Portal through Mathematics: Journey to Advanced Thinking (Anneli Lax New Mathematical Library)

320 Pages·4.019 MB·English

by  Oleg A. Ivanov

AMS / MAA ANNELI LAX NEW MATHEMATICAL LIBRARY VOL 52
Portal through Mathematics
Journey to Advanced Thinking
O.A. Ivanov
Translated
by Robert G. Burns
Portal through
Mathematics
Originallypublishedby
TheMathematicalAssociationofAmerica,2017.
ISBN:978-1-4704-4876-9
LCCN:2016960274
Copyright©2018,heldbytheAmercanMathematicalSociety
PrintedintheUnitedStatesofAmerica.
ReprintedbytheAmericanMathematicalSociety,2018
TheAmericanMathematicalSocietyretainsallrights
exceptthosegrantedtotheUnitedStatesGovernment.
⃝1Thepaperusedinthisbookisacid-freeandfallswithintheguidelines
establishedtoensurepermanenceanddurability.
VisittheAMShomepageathttps://www.ams.org/
1098765432 232221201918
AMS/MAA ANNELILAXNEWMATHEMATICALLIBRARY
VOL 52
Portal through
Mathematics
O. A. Ivanov
CouncilonPublicationsandCommunications
JenniferJ.Quinn,Chair
CommitteeonBooks
JenniferJ.Quinn,Chair
AnneliLaxNewMathematicalLibraryEditorialBoard
KarenSaxe,Editor
TimothyG.Feeman
JohnH.McCleary
KatharineOtt
KatherineS.Socha
JamesS.Tanton
JenniferM.Wilson
ANNELILAXNEWMATHEMATICALLIBRARY
1. Numbers:RationalandIrrationalbyIvanNiven
2. WhatisCalculusAbout?byW.W.Sawyer
3. AnIntroductiontoInequalitiesbyE.F.BeckenbachandR.Bellman
4. GeometricInequalitiesbyN.D.Kazarinoff
5. TheContestProblemBookIAnnualHighSchoolMathematicsExaminations
1950–1960.CompiledandwithsolutionsbyCharlesT.Salkind
6. TheLoreofLargeNumbersbyP.J.Davis
7. UsesofInfinitybyLeoZippin
8. GeometricTransformationsIbyI.M.Yaglom,translatedbyA.Shields
9. ContinuedFractionsbyCarlD.Olds
10. (cid:2) ReplacedbyNML-34
11. HungarianProblemBooksIandII,BasedontheEo¨tvo¨sCompetitions
12. 1894–1905and1906–1928,translatedbyE.Rapaport
13. EpisodesfromtheEarlyHistoryofMathematicsbyA.Aaboe
14. GroupsandTheirGraphsbyE.GrossmanandW.Magnus
15. TheMathematicsofChoicebyIvanNiven
16. FromPythagorastoEinsteinbyK.O.Friedrichs
17. TheContestProblemBookIIAnnualHighSchoolMathematicsExaminations
1961–1965.CompiledandwithsolutionsbyCharlesT.Salkind
18. FirstConceptsofTopologybyW.G.ChinnandN.E.Steenrod
19. GeometryRevisitedbyH.S.M.CoxeterandS.L.Greitzer
20. InvitationtoNumberTheorybyOysteinOre
21. GeometricTransformationsIIbyI.M.Yaglom,translatedbyA.Shields
22. Elementary Cryptanalysis by Abraham Sinkov, revised and updated by
ToddFeil
23. IngenuityinMathematicsbyRossHonsberger
24. GeometricTransformationsIIIbyI.M.Yaglom,translatedbyA.Shenitzer
25. TheContestProblemBookIIIAnnualHighSchoolMathematicsExamina-
tions 1966–1972. Compiled and with solutions by C. T. Salkind and J. M.
Earl
26. MathematicalMethodsinSciencebyGeorgePo´lya
27. International Mathematical Olympiads—1959–1977. Compiled and with
solutionsbyS.L.Greitzer
28. The Mathematics of Games and Gambling, Second Edition by Edward
W.Packel
29. TheContestProblemBookIVAnnualHighSchoolMathematicsExamina-
tions1973–1982.CompiledandwithsolutionsbyR.A.Artino,A.M.Gaglione,
andN.Shell
30. TheRoleofMathematicsinSciencebyM.M.SchifferandL.Bowden
31. InternationalMathematicalOlympiads1978–1985andfortysupplementary
problems.CompiledandwithsolutionsbyMurrayS.Klamkin
32. RiddlesoftheSphinxbyMartinGardner
33. U.S.A. Mathematical Olympiads 1972–1986. Compiled and with solutions
byMurrayS.Klamkin
34. Graphs and Their Uses by Oystein Ore. Revised and updated by Robin
J.Wilson
35. ExploringMathematicswithYourComputerbyArthurEngel
36. GameTheoryandStrategybyPhilipD.Straffin,Jr.
37. Episodes in Nineteenth and Twentieth Century Euclidean Geometry by Ross
Honsberger
38. TheContestProblemBookVAmericanHighSchoolMathematicsExaminations
and American Invitational Mathematics Examinations 1983–1988. Compiled
andaugmentedbyGeorgeBerzsenyiandStephenB.Maurer
39. OverandOverAgainbyGengzheChangandThomasW.Sederberg
40. TheContestProblemBookVIAmericanHighSchoolMathematicsExamina-
tions1989–1994.CompiledandaugmentedbyLeoJ.Schneider
41. TheGeometryofNumbersbyC.D.Olds,AnneliLax,andGiulianaP.Davidoff
42. Hungarian Problem Book III, Based on the Eo¨tvo¨s Competitions 1929–1943,
translatedbyAndyLiu
43. MathematicalMiniaturesbySvetoslavSavchevandTituAndreescu
44. GeometricTransformationsIVbyI.M.Yaglom,translatedbyA.Shenitzer
45. WhenLifeisLinear:fromcomputergraphicstobracketologybyTimChartier
46. The Riemann Hypothesis: A Million Dollar Problem by Roland van der Veen
andJanvandeCraats
47. PortalthroughMathematics:JourneytoAdvancedThinkingbyOlegA.Ivanov.
TranslatedbyRobertG.Burns.
Othertitlesinpreparation.
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Contents
Foreword ix
PrefaceforanAmericanReadership xi
Author’sPreface xiii
PartI SurprisingandEasy 1
1 Surprisingrighttriangles 3
2 Surprisinglyshortsolutionsofgeometricproblems 7
3 Anaturalassertionwithasurprisingproof 11
4 Surprisinganswers 17
5 Asurprisingconnectionbetweenthreesequences 23
PartII Algebra,Calculus,andGeometry:problems 27
6 Fiveproblemsandafunction 29
7 Fivesolutionsofaroutineproblem 33
8 Equationsoftheformf(x,y)=g(x,y)andtheirgeneralizations 39
9 ThegeneralizedversionofVie`te’sformula 49
10 Multiplerootsofpolynomials 55
11 Non-routineapplicationsofthederivative 63
12 Complexnumbers,polynomials,andtrigonometry 71
13 Complexnumbersandgeometry 79
14 Areasoftrianglesandquadrilaterals 85
15 Constructionsinsolidgeometry 93
16 Inequalities 101
17 Diophantineequations 111
18 Combinatorialtales 119
19 Integrals 129
vii
viii Contents
PartIII Algebra,Calculus,andGeometry:theory(alittle
waybeyondhighschoolmathematics) 139
20 Functionalequationsofelementaryfunctions 143
21 Sequencesgivenbyrecurrencerelations 151
22 The“goldenratio”orsolvingequationsoftheformf(x)=x 161
23 Convexfunctions:inequalitiesandapproximations 167
24 Taylor’sformula,Euler’sformula,andacombinatorialproblem 177
25 Derivativesofvector-functions 187
26 Polynomialsandtrigonometricrelations 199
27 Areasandvolumesasfunctionsofco-ordinates 207
28 Valuesoftrigonometricfunctionsandsequencessatisfyinga
certainrecurrencerelation 217
29 Dothereexistfurther“numbers”beyondcomplexnumbers? 223
Solutionsofthesupplementaryproblems 231
Index 303
Foreword
Theteachingofelementarymathematicsisoftenpresentedasatraintrack.
Studentswithearlymasteryofthemathematics“attheirstop”aresentonthe
next stop (bumped up a grade) without getting the chance to develop what
theyknowandseethelandscapebetweenstations.Oneofthebiggestleaps
isbetweenhighschoolanduniversitymathematics.Whatarewemissingby
jumpingtothenextstation?
Oleg Ivanov introduces us to the rich world between grade 12 plus or
minus epsilon mathematics (and here epsilon can be small or large!) and
universitymathematicswithawonderfulcollectionofmathematicaltidbits
to intrigue, propel, and delight. Is there a natural way to find an explicit
formulaforthenthFibonaccinumber?Thesetofcomplexnumbersisafield
that contains the real numbers; is there another such field? A rope is tied
aroundtheEarth’sequatorandthenlengthenedby6feet.Howhighcanthe
rope be raised off the equator to the same height all the way round? Now
suppose that the rope is pulled away from the Earth’s surface at just one
point.Howhighabovethesurfacecanthatpointoftheropebepulled?
The 29 mathematical themes presented in this text range in style and
content, background and outlook. Teachers and other life-long students of
mathematics occupy different places along the track between school and
beyond-schoolmathematics,andeachwillrespondtotheessayshereintheir
own ways. Ivanov has provided us a wide selection of deep and surprising
mathematical delights to reflect upon and savor. Wherever you are on the
trackbetweenhighschoolmathematicsanduniversitycourses,youcanstop
andexplorethelandscapeawayfromthetrack.Somethingswillbefamiliar
immediately, some willbecome clear a littlelater on. This book is a guide
tothelandscape,withwonderfulhikesmappedout,andpromisestodelight
thereaderagainandagain.
Stop the train, pack your backpack, and follow Ivanov to the joy that
evenelementarymathematicsaffordstheexplorer.
KarenSaxefortheNMLEditorialBoard
ix