ifIamawarethatonly twopeoplearecurrentlystayingathome.Thenthe probabilityofanotherpersonintheapartmentcalling thecompanyisonly 1 2 ,whichmakesmemorelikelyto callthecompany.

6Prisoner’sDilemma

Oneofthemostfamousexamplesofgametheoryis theprisoner’sdilemma.

Forexample,twocriminalsarearrestedforcommittingacrime.However,prosecutorsdonothaveappropriateevidencetoconvictthem.Togainaconfession, officialsoftenseparatetheprisonersandquestionthem indifferentareas,providingprisonerswithnomeansof communication.Therearetwooptionsfortheprisoners,toconfessornot.However,thisleadstodifferent consequences.

Forexample,

1. Ifbothconfess,theywilleachreceiveafive-year prisonsentence;

2. IfPrisoner1confesses,butPrisoner2doesnot, Prisoner1willnotgotojail,butPrisoner2will getaten-yearsentenceorviceversa.

3. Ifneitherconfesses,eachwillservetwoyearsin prison.

bothprisonersdonotconfess.However,asbothprisonersareunawareoftheother’sstrategy,theprobability ishelpfulforthemtopredictwhatothersmightdo.In thiscase,thenumberoffavourableoutcomestoAis1, butthetotalnumberofpossibleoutcomesis3.Therefore,theopponent’sprobabilityofstayingsilentisonly 13.Thislackofcertaintythattheotheronewillnotremainsilentcausestheprisonertoconfess,whichmakes itmorelikelyforbothprisonerstoserveinprisonfor fiveyears.

Assuch,gametheoryiscloselyrelatedtoourlives. Thistheoryisfrequentlyusednotonlyintheprisoner’s dilemmabutalsoinareassuchasbusinessandeconomics.

Asbusinessesrequiremanystrategicchoicesthat affecttheireconomicgain,theymayfacemanydilemmaswhendecidingwhethertodevelopnewproducts, stopproducingoldones,orusemarketingstrategies.

7RepeatedOligopolyGames

Unliketheprisoner’sdilemmawhichisonlyplayed bytwoplayerswitheachgivenapayoffmatrix,only beingabletomakeonechoice,andthegameendsafter thefirstroundofchoices,therealworldoligopolyhas asmanyplayers.

Asoligopolygamesmayhavemorethantwoplayers,theyarefarmorecomplex.However,itdoesnot changeitsbasicstructure.Thefactthattheverygames arerepeatedintroducesnewstrategicconsiderations. Playersmustconsiderhowtheirchoicesmightaffect theirfuture,nottherivals.

8Bibliography

1. britannica.com.2022.GameTheory-Britannica.[online]Availableat: <https://www.britannica.com/science/gametheory>

2. hal.scence.2019.Probabilityandsocialscience. [online]Availableat:<https://hal.science/hal02065664/document>

3. investopedia.com.2022.GameTheory-Investopedia[online]Availableat: <https://www.investopedia.com/ terms/g/gametheory.asp>

4. innovationmanagement.se2020.IntroductiontoGameTheory[online]Availableat: <https://innovationmanagement.se/2020/11/06/ introduction-to-game-theory/>

Fig.1:Prisoner’sDilemma

Themostfavourablestrategyischoice3,where

EquationofCircleandGPS

JungseoPark

Year9Geomun

MemberoftheMathematicsSociety

Email:js2park27@pupils.nlcsjeju.kr

Editor

TaeheeKim ∗

RecommendedYearLevel:KS4,KS5

Keywords:Equationofacircle,GPS

1Introduction

Nowadays,GPSisusedinvariousplaces.GPStechnologyisusedintrafficaccidentnotificationdevices, vehiclenavigationsystems,vehicleenforcementdevices, anddatacollectiondevicesforroadusagefeescollection.

2GPS

GPS,ortheGlobalPositioningSystem,isaglobal navigationsatellitesystemthatprovideslocation,velocityandtimesynchronization.

2.1GPSprinciple

AGPSreceiverreceivessignalstransmittedfrom threeormoreGPSsatellitestodeterminethepositions ofthesatellitesandthereceiver.Bymeasuringthetime differencebetweenthesignaltransmittedfromthesatelliteandthesignalreceivedbythereceiver,thedistance betweenthesatelliteandthereceivercanbeobtained. Inthiscase,thetransmittedsignalcontainsinformation aboutthepositionofthesatellite.Inmostcases,the locationisdeterminedusingthreesatellitestoreduce theerror.

3Equationofacircle

Thestandardequationofacircleisgivenby:

Fig.1:.

Where(h,k)isthecoordinatesofthecentreofthe circleandristheradius.

Fig.2:.

4TheusesofcircleofequationinGPS

Nowthisisthemethodtofindthelocationbyusing theequationofthecircle.Tounderstandeasily,itwill findthelocationofArchimedeswhoisrandomlylocated onEarth.

First,thesatellitesendsradiowavestodetermine thedistancetoArchimedes.Atthistime,Archimedes liesonacirclewhoseradiusistheknowndistance.However,onecirclecannotdeterminetheexactlocation,so othersatellitesareusedtomeasurethedistance.

Second,measurethedistancefromanothersatellite towhereArchimedesis.Thistimeasecondcircleis createdandArchimedesisalsopositionedonthesecond circle.

Amongthecorrectionsofthetwocircles,thelower intersectionpointtouchingtheearthisArchimedes’sposition.Nowconnectthecircletothecoordinateplane. Thisconfirmsthatthegpsmeasurementprincipleis linkedtotheequationproblemofacircle.

ThemidpointofcircleAis(-3,0)andthemidpoint ofcircleBis(0,3).Therefore,theradiusofcircleAis 2kmandtheradiusofcircleBis4km.Now,ifeach circleisexpressedasanequation, A :(x +3)2 + y 2 =4 (2) B : x 2 +(y 3)2 =16 (3)

Fig.3:.

Fig.4:.

Fig.5:.

Fig.6:.

OpticalIllusions

SangyoonLee

Year10Jeoji

MemberoftheMathematicsSociety

Email:sy2lee26@pupils.nlcsjeju.kr

Editor

DanielSoohyukChang∗

RecommendedYearLevel:KS3,KS4

Keywords:OpticalIllusion

1Introduction

mostfamousopticalillusions,asshownin Figure1.This trianglecannotexistintherealworld.Exceptforthis triangle,therearemanyopticalillusionsexisting.

2Howdoesopticalillusionwork

Fig.1:PenroseTriangle

Inthedictionary,opticalillusionmeanssomething thatdeceivestheeyebyappearingtobeotherthanit is.Thesedays,manyopticalillusionsaremadebymany famousartists.PenroseTriangleisoneoftheworld’s

∗ MemberofJPAMEditingTeam

Fig.2:Exampleofbrainfillingtheinformationwhich isnotreal

Iwillexplainthisinaverysimpleway.Whenwe seeorfeelthingsandgatherimagesfromthesenses,it sendstheinformationtothebrain.Inthisway,they domorethanjustgatherthedata.Whenthereis incompleteinformation,ourbrainfillsthisgapwithour ownperception.Whenourbraincreatesanimagethat isnotthere,theopticalillusionoccurs.Forexamplein Figure2,therearefoursectorswithanangleof270° However,inourbrains,theyfindthemissinggapinthe image.Assaidatthetop,whenourbrainfindsthegap, theytrytofillthegapwithperception.Theconclusion

ofthiscreatingimageismakingawhitesquare.Our brainmakesawhitesquareandseemslikeitisonthe topoffourcircles.Thisishowtheopticalillusionworks.

calillusionartistsintheworld.HeisaHungarianFrenchartistwhoobtainedtheentrancequalificationto theBudapestUniversityMedicalFacultytobecomea doctorbutgaveitupandwenttoDesignSchool.He wasgreatlyinfluencedbytheworkofMoholyNagyand wasinfluencedbytheworkofMalevich,Mondrian,and Kandinsky.Hisgrid-likepaintingsandsculpturesare hismostfamousworks.

thatthecoloroftheshoeispinkwithwhitehighlights. Somescientistssaythatthisdependsonwhichbrained personyouare.Ontheotherhand,theysayitdepends onlighting,background,andanythingrelated.

4Conclusion

Byresearchinghowopticalillusionworks,examples ofopticalillusions,andfamousopticalillusionartists,I wasabletounderstandopticalillusions.Also,bystudyinghowopticalillusionworks,Ihavelearnedthatthere

3Examplesofopticalillusionandfamous artists

Fig.3:Coloroftheshoe

Figure3 isoneofthefamousphotosshowingoptical illusion.Inthisphoto,peoplecanbedividedintotwo groups.Onegroupofpeoplesaysthecoloroftheshoe isgraywithtealhighlights.Andtheothergroupsays

Fig.4:VictorVasarely(1906-1997)

VictorVasarelyisoneofthemostfamousopti-

Fig.5:VictorVasarely’s‘Zebra’

Fig.6:VictorVasarely’s‘KepleGestalt’

Figure5 and Figure5 aretwofamousartworksby VictorVasarelythatshowopticalillusions.In Figure 5,heexpressesthezebrawithopticalillusion,andin Figure5,hemadeanopticalillusionwithasquareand thedifferenceofcolorofeachsquare.

aremanydifferenttypesofopticalillusionsandtheprinciplesofopticalillusionsarealldifferent.Itmightbe betterifIstudymoreaboutmakingdifferenttypesof opticalillusions.

5Bibliography

1. Britannica‘illusion’

https://www.britannica.com/topic/illusion

2. NationalGeographic‘Unravelingopticalillusions withmaths’

https://www.nationalgeographic.com/science/article/partnercontent-when-circles-are-finally-squared

3. QueenslandBrainInstitute‘Howdoesanoptical illusionwork’

https://qbi.uq.edu.au/blog/2019/10/how-doesoptical-illusion-work

4. AmericanMuseumofNaturalHistory‘Optical IllusionsandHowTheyWork’

https://www.amnh.org/explore/ology/brain/opticalillusions-and-how-they-work/filling-in

5. Health‘IsThisShoePinkorGray—andCanYour AnswerReallyTellYouAnythingAboutYour Brain?’

https://www.health.com/mind-body/sneakeroptical-illusion

6. PanArt‘VictorVasarely’

https://pan-art-connections.com/content/vasarely/

ApplicationsofGraphTheory

DanielTaehyeonJun Year10Geomun

MemberofMathematicsSociety

Email:thjun26@pupils.nlcsjeju.kr

Editor AarenJoonseokKang∗ DanielSoohyukChang†

RecommendedYearLevel:KS4andabove

Keywords:GraphTheory,Dijkstra’sAlgorithm, Floyd-WarshallAlgorithm

1Introduction

LeonhardEulerintroducedgraphtheoryinthepaperSevenBridgesofKönigsbergin1736.Thefundamentalofthispaperwastodeviseawalkthrough thecityofKönigsberg-crossingeachofthebridgesin PregelRiveronlyonce.Interestingly,Eulerprovedthat theproblemhadnosolutioninitiallybecausetheroute choiceandlandmasswereirrelevant.Therefore,he hadtoreformulatetheprobleminmoreabstractterms, namely"edge"and"vertex",whichlaidthefoundation ofgraphtheory.Graphtheoryisaboutpracticaluse. Mostofthedataweuseoncomputersheavilyrelieson graphtheory.Forinstance,thelinkstructureofawebsiteisrepresentedbythedirectgraphinwhichvertices representthewebpages,andedgesrepresentthelink fromonepagetoanother.Moreover,incomputerscience,graphtheoryisusedinchemistryandphysicsto studymoleculesandhowtheyinteract.

Thispaperwillgoin-depthaboutvariousterminologiesingraphtheoryandhowthedifferenttypesof algorithmsworkaswellastheapplicationsinreallife.

2Terminology

Inunderstandinggraphtheory,preciseknowledge ofeachkeyterminologyisveryimportant.

1. Graph -Agraphisanorderedpairwhichcanbe representedas G =(V,E ) where G standsforgraph,

∗ ChairofMathematicsSociety,LeaderofJPAMEditingTeam

† MemberofMathematicsSociety

V isasetofverticesand E isasetofedges.

2. WeightedGraph -Ifagraphhasanumberassociatedwitheachedge,itiscalledaweightedgraph. Ifnot,itiscalledanunweightedgraph.(Referto Fig.3)

3. DirectedGraph -Ifagraphhasadirectionassociatedwitheachedgeinsteadofanumber,itis calledadirectedgraph.(RefertoFig.5)

4. Walk -Awalkisaroutethroughagraphalongthe edgesfromonevertextothenext

5. Path -Apathisawalkinwhichnovertexisvisited morethanonce.

6. Trail -Atrailisawalkinwhichnoedgeisvisited morethanonce.

7. Cycle -Acycleisawalkinwhichtheendvertex isthesameasthestartvertexandnoothervertex isvisitedmorethanonce.

8. Degree -Adegreeisthenumberofedgesincident tothevertex.

9. AdjacencyMatrix -A nxn matrixwhichrepresentstheweightofthegraphforeverypossible route.Forexample,iftheweightoftheedgelinkingthevertices C and D is 5,thecorresponding values, aCD and aDC willbelabelled 5.Ifitwas adirectededgefrom C to D ,thematrixvaluefor aCD wouldbe 5 but aDC doesnothaveavalue.

10. DistanceTable -Atableformofadjacencymatrix thatshowstheweightorthedistanceoftheshortest route.(RefertoFig.1)

11. RouteTable -Atablethatshowstheshortest routefromeverysinglepossiblerouteusingthealphabetofthevertex.(RefertoFig.4)

Inordertocalculatetheshortestroutefromthe startvertextothedestinationvertex,Dijkstra’salgorithm,Bellman-FordalgorithmandFloydalgorithmare themostsuitable.ThedifferencebetweenthethreealgorithmsisthatDijkstra’salgorithmcanonlybeused

ingraphswithpositiveweightmeaningpositiveedges whereastheBellman-FordalgorithmandFloydalgorithmcanhandlebothpositiveandnegativeweights oftheedges.So,whatisthepointofusingDijkstra’salgorithm?TheadvantageofDijkstra’salgorithm overtheBellman-FordalgorithmandFloydalgorithm isthatittakeslesstimetocalculatetheshortestdistance.ThedifferencebetweentheFloydalgorithmand Dijkstra’salgorithm,excludingthatonecanhandlenegativeweightandtheothercannot,isthatDijkstra’salgorithmonlycalculatestheshortestroutewhereasthe Floydalgorithmcalculatesallofthepossibleroutesand comparesthedistancetofindtheshortestroute.This allowstheFloydalgorithmtobeusedinplaceswhere Dijkstra’salgorithmcannotbeused.

3Dijkstra’sAlgorithm

3.1Definition

Dijkstra’salgorithmcanbeappliedtoanytype ofgraphwithpositiveweightincludingdirectededges, suchasaroutewithone-waystreets.

Thefirststepistocreateaboxlikethefigurebelow foreachvertex.Inthetopleftspace,writedownthe vertex,inthespacetotheright,writedowntheorder oflabellingandinthespacefarright,rightdownthe finalvalue.Inthespacebelowisfortheworkingspace incalculatingtheorderoflabellingandfinallabel.

Repeatthesestepsuntilthedestinationvertexreceivesitsfinallabel.

Findtheshortestpathbytrackingbackfromthe destinationvertextothestartvertex.Includeanedge AB ontherouteif B isalreadyontherouteandthis becomes:

finallabelB finallabelA = weightofedgeAB (2)

Byusinganexampleofthefollowingfigurebelow, wewillcalculatetheshortestpathfromvertex S to vertex T

Followingthesteps,thefirstthingtodoistosubstitutethevectorswithboxesandfillthevertexcellwith thecorrespondingvertexalphabet.Thenextstepisto assign 0 inthefinallabelboxofvertex S and 1 inthe orderoflabellingboxsincethequestionisrequiringus tofindtheshortestpathfrom S to T

Wethenfindtheweightofeveryvertexthatisdirectlyconnectedtovertex S whichinthiscasearevertex

A,vertex B andvertex C .Comparetheweightofthe threeverticesandassign 2 intheorderoflabellingbox ofthevertexwiththesmallestweightandtheweightin thefinallabelbox.Forourexample,thisbecomes:

Thesecondstepistoassign0forthefinallabelof thefirstvertex.

Thethirdstepistorecordaworkingvalueatevery vertex, Y ,thatisdirectlyconnectedtothevertexthat hasjustreceiveditsfinallabel, X :

finallabelatX + weightofedgeXY (1)

Fromeveryworkingvaluefound,selectthesmallest workingvalueandassignittothefinallabel.

Thisprocessisrepeated.Fromvertex A,calculate thefinallabelofthedirectlyconnectedverticesexcludingvertex S .Inthisstage,wealreadyhavethevalue forvertex B whichis 12 butsincetheweightfromvertex A tovertex B is 3,thenewworkingvalue, 11,by adding 8 and 3 replacestheworkingvalue.Comparing thefinallabelofvertex B andvertex D , 11 and 21 respectively,wecanconcludethatproceedingtovertex B istheshorterroutethanproceedingtovertex D .The productofvertex B becomes:

Aftergoingthroughthisrepetitiveprocedureuntil wereachthedestination,vertex T ,thefinalstepisto tracebacktheroutefromthedestination.Thisisnot thesamewiththeorderoflabellingandwehaveto matchtheweightandthefinallabel.RefertoAppendix B forthefinallabelofeachvertex.

Fig.1:Exampleofadistancetable

Fig.2:BoxinDijkstra’salgorithmreplacingeachvertex

Bymatchingtheweightandthefinallabel,select thevertexwhichmatchesitsfinallabelandthevalue offinallabelofvertex T -weightlinkingthevertexwith vertex T .Inthisexample,therearethreeoptions,vertex F ,vertex G andvertex H .Vertex F andvertex G cannotbethenextvertexsincethetwovaluesdonot match.Thisleaveswithvertex H ,whichbecomesthe nextvertex.Repeatthisprocessuntilwereachbackto thestartvertex.Therecanbemultipleroutesforthe shortestdistanceandinthiscase,therearetwopossibleroutes. SCEHT and SABDGEHT .Eitherofthem arecorrect.

route,includingnegativeweightedgedgraph.Thefirst stepistodrawadistancetablethatmatchesthegraph. Wherethereisnodirectedroute,whetherduetobeing adirectededgeorhavingnopath,theinfinitysymbol ()isusedtorepresenttheweight.Afterdrawingthe distancetable,drawaroutetablewhicheveryinitial onewouldlooklikethefollowing:

Sincegraphtheoryisaboutalgorithms,asmentionedpreviously,itisalsorelatedtocomputerscience. ForanexamplePythoncodeofDijkstra’salgorithm, refertoAppendixC.

3.2ApplicationoftheDijkstra’salgorithm

Thepurposeofthenavigatingsystemistofindthe shortestorthefastestrouteonecantakefromastarting pointtothedestinationpoint.Theprogramsetsthedeterminedstartinglocationasthestartingvertexandthe stoppinglocationasthedestinationvertex.Eachstreet betweenthestartingpointandthestoppingpointwill berepresentedbyedgesandtheotherpossibledestinationsarerepresentedbyvertices.Theprogramrunsthe algorithmandfindstheshortestpossibleroutesomeone cantakebyfollowingthemechanismaforementioned. Also,navigationmustbeabletodeterminetheshortest drivingtimesaccordingtotheupdatingreal-timedata. Usingtheexampleabove,assumethattheweightofthe graphisthetimetakenfortheedge.Iftheweightof theedgefromvertex H tovertex T waschangedfrom 5 to 13,thefinalvaluewouldalsochangefrom 31 to 32.Thismeansthatthetotaltimetakenfrom S to T hasincreasedbyoneminuteandalsotheshortestroute wouldhavechanged.Theoriginalshortestroutewas SCEHT from SABDGT or SABDFGT

4Floydalgorithm 4.1Definition

Asintroducedseveraltimesbefore,Floydalgorithm isusedtofindtheshortestrouteamongsteverypossible

Ifthetwotablesareready,shadethefirstrowand columnofthedistancetable.Ineachoftheunshaded positions,comparethesumoftheshadedvaluesthatare inthesamerowandcolumnoftheselectedunshaded position.Ifthesumissmallerthanthevalueassignedin theunshadedposition,replacethevaluewiththesum andifthesumisgreater,leaveitasitis.

Intheroutetable,replacethealphabetwith[A]for anyvaluesthathavebeenreplacedfromtheprevious step.Thisistheendofthefirstiteration.

Theseconditerationstartsbyshadingthesecond rowandcolumnofthedistancetable.Oncemore,comparethesumoftheshadedvaluesforeveryunshaded positionandifthevalueissmaller,replaceandifnot, donot.Intheseconditeration,replacethealphabet with[B]foranyvaluesthathavebeenreplacedduring theseconditerationonly.

Repeatthethirdandfourthiterationjustlikethe previoustwobutinstead,shadethethirdrowandcolumnforthethirditerationandreplacethecorrespondingvalueintheroutetablewith[C]andshadethefourth rowandcolumnforthefourthiterationandreplacethe correspondingvalueintheroutetablewith[D].

Finally,supposethattheshortestroutewewantto findisfromvertexAtovertexC.Findthevaluefrom thedistancetablewiththestartingvertexontherow andthedestinationvertexonthecolumn.Thisvalue representstheshortestdistancefromvertexAtovertex C.Inordertofindtheroute,findthealphabetwith thesamemethod.IfthealphabetisB,thisissuggestingthattheshortestroutefromvertexAtovertexC requirestopassthroughvertexB.

UsingFig.5asanexample,wewillnowinvestigate howtocalculatetheshortestdistancebetweenavertex toanotheraswellastheroutebyusingFloydalgorithm.

Thefirstthingtodoistocreateadistancetable andaroutetable.Thedistancetablewilllooklikethe

Fig.3:Exampleofaweightedgraph

Fig.4:Exampleofaroutetable

onebelow:

Andtheroutetablewillalwaysbeliketheonein Fig.4.

Forthefirstiteration,thenextstepistoshadethe firstrowandcolumnlikethefollowing:

Afterthefirstiterationisover,starttheseconditeration.Usingthetablewiththeupdatedvalues,shade thesecondrowandcolumn.

Replaceanyvaluethatislargerthanthesumofthe twosuitablevalues,onefromarowandacolumn.After replacing,thedistancetableandroutetableshouldlook likethis:

Repeattheprocesswedidinthefirstiteration.Replacingallofthevaluesthataresmallerthantheoriginalvaluewegetanupdatedtablethatlookslike:

Nowforthethirditeration,shadethethirdrow andcolumnandafterfollowingtheprocess,thetables become:

IfweseethepositionRowQ,ColumnSfromthe distancetable,wecanseethatavaluethathasbeen replacedinthepreviousiterationhasalowervaluein thethirditeration.Inthiscase,replacethevalueonce morelikethetableabove.

Afterthefourthiteration,wegetthefinaldistance androutetablewhichisshownunderneath.

5Conclusion

Thispapercoveredtwotypesofalgorithmsoutof manyalgorithmsingraphtheory.Thebeautyofthis theoryisthatitisnotonlylimitedtomathematics,but alsoseveralothersubjectsthatseemtohavenorelationshipwithmathematics.Theimportanceofgraphtheorycannotbestressedenoughasitisthebasicprincipleinnavigatingsystemsbutalsoincomputing,search engines,linguisticmodellinginstructureoflanguage, epidemiology,socialmediaandsomuchmoreothersystemsthatweencountereveryday.Letusnotforgethow graphtheoryimpactsourlivesandunderrateitsquality.

Assumingthatwewanttofindtheshortestpath anddistancefromvertexRtovertexP,weshade inRowRandColumnPonboththefinaldistance androutetable.Whatwecaninterpretfromthe distancetableisthatthecellthatisoverlappedbythe twoshadedregionsistheshortestdistancefromthe designatedstartinganddestinationvertex.Forthe routetable,theoverlappingcellmustbetravelled.In thiscase,vertexSisthevalueintheoverlappingcell andtherefore,theshortestroutemusttravelthrough vertexS.

Fig.5:Exampleofadirectedgraph

1stIteration-Distancetable

2ndIteration-Routetable

3rdIteration-Distancetable

1stIteration-Routetable

2ndIteration-Distancetable

3rdIteration-Routetable

4thIteration-Distancetable

4thIteration-Routetable

5thIteration-Distancetable

6thIteration-Routetable

7AppendixB

5thIteration-Routetable

6thIteration-Distancetable

import sys

class Graph( object ): def __init__(self,nodes,init_graph): self.nodes=nodes self.graph=self.construct_graph(nodes ,init_graph)

def construct_graph(self,nodes,init_graph ):

graph={} for node in nodes: graph[node]={}

graph.update(init_graph)

for node,edges in graph.items(): for adjacent_node,value in edges. items(): if graph[adjacent_node].get( node,False)==False: graph[adjacent_node][node] =value

return graph

def get_nodes(self): return self.nodes

def get_outgoing_edges(self,node): connections=[] for out_node in self.nodes:

if self.graph[node].get(out_node, False)!=False: connections.append(out_node)

return connections

def value(self,node1,node2): "Returns␣the␣value␣of␣an␣edge␣between␣ two␣nodes."

return self.graph[node1][node2]

def dijkstra_algorithm(graph,start_node): unvisited_nodes= list (graph.get_nodes())

shortest_path={}

previous_nodes={}

max_value=sys.maxsize for node in unvisited_nodes:

shortest_path[node]=max_value

shortest_path[start_node]=0

while unvisited_nodes:

current_min_node=None for node in unvisited_nodes:

if current_min_node==None: current_min_node=node elif shortest_path[node]< shortest_path[current_min_node

]:

current_min_node=node neighbors=graph.get_outgoing_edges( current_min_node) for neighbor in neighbors:

tentative_value=shortest_path[ current_min_node]+graph.value (current_min_node,neighbor)

if tentative_value<shortest_path[ neighbor]: shortest_path[neighbor]= tentative_value previous_nodes[neighbor]= current_min_node

unvisited_nodes.remove(current_min_node ) return previous_nodes,shortest_path

def print_result(previous_nodes,shortest_path, start_node,target_node):

path=[] node=target_node

while node!=start_node: path.append(node) node=previous_nodes[node]

path.append(start_node)

print ("We␣found␣the␣following␣best␣path␣ with␣a␣value␣of␣{}.". format ( shortest_path[target_node])) print ("␣ >␣".join( reversed (path)))

nodes=["Reykjavik","Oslo","Moscow","London ","Rome","Berlin","Belgrade","Athens", "Vaduz","Seoul","Toronto","Warsaw"]

init_graph={}

for node in nodes: init_graph[node]={}

starting_city= input ("Enter␣the␣initial␣ location:␣")

destination_city= input ("Enter␣the␣final␣ location:␣")

init_graph["Reykjavik"]["Oslo"]=5 init_graph["Reykjavik"]["London"]=4 init_graph["Oslo"]["Berlin"]=1 init_graph["Oslo"]["Moscow"]=3 init_graph["Moscow"]["Belgrade"]=5 init_graph["Moscow"]["Athens"]=4 init_graph["Athens"]["Belgrade"]=1 init_graph["Rome"]["Berlin"]=2 init_graph["Rome"]["Athens"]=2 init_graph["Vaduz"]["Warsaw"]=2 init_graph["Vaduz"]["London"]=2 init_graph["Warsaw"]["Athens"]=5 init_graph["Seoul"]["Toronto"]=20 init_graph["Toronto"]["Reykjavik"]=8 graph=Graph(nodes,init_graph) previous_nodes,shortest_path= dijkstra_algorithm(graph=graph,start_node= starting_city)

print_result(previous_nodes,shortest_path, start_node=starting_city,target_node= destination_city)