学术英语代数问题重要性ppt省名师优质课赛课获奖课件市赛课百校联赛优质课一等奖课件

ThePowerofGoodQuestionProvideby:BME1411/17Whatmakesagoodquestion?

Ifyou’redoingmathsforfun,orareaprofessionalmathematician,youanswerisgoingtobedifferent.（Aneasyquestionisboring）Ifyouareastudentfacingexams,youmight(understandably)saythatgoodmeanseasy.

2/17Wheredothesegoodquestionscomefrom?

Generalize

Simplifyandvary

Lookfornewtools

Takerisks3/17Generalizean

+

bn

=

cn?In

numbertheory,

Fermat'sLastTheorem

(sometimescalled

Fermat'sconjecture,especiallyinoldertexts)statesthatnothree

positiveintegers

a,

b,and

c

satisfytheequation

an

+

bn

=

cn

foranyintegervalueof

n

strictlygreaterthantwo.Thecases

n

=

1and

n

=

2havebeenknowntohaveinfinitelymanysolutionssinceantiquity.4/17350years!?

Thistheoremwasfirst

conjecturedby

PierredeFermat

in1637inthemarginofacopyof

Arithmetica(算术)whereheclaimedhehadaproofthatwastoolargetofitinthemargin.

Thefirstsuccessfulproof

wasreleasedin1994by

AndrewWiles,andformallypublishedin1995,after358yearsofeffortbymathematicians..Itisamongthemostnotabletheoremsinthe

historyofmathematicsandpriortoitsproof,itwasinthe

GuinnessBookofWorldRecords

asthe"mostdifficultmathematicalproblem",oneofthereasonsbeingthatithasthelargestnumberofunsuccessfulproofs.5/17万畅高清摄像机万畅高清摄像机万畅传输接入模块万畅局端模块和设备视频枢纽万局端模块Fermat’ssimplequestionturnedouttobeincrediblyfruitful:itgeneratednewmathematics,newinsightsandnewwaysoflookingatthings.Thoughhard,manymathematicianswouldregardthisasa“good”question.Togetherwith

RenéDescartes（笛卡尔）,Fermatwasoneofthetwoleadingmathematiciansofthefirsthalfofthe17thcentury.

6/17SimplifyandvaryGalleryproblemAniceexampleistheartgalleryproblem:howmanysecurityguardsdoyouneedtobesurethattogethertheycanoverseethewholeinteriorofanartgallery?7/17AnswerThefirstanswer,givenin1978fiveyearsafertheproblemwasposed.Usinganingeniouslineofattack,themathematicianS.Fiskprovedthatyouneverneedmorethan1/3guards,wherenisthenumberofvertices(corners)ofthepolygon.8/1730yearson,theseproblemisstillgoing

Whatiftheguardsarenotconfinedtothecornersofthegallery?

GalleryproblemsWhatiftheyareallowedtomovearound?

Whatifthereareobstaclesinthemiddleofthegallerythatyoucannotseethrough?

Thewallsarecurved?

Whatif,insteadofguardingatwo-dimensionalpolygon,youaretryingtoguardathree-dimensionalpolyhedron?

9/17LookfornewtoolsCalculusTherearealsoquestionsthatarebeingasked,notbyindividuals,butbyawholeage，cryingoutfornewmathematicaltools.Theiranswerscanspawnsomethingofarevolution.Agreatexampleistheinventionofcalculusintheseventeenthcentury.10/17CalculusHowcanwedescribecontinuouschange?Ajourney:speedistherateofchangeofdistancepertime,soyousimplydividethedistanceyoutraveledbythetimeittooktotravelit.(S/T)Butofcourse,youdidn‘ttravelatthataveragespeedateverymomentofyoujourney.Atsometimesyouwillhavebeengoingslowerandatsometimesfaster,withthespeedvaryingcontinuously.Toworkoutyourexactspeedataparticularmomentintime,youhavetocalculatetheinstantaneousrateofchangeofdistancewithrespecttotime.11/17ApplicationsofcalculusThemethodsfordoingthiswereinventedprimarilybyGottfriedLeibnizandIsaacApplicationsofintegralcalculusincludecomputationsinvolvingarea,volume,arclength,centerofmass,work,andpressure.MoreadvancedapplicationsincludepowerseriesandFourierseries.Calculusisalsousedtogainamorepreciseunderstandingofthenatureofspace,time,andmotion.GottfriedLeibniz(left)IsaacNewton(right)12/17TakerisksFourcolourtheoremNotallquestionsturnouttohaveinterestinganswers.Mathematicianssimplyhavetoaccepttheriskthataquestiontheychoosetoworkonmaynotbesolvedintheirlifetime,orthatitmayturnouttohaveaboringanswer.It‘sallpartofthecreativeprocess.Aquestionthatnotbesolvedintheirlifetime-Fermat’sLastTheorem.13/17FourcolortheoremItsaysthatfourcoloursareenoughtocolouramapdrawnontheplanesothatnotwoneighbouringcountrieshavethesamecolour.Theproofofthistheorem,whenitfinallycameinthe1970saftermathematicianshadbeenwrestlingwiththetheoremforoveracentury,wasdisappointing.Itusedabruteforceapproachinvolvingacomputercheckingthroughahugenumberofpossibilities,makingsuretheydidnotprovideacounterexampletothetheorem.Theapproachdeliverednonewinsightsatall.Asimplemapcolouredcorrectlywithfourcolours.14/17万畅高清摄像机万畅高清摄像机万畅传输接入模块万畅局端模块和设备视频枢纽万局端模块However,thepartoftheirproofwasactuallydonebyacomputer.Nohumanbeingcouldintheirlifetimeeveractuallyreadtheentireprooftocheckthatitwascorrect.Severalmathematiciansofthetimecomplainedthatthismeantthatitwasn'treallyaproofatall!Ifnobodycouldchecktheproof,howcouldweeverknowwhetheritwasrightorwrong?Partoftheworldmap,colouredin