Post on 08-Jan-2016
description
transcript
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 1/31
Ihaveseveralwebpagesintendedforstudentsthisseemstobethemostpopularone.FONTSFINALLYREPAIREDNovember2009.
Browseradjustments:Thiswebpageusessubscripts,superscripts,andunicodesymbols.Thelattermaydisplayincorrectlyonyourcomputerifyouareusinganold
browserand/oranoldoperatingsystem.Notetoteachers(andanyoneelsewhoisinterested):Feelfreetolinktothispage(around500peoplehavedoneso),tellyourstudentsaboutthispage,orcopy(withappropriatecitation)partsorallofthispage.Youcandothosethingswithoutwritingtome.Butifyouhaveanythingelsetosayaboutthispage,pleasewritetomewithyourquestions,comments,orsuggestions.IwillreplywhenIhavetime,thoughthatmightnotbeimmediatelyrecentlyI'vebeenswampedwithotherwork.EricSchechter,versionof11Nov2009.
THEMOSTCOMMONERRORSINUNDERGRADUATEMATHEMATICS
ThiswebpagedescribestheerrorsthatIhaveseenmostfrequentlyinundergraduatemathematics,thelikelycausesofthoseerrors,andtheirremedies.Iamtiredofseeingthesesameolderrorsoverandoveragain.(Iwouldratherseenew,originalerrors!)Icautionmyundergraduatestudentsabouttheseerrorsatthebeginningofeachsemester.Outlineofthiswebpage:
ERRORSINCOMMUNICATION,includingteacherhostilityorarrogance,studentshyness,unclearwording,badhandwriting,notreadingdirections,lossofinvisibleparentheses,termslostinsideanellipsisALGEBRAERRORS,includingsignerrors,everythingisadditive,everythingiscommutative,undistributedcancellations,dimensionalerrorsCONFUSIONABOUTNOTATION,includingidiosyncraticinverses,squareroots,orderofoperations,ambiguouslywrittenfractions,streamofconsciousnessnotations.ERRORSINREASONING,includinggoingoveryourwork,overlookingirreversibility,notcheckingforextraneousroots,confusingastatementwithitsconverse,workingbackward,difficultieswithquantifiers,erroneousmethodsthatwork,unquestioningfaithincalculators.UNWARRANTEDGENERALIZATIONS,includingEuler'ssquarerooterror,xx.OTHERCOMMONCALCULUSERRORS,includingjumpingtoconclusionsaboutinfinity,lossormisuseofconstantsofintegration,lossofdifferentials.
(Thereissomeoverlapamongthesetopics,soIrecommendreadingthewholepage.)...Ofrelatedinterest:PaulCox'swebpage,andthebooksofBradis,Minkovskii,andKharchevaandE.A.Maxwell.
Ultimately,whatarethesourcesoferrorsandofmisunderstanding?Whatkindsofbiasesanderroneouspreconceptionsdowehave?TwoofmyfavoritehistoricdiscoveriesareEinstein'sdiscoveryofrelativityandCantor'sdiscoveriesofsomeofthemostbasicrulesofinfinities.Thesediscoveriesareremarkableinthatneitherinvolvedlong,involved,complicatedcomputations.Botharefairlysimple,inretrospect,toanyonewhohasstudiedthem.Butbothinvolved"thinkingoutsidethebox"parexcellencei.e.,seeingpasttheassumptionsthatwereinherentinourcultureandourlanguage.AsphilosopherJohnCulkinsaid,"Wedon'tknowwhodiscoveredwater,butwearecertainitwasn'tafish."Thatcertainmathematicalerrorsarecommonamongstudentsmaybepartlyaconsequenceofbiasesthatarebuiltintoourlanguageandculture,someofwhichwearen'tevenawareof.
ErrorsinCommunication
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 2/31
Someteachersarehostiletoquestions.Thatisanerrormadebyteachers.Teachers,youwillbemorecomfortableinyourjobifyoutrytodoitwell,anddon'tthinkofyourstudentsastheenemy.Thismeanslisteningtoyourstudentsandencouragingtheirquestions.Ateacherwhoonlylectures,anddoesnotencouragequestions,mightaswellbereplacedbyabookoramovie.Toteacheffectively,youhavetoknowwhenyourstudentshaveunderstoodsomethingandwhentheyhaven'tthemostefficientwaytodiscoverthatistolistentothemandtowatchtheirfaces.Perhapsyouidentifywithyourbrighteststudents,becausetheyaremostabletoappreciatethebeautyoftheideasyouareteachingbuttheotherstudentshavegreaterneedofyourhelp,andtheyhavearighttoit.
Avariantofteacherhostilityisteacherarrogance.Initsmildestform,thismaysimplymeanateacherwho,despitebeingpoliteandpleasant,isunabletoconceiveoftheideathathe/shecouldhavemadeanerror,evenwhenthaterrorisbroughtdirectlytohis/herattention.Actually,mostoftheerrorslistedbelowcanbemadebyteachers,notjustbystudents.(However,mostteachersarerightfarmoreoftenthantheirstudents,sostudentsshouldexercisegreatcautionwhenconsideringwhethertheirteacherscouldbeinerror.)
Ifyou'reastudentwithahostileteacher,thenI'mafraidIdon'tknowwhatadvicetogiveyoutransfertoadifferentsectionordropthecoursealtogetherifthatisfeasible.Theremarksoncommunicationinthenextfewparagraphsareforstudentswhoseteachersarereceptivetoquestions.Forsuchstudents,acommonerroristhatofnotaskingquestions.
Whenyourteachersayssomethingthatyoudon'tunderstand,don'tbeshyaboutaskingthat'swhyyou'reinclass!Ifyou'vebeenlisteningbutnotunderstanding,thenyourquestionisnota"stupidquestion."Moreover,youprobablyaren'taloneinyourlackofunderstandingthereareprobablyadozenotherstudentsinyourclassroomwhoareconfusedaboutpreciselythesamepoint,andareevenmoreshyandinarticulatethanyou.Thinkofyourselfastheirspokespersonyou'llbedoingthemallafavorifyouaskyourquestion.You'llalsobedoingyourteacherafavoryourteacherdoesn'talwaysknowwhichpointshavebeenexplainedclearlyenoughandwhichpointshavenotyourquestionsprovidethefeedbackthatyourteacherneeds.
Ifyouthinkyourteachermayhavemadeamistakeonthechalkboard,you'dbedoingthewholeclassafavorbyaskingaboutit.(Tosaveface,justincasetheerrorisyourown,formulateitasaquestionratherthanastatement.Forinstance,insteadofsaying"that5shouldbea7",youcanask"shouldthat5bea7?")
Andtrytoaskyourquestionassoonaspossibleafteritcomesup.Don'twaituntiltheveryendoftheexample,oruntiltheendofclass.Asateacher,Ihateitwhenclasshasendedandstudentsareleavingtheroomandsomestudentcomesuptomeandsays"shouldn'tthat5havebeena7?"ThenIsay"Yes,you'reright,butIwishyouhadaskedaboutitoutsooner.Nowallyourclassmateshaveanerrorinthenotesthattheytookinclass,andtheymayhavetroubledecipheringtheirnoteslater."
MarcMimssentmethisanecdoteaboutunaskedquestions:
Intheearly1980s,Imanagedacomputerretailstore.Severalofmyemployeeswerecollegestudents.OnebrightyourmanwashavingdifficultywithhisFreshmancollegealgebraclass.Itutoredhimandhedidverywell,butinvariably,hewouldsay,"theprofessorworkedthroughthisproblemontheboard,anditwasnothinglikethis.Isurehopewegotthecorrectanswer."
Iaccompaniedhimtoclassonemorninganddiscoveredthesourceofhisfrustration.Theprofessorwasfromthemusicdepartment,anddidn'tnormallyteachcollegealgebrahehadbeenpressedintodutywhenoverenrollmentforcedtheclasstobesplit.
Duringtheclass,hepickedaproblemfromtheassignmenttoworkoutontheboard.Veryearlyintheproblem,hemadeanerror.Idon'trecallthespecifics,butI'msureitwasoneofthemanytypicalalgebraerrorsyoulist.
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 3/31
Becauseoftheerror,heeventuallyreachedapointfromwhichhecouldnolongerproceed.Ratherthanadmittinganerrorandgoingtoworktofindit,hepausedstaringattheboardforseveralseconds,thenturnedtotheclassandsaid,"...andtherest,youngpeople,shouldbeobvious."
Unclearwording.TheEnglishlanguagewasnotdesignedformathematicalclarity.Indeed,mostoftheEnglishlanguagewasnotreallydesignedatallitsimplygrew.Itisnotalwaysperfectlyclear.MathematiciansmustbuildtheircommunicationontopofEnglish[orreplaceEnglishwithwhateverisyournativeorlocallanguage],andsotheymustworktoovercometheweaknessesofEnglish.Communicatingclearlyisanartthattakesgreatpractice,andthatcanneverbeentirelyperfected.
Lackofclarityoftencomesintheformofambiguityi.e.,whenacommunicationhasmorethanonepossibleinterpretation.Miscommunicationcanoccurinseveralwaysherearetwoofthem:
Oneofthethingsthatyou'vesaidhastwoormorepossiblemeanings,andyou'reawareofthatfact,butyou'resatisfiedthatit'sclearwhichmeaningyouintendedeitherbecauseit'sclearfromthecontext,orbecauseyou'veaddedsomefurther,clarifyingwords.Butyouraudienceisn'tasknowledgeableasyouaboutthissubject,andsothedistinctionwasnotcleartothemfromthecontextorfromyourfurtherclarifyingwords.Or,Oneofthethingsthatyou'vesaidhastwoormorepossiblemeanings,andyou'renotawareofthatfact,becauseyouweren'twatchingyourownchoiceofwordscarefullyenoughand/orbecauseyou'renotknowledgeableenoughaboutsomeoftheothermeaningsthatthosewordshavetosomepeople.
Onewaythatambiguitycanoccuriswhentherearemultipleconventions.Aconventionisanagreeduponwayofdoingthings.Insomecases,onegroupofmathematicianshasagreedupononewayofdoingthings,andanothergroupofmathematicianshasagreeduponanotherway,andthetwogroupsareunawareofeachother.Thestudentwhogetsateacherfromonegroupandlatergetsanotherteacherfromtheothergroupissuretoendupconfused.Anexampleofthisisgivenunder"ambiguouslywrittenfractions,"discussedlateronthispage.
Choosingprecisewordingisafineart,whichcanbeimprovedwithpracticebutneverperfected.Eachtopicwithinmath(orwithinanyfield)hasitsowntrickyphrasesfamiliaritywiththattopicleadstoeventuallymasteringthosephrases.
Forinstance,onestudentsentmethisexamplefromcombinatorics,atopicthatrequiressomewhatawkwardEnglish:
Howmanydifferentwordsoffiveletterscanbeformedfromsevendifferentconsonantsandfourdifferentvowelsifnotwoconsonantsandvowelscancometogetherandnorepetitionsareallowed?Howmanycanbeformedifeachlettercouldberepeatedanynumberoftimes?
Thereareanumberofplaceswherethisproblemisunclear.Inthefirstsentence,I'mnotsurewhat"cancometogether"means,butIwouldguessthattheintendedmeaningis
Howmanydifferentwordsoffiveletterscanbeformedfromsevendifferentconsonantsandfourdifferentvowelsifnotwoconsonantscanoccurconsecutivelyandnotwovowelscanoccurconsecutivelyandnorepetitionsareallowed?
Thesecondsentenceisabitworse.Thestudentmisinterpretedthatsentencetomean
Howmanydifferentwordsoffiveletterscanbeformedfromsevendifferentconsonantsandfourdifferentvowelsifeachlettercouldberepeatedanynumberoftimes?
Butusually,whenamathbookaskstwoconsecutivequestionsrelatedinthisfashion,thesecondquestionisintendedasamodificationofthefirstquestion.Wearetoretainallpartsofthefirstquestionthatarecompatiblewiththenewconditions,andtodiscardallpartsofthefirstquestionthatwouldbecontradictedbythenewconditions.Thus,thesecondsentencein
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 4/31
ourexampleshouldbeinterpretedinthisratherdifferentfashion,whichyieldsadifferentanswer:
Howmanydifferentwordsoffiveletterscanbeformedfromsevendifferentconsonantsandfourdifferentvowelsifnotwoconsonantscanbeconsecutive,notwovowelscanbeconsecutive,andeachlettercouldberepeatedanynumberoftimes?
Badhandwritingisanerrorthatthestudentmakesincommunicatingwithhimselforherself.Ifyouwritebadly,yourteacherwillhavedifficultyreadingyourwork,andyoumayevenhavedifficultyreadingyourownworkaftersometimehaspassed!
UsuallyIdonotdeductpointsforasloppyhandwritingstyle,providedthatthestudentendsupwiththerightanswerattheendbutsomestudentswritesobadlythattheyendupwiththewronganswerbecausetheyhavemisreadtheirownwork.Forinstance,
(5x4+2)dx x5+7x+C(shouldbex5+2x+C)Thisstudent'shandwritingwassobadthathemisreadhisownwritinghetookthe"2"fora"7".You'llhavetouseyourimaginationhere,sincethiselectronictypesettingcannotduplicatesloppyhandwriting.Youdonotneedtomakeyourhandwritingasneatasthistypesetdocument,butyouneedtobeneatenoughsothatyouoranyoneelsecandistinguisheasilybetweencharactersthatareintendedtobedifferent.Moststudentswouldfarebetteriftheywouldprinttheirmathematics,insteadofusingcursivewriting.
Bytheway,writeyourplussign(+)andlowercaseletterTee(t)sothattheydon'tlookidentical!Oneeasywaytodothisistoputalittle"tail"atthebottomofthet,justasitappearsinthistypesetdocument.(Iassumethatthefontsyou'reusingonyourbrowseraren'tmuchdifferentfrommyfonts.)
Notreadingdirections.Studentsoftendonotreadtheinstructionsonatestcarefully,andsoinsomecasestheygivetherightanswertothewrongproblem.
Lossofinvisibleparentheses.Thisisnotanerroneousbeliefrather,itisasloppytechniqueofwriting.Duringoneofyourcomputations,ifyouthinkapairofparenthesesbutneglecttowritethem(forlackoftime,orfromsheerlaziness),andtheninthenextstepofyourcomputationyouforgetthatyouomittedaparenthesisfromthepreviousstep,youmaybaseyoursubsequentcomputationsontheincorrectlywrittenexpression.Hereisatypicalcomputationofthissort:
3(5x4+7)dx 3x5+7x+CButthatshouldbe
3(5x4+7)dx=3(x5+7x)+C=3x5+21x+CThat'sanentirelydifferentanswer,andit'sthecorrectanswer.Toseewheretheerrorcreepsin,justtryerasingthelastpairofparenthesesinthelineabove.
Apartiallossofparenthesesresultsinunbalancedparentheses.Forexample,theexpression"3(5x4+2x+7"ismeaningless,becausetherearemoreleftparenthesesthanrightparentheses.Moreover,itisambiguousifwetrytoaddarightparenthesis,wecouldgeteither"3(5x4+2x)+7"or"3(5x4+2x+7)"thosearetwodifferentanswers.
Lossofparenthesesisparticularlycommonwithminussignsand/orwithintegralsforinstance,
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 5/31
(5x47)dx x57x+C(shouldbex5+7x+C)
Termslostinsideanellipsis.Anellipsisisthreedots(...),usedtodenote"continuethepattern".Thisnotationcanbeusedtowritealonglist.Forinstance,"1,2,3,...,100"representsalltheintegersfrom1to100that'smuchmoreconvenientthanactuallywritingall100numbers.Andforsomepurposes,anellipsisisnotjustaconvenience,it'sanecessity.Forinstance,"1,2,3,...,n"representsalltheintegersfrom1ton,wherenissomeunspecifiedpositiveintegerthere'snowaytowritethatwithoutanellipsis.
Theellipsisnotationconcealssometermsinthesequence.Butcanonlybeusedifenoughtermsareleftunconcealedtomakethepatternevident.Forinstance,"1,...,64"isambiguousitmighthaveanyoftheseinterpretations:
"1,2,3,4,...,64"(alltheintegersfrom1to64)"1,4,9,16,25,36,49,64"(that'sn2asngoesfrom1to8)"1,2,4,8,16,32,64"(that's2nasngoesfrom0to6)
Ofcourse,insomecasesoneofthesemeaningsmightbeclearfromthecontext.Andjusthowmuchinformationisneeded"tomakeapatternevident"isasubjectivematteritmayvaryfromoneaudiencetoanother.Besttoerronthesafeside:giveatleastasmuchinformationaswouldbeneededbytheleastimaginativememberofyouraudience.
IhaveseenmanyerrorsinusingellipseswhenI'vetriedtoteachinductionproofs.Forinstance,supposethatwe'dliketoprove
[*n]12+22+32+...+n2=n(n+1)(2n+1)/6
forallpositiveintegersn.Theprocedureisthis:Verifythattheequationistruewhenn=1(that'sthe"initialstep)thenassumethat[*n]istrueforsomeunspecifiedvalueofnandusethatfacttoprovethatit'strueforthenextvalueofni.e.,toprove[*(n+1)](that'sthe"transitionstep").Hereisatypicalerrorinthetransitionstep:Add2n+1tobothsidesof[*n].Thusweobtain
[i]12+22+32+...+n2+2n+1=(2n+1)+n(n+1)(2n+1)/6.
Butthatsays
[ii]12+22+32+...+(n+1)2=(2n+1)+n(n+1)(2n+1)/6.
We'vemadeamistakealready,intheleftsideoftheequation.(Canyoufindit?I'llexplainitinamoment.)Nowmakesomealgebraerrorwhilerearrangingtherightsideoftheequation,toobtain
[*(n+1)]12+22+32+...+(n+1)2=(n+1)(n+2)(2n+3)/6.
Andnowitappearsthatwe'redone.Buttherewasanalgebraerrorontherightside:(2n+1)+n(n+1)(2n+1)/6actuallyisnotequalto(n+1)(n+2)(2n+3)/6.(Youcancheckthateasily.)
Theerrorontheleftsidewasmoresubtle.Itisbasedonthefactthattoomanytermswereconcealedintheellipsis,andsothepatternwasnotrevealedaccurately.Toseewhatisreallygoingon,let'srewriteequations[i]and[ii],puttingmoretermsin:
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 6/31
[i]12+22+32+...+(n2)2+(n1)2+n2+2n+1=(2n+1)+n(n+1)(2n+1)/6.
[ii]12+22+32+...+(n2)2+(n1)2+(n+1)2=(2n+1)+n(n+1)(2n+1)/6.
Andnowyoucanseethattheleftsideismissingitsn2term,sotheleftsideof[ii]isnotequaltotheleftsideof[*(n+1)].
AlgebraErrors
Signerrorsaresurelythemostcommonerrorsofall.Igenerallydeductonlyonepointfortheseerrors,notbecausetheyareunimportant,butbecausedeductingmorewouldinvolveswimmingagainstatidethatisjusttoostrongforme.Thegreatnumberofsignerrorssuggeststhatstudentsarecarelessandunconcernedthatstudentsthinksignerrorsdonotmatter.Butsignerrorscertainlydomatter,agreatdeal.Yourtrainswillnotrun,yourrocketswillnotfly,yourbridgeswillfalldown,iftheyareconstructedwithcalculationsthathavesignerrors.
Signerrorsarejustthesymptomtherecanbeseveraldifferentunderlyingcauses.Onecauseisthe"lossofinvisibleparentheses,"discussedinalatersectionofthiswebpage.Anothercauseisthebeliefthataminussignmeansanegativenumber.Ithinkthatmoststudentswhoharborthisbeliefdosoonlyonanunconsciousleveltheywouldgiveitupifitwerebroughttotheirattention.[MythankstoJonJacobsenforidentifyingthiserror.]
Isxanegativenumber?Thatdependsonwhatxis.
Yes,ifxisapositivenumber.No,ifxitselfisanegativenumber.Forinstance,whenx=6,thenx=6(or,foremphasis,x=+6).
That'ssomethinglikea"doublenegative".Wesometimesneeddoublenegativesinmath,buttheyareunfamiliartostudentsbecausewegenerallytrytoavoidtheminEnglishtheyareconceptuallycomplicated.Forinstance,insteadofsaying"Idonothavealackoffunds"(twonegatives),itissimplertosay"Ihavesufficientfunds"(onepositive).
Anotherreasonthatsomestudentsgetconfusedonthispointisthatweread"x"aloudas"minusx"oras"negativex".Thelatterreadingsuggeststosomestudentsthattheanswershouldbeanegativenumber,butthat'snotright.[SuggestedbyChrisPhillips.]
Misunderstandingthispointalsocausessomestudentstohavedifficultyunderstandingthedefinitionoftheabsolutevaluefunction.Geometrically,wethinkof|x|asthedistancebetweenxand0.Thus|3|=3and|27.3|=27.3,etc.Adistanceisalwaysapositivequantity(ormoreprecisely,anonnegativequantity,sinceitcouldbezero).Informallyandimprecisely,wemightsaythattheabsolutevaluefunctionisthe"makeitpositive"function.
Thosedefinitionsofabsolutevalueareallgeometricorverbaloralgorithmic.Itisusefultoalsohaveaformulathatdefines|x|,buttodothatwemustmakeuseofthedoublenegative,discussedafewsentencesago.Thusweobtainthisformula:
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 7/31
whichisabitcomplicatedandconfusesmanybeginners.Perhapsit'sbettertostartwiththedistanceconcept.
Manycollegestudentsdon'tknowhowtoaddfractions.Theydon'tknowhowtoadd(x/y)+(u/v),andsomeofthemdon'tevenknowhowtoadd(2/3)+(7/9).Itishardtoclassifythedifferentkindsofmistakestheymake,butinmanycasestheirmistakesarerelatedtothisone:
Everythingisadditive.Inadvancedmathematics,afunctionoroperationfiscalledadditiveifitsatisfiesf(x+y)=f(x)+f(y)forallnumbersxandy.Thisistrueforcertainfamiliaroperationsforinstance,
thelimitofasumisthesumofthelimits,thederivativeofasumisthesumofthederivatives,theintegralofasumisthesumoftheintegrals.
Butitisnottrueforcertainotherkindsofoperations.Nevertheless,studentsoftenapplythisadditionruleindiscriminately.Forinstance,contrarytothebeliefofmanystudents,
Wedogetequalityholdingforafewunusualandcoincidentalchoicesofxandy,butwehaveinequalityformostchoicesofxandy.(Forinstance,allfourofthoselinesareinequalitieswhenx=y=/2.Thestudentwhoisnotsureaboutallthisshouldworkoutthatexampleindetailheorshewillseethatthatexampleistypical.)
Oneexplanationfortheerrorwithsinesisthatsomestudents,seeingtheparentheses,feelthatthesineoperatorisamultiplicationoperatori.e.,justas6(x+y)=6x+6yiscorrect,theythinkthatsin(x+y)=sin(x)+sin(y)iscorrect.
The"everythingisadditive"errorisactuallythemostcommonoccurrenceofamoregeneralclassoferrors:
Everythingiscommutative.Inhighermathematics,wesaythattwooperationscommuteifwecanperformthemineitherorderandgetthesameresult.We'vealreadylookedatsomeexampleswithadditionherearesomeexampleswithotheroperations.Contrarytosomestudents'beliefs,
etc.Anothercommonerroristoassumethatmultiplicationcommuteswithdifferentiationorintegration.Butactually,ingeneral(uv)doesnotequal(u)(v)and(uv)doesnotequal(u)(v).
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 8/31
However,tobecompletelyhonestaboutthis,Imustadmitthatthereisoneveryspecialcasewheresuchamultiplicationformulaforintegralsiscorrect.Itisapplicableonlywhentheregionofintegrationisarectanglewithsidesparalleltothecoordinateaxes,and
u(x)isafunctionthatdependsonlyonx(notony),andv(y)isafunctionthatdependsonlyony(notonx).
Underthoseconditions,
(IhopethatIamdoingmoregoodthanharmbymentioningthisformula,butI'mnotsurethatthatisso.Iamafraidthatafewstudentswillwritedownanabbreviatedformofthisformulawithouttheaccompanyingrestrictiveconditions,andwillendupbelievingthatItoldthemtoequate(uv)and(u)(v)ingeneral.Pleasedon'tdothat.)
Undistributedcancellations.HereisanerrorthatIhaveseenfairlyoften,butIdon'thaveaveryclearideawhystudentsmakeit.
(3x+7)(2x9)+(x2+1) (3x+7)(2x9)+(x2+1) (2x9)+(x2+1)
f(x)= =
(3x+7)(x3+6) (3x+7)(x3+6) (x3+6)
Inasense,thisisthereverseofthe"lossofinvisibleparentheses"mentionedearlieryoumightcallthiserror"insertionofinvisibleparentheses."Toseewhy,comparetheprecedingcomputation(whichiswrong)withthefollowingcomputation(whichiscorrect).
(3x+7)[(2x9)+(x2+1)] (3x+7)[(2x9)+(x2+1)] (2x9)+(x2+1)g(x)= = =
(3x+7)(x3+6) (3x+7)(x3+6) (x3+6)
Apparentlysomestudentsthinkthatf(x)andg(x)arethesamethingorperhapstheysimplydon'tbothertolookcarefullyenoughatthetoplineoff(x),todiscoverthatnoteverythinginthetoplineoff(x)hasafactorof(3x+7).Ifyoustilldon'tseewhat'sgoingon,hereisacorrectcomputationinvolvingthatfirstfunctionf:
x2+12x9+
(3x+7)(2x9)+(x2+1) 3x+7f(x)= =
(3x+7)(x3+6) x3+6Whywouldstudentsmakeerrorslikethese?Perhapsitispartlybecausetheydon'tunderstandsomeofthebasicconceptsoffractions.Herearesomethingsworthnoting:
Multiplicationiscommutativethatis,xy=yx.Consequently,mostrulesaboutmultiplicationare
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 9/31
symmetric.Forinstance,multiplicationdistributesoveradditionbothontheleftandontheright:
(x1+x2)y=(x1y)+(x2y)andx(y1+y2)=(xy1)+(xy2).
Divisionisnotcommutativeingeneral,x/yisnotequaltoy/x.Consequently,rulesaboutdivisionarenotsymmetric(thoughperhapssomestudentsexpectedthemtobesymmetric).Forinstance,
(x1+x2)/y=(x1/y)+(x2/y)butingeneralx/(y1+y2)(x/y1)+(x/y2).
Fractionsrepresentdivisionandgrouping(i.e.,parentheses).Forinstance,thefractiona+bc+d
isthesamethingas(a+b)/(c+d).Ifyouomiteitherpairofparenthesesfromthatlastexpression,yougetsomethingentirelydifferent.(ThankstoMarkMeckesforpointingoutthispossibleexplanationoftheoriginofsucherrors.)Perhapssomeofthestudents'errorsstemfromsuchanomissionofparentheses?oralackofunderstandingofhowimportantthoseparenthesesare?Thatwouldseemtobeindicatedbytheprevalanceofanothertypeoferrordescribedelsewhereonthispage,"lossofinvisibleparentheses".
Dimensionalerrors.Mostofthiswebpageisdevotedtothingsthatyoushouldnotdo,butdimensionalanalysisissomethingthatyoushoulddo.Dimensionalanalysisdoesn'ttellyoutherightanswer,butitdoesenableyoutoinstantlyrecognizethewrongnessofsomekindsofwronganswers.Justkeepcarefultrackofyourdimensions,andthenseewhetheryouranswerlooksright.Herearesomeexamples:
Ifyou'reaskedtofindavolume,andyouranswerissomenumberofsquareinches,thenyouknowyou'vemadeanerrorsomewhereinyourcalculations.(Ifyoufindthiskindoferrorinyouranswer,don'tjustchange"squareinches"to"cubicinches"inyouranswerandleavethenumericalpartunchanged.Thestepinyourcalculationwhereyougotthewrongunitsmayalsobeastepwhereyoumadeanumericalerror.Trytofindthatstep,)Ifyou'reaskedtofindanareaoravolume,andyouranswerisanegativenumber,thenyouknowyou'vemadeanerrorsomewhere.(Again,don'tjustchangethesigninyouranswertheremaybemoretoyourerrorthanthat.)Ifthequestionisawordproblem,thinkaboutwhetheryouranswermakessense.Forinstance,ifyou'regiventhedimensionsofacoinandyou'reaskedtofinditssurfacearea,andyoucomeupwithananswerof3000squaremiles,youshouldrealizethatyou'veprobablymadeanerror(eventhoughyouranswerhastherightunits),andyoushouldlookforthaterror.(Thisisnotreallyanexampleofdimensionalanalysis,butIdidn'tknowwhereelsetoputit.ThankstoSandeepKanabarforthisexample.)Evenifyoudon'tremembertheformulaforthevolumeofasphereofradiusr,youknowthatithastohaveafactorofr3,notr2,sotheanswercouldn'tpossiblyber2.Evenifyoudon'tremembertheformulaforthesurfaceareaofasphereofradiusr,youknowithastogetsmallwhenrgetssmall.Soitcouldn'tpossiblybesomethinglike2+3r2.
Hereisacuteexampleofdimensionalanalysis(submittedbyBenjaminTilly).
Problem:Wherehasmymoneygone?Mydollarseemstohaveturnedintoapenny:
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 10/31
$1=100=(10)2=($0.10)2=$0.01=1
Explanation:Ofcourse,theproblemisadisregardfordimensionalunits.Strictlyspeaking,ifyousquareadollar,youshouldgetasquaredollar.Idon'tknowwhata"squaredollar"is,butIstillknowhowtocomputewithit,andIknowthata"squaredollar"mustbeequalto10,000"squarepennies",sinceonedollaris100pennies.Dimensionalcomputationswillnotyielderrorsifwehandlethedimensionalunitscorrectly.Hereisacorrectcomputation:
$21=($1)2=(100)2=10022=10,0002.
Itshouldnowbeevidentwhatwaswrongwiththefirstcalculation:100isnotequalto(10)2.It'struethatthe100isequaltothe102,buttheisnotequalto2.Likewise,laterinthecomputation,$2isnotequalto$.
ConfusionaboutNotation
Idiosyncraticinverses.Weneedtobesympatheticaboutthestudent'sdifficultyinlearningthelanguageofmathematicians.ThatlanguageisabitmoreconsistentthanEnglish,butitisnotentirelyconsistentittoohasitsidiosyncrasies,which(likethoseofEnglish)arelargelyduetohistoricalaccidents,andnotreallyanyone'sfault.Hereisonesuchidiosyncrasy:Theexpressionssinnandtanngetinterpretedindifferentways,dependingonwhatnis.
sin2x=(sinx)2andtan2x=(tanx)2
but
sin1x=arcsin(x)andtan1x=arctan(x).
Somestudentsgetconfusedaboutthissomeevenendupsettingarctan(x)equalto1/(tanx).WhenIteach,Itrytoreduceconfusionbyalwayswritingarcsinorarctan,ratherthansin1ortan1.Butthesin1andtan1notationstillneedstobediscussed,asitisusedonnearlyallhandheldcalculators.ThankstoIanMorrisonandJohnArmerdingforpointingthisoneout.
Confusionaboutthesquarerootsymbol.Everypositivenumberbhastwosquareroots.Theexpressionbactuallymeans"thenonnegativesquarerootofb,"butunfortunatelysomestudentsthinkthatthatexpressionmeans"eitherofthesquarerootsofb"i.e.,theythinkitrepresentstwonumbers....Thiserrorismademorecommonbecauseoftheunfortunatefactthatwemathteachersaremerelyhuman,andsometimesalittlesloppy:Whenwewritebontheblackboard,whatwesayaloudmightjustbe"thesquarerootofb."Butthat'sjustlaziness.Ifyouaskusspecificallyaboutthat,we'lltellyou"Oh,I'msorry,ofcourseImeantthenonnegativesquarerootofbIthoughtthatgoeswithoutsaying."...Ifyoureallydowanttoindicatebothsquarerootsofb,youusetheplusorminussign,asinthisexpression:b.
Problemswithorderofoperations.Itiscustomarytoperformcertainmathematicaloperationsincertainorders,andsowedon'tneedquitesomanyparentheses.Forinstance,everyoneagreesthat
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 11/31
"6w+5"means"(6w)+5",andnot"6(w+5)"themultiplicationisperformedbeforetheaddition,andsotheparenthesesarenotneededif"(6w)+5"iswhatyoureallymeantosay.Unfortunately,somestudentshavenotlearnedthecorrectorderofsomeoperations.
HereisanexamplefromIanMorrison:Whatis32?Manystudentsthinkthattheexpressionmeans(3)2,andsotheyarriveatananswerof9.Butthatiswrong.Theconventionamongmathematiciansistoperformtheexponentiationbeforetheminussign,andso32iscorrectlyinterpretedas(32),whichyields9.
Ambiguouslywrittenfractions.Incertaincommonsituationswithfractions,thereisalackofconsensusaboutwhatordertoperformoperationsin.Forinstance,does"3/5x"mean"(3/5)x"or"3/(5x)"?
Forthisconfusion,teachersmustsharetheblame.Theycertainlymeanwellmostmathteachersbelievethattheyarefollowingtheconventionalorderofoperations.Theyarenotawarethatseveralconventionsarewidelyused,andnooneofthemisuniversallyaccepted.Studentsmaylearnonemethodfromoneteacherandthengoontoanotherteacherwhoexpectsstudentstofollowadifferentmethod.Bothteacherandstudentmaybeunawareofthesourceoftheproblem.
Herearesomeofthemostwidelyusedinterpretations:
The"BODMASinterpretation"(bracketedoperations,division,multiplication,addition,subtraction):Performdivisionbeforemultiplication.Forinstance,thefunctionf(x)=3/5xgets
interpretedas(3/5)x= .Inparticular,f(5)=3andf(1/5)=3/25.
The"MyDearAuntSally"interpretation(multiplication,division,addition,subtraction):Performmultiplicationbeforedivision.Forinstance,thefunctionf(x)=3/5xgetsinterpretedas3/(5x)=
.Inparticular,f(5)=3/25andf(1/5)=3.Likewise,"ax/by"wouldbeinterpretedas
"(ax)/(by)".
TheinterpretationusedbyFORTRANandsomeothercomputerlanguages(aswellassomehumans):Multiplicationanddivisionaregivenequalpriorityastringofsuchoperationsisprocessedfromlefttoright.Forinstance,"ax/by"wouldbeinterpretedas"((ax)/b))y",ormoresimply"(axy)/b".
Somestudentsthinkthattheirelectroniccalculatorscanberelieduponforcorrectanswers.Butsomecalculatorsfollowoneconvention,andothercalculatorsfollowanotherconvention.Infact,someoftheTexasInstrumentscalculatorsfollowtwoconventions,accordingtowhethermultiplicationisindicatedbyjuxtapositionorasymbol:
3/5xisinterpretedas3/(5x),but3/5*xisinterpretedas(3/5)x.
(ThankstoChrisPhillipsandThomasCowderyforsomeoftheseexamplesandcomments.)
Becausethereisnoconsensusofinterpretation,Irecommendthatyoudonotwriteexpressionslike"3/5x"i.e.,donotwriteafractioninvolvingadiagonalslashfollowedbyaproduct,withoutany
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 12/31
parentheses.Instead,useoneofthesefournonambiguousexpressions:(3/5)x, ,3/(5x), .
Insomecases,additionalinformationisevidentfromthecontextifoneisfamiliarwiththecontext.Forinstance,anexperiencedmathematicianwillrecognizedy/dxasaderivativeitisthequotientoftwodifferentials.Theletterdrepresentsthedifferentialoperator,notavariable.Theexpressiondxrepresentsthedifferentialofx,nottheproductoftwovariables.Thus,parenthesesarenotneeded,andwouldlookratherstrangeifused.Wedonotwritedy/(dx)or(dy)/(dx).
Hereisanothercommonerrorinthewritingoffractions:Ifyouwritethehorizontalfractionbartoohigh,
itcanbemisread.Forinstance, or areacceptableexpressions(withdifferentmeanings),but
isunacceptableithasnoconventionalmeaning,andcouldbeinterpretedambiguouslyaseither
ofthepreviousfractions.Iwillnotgivefullcreditforambiguousanswersonanyquizortest.Inthistypeoferror,sloppyhandwritingistheculprit.Whenyouwriteanexpressionsuchas ,besuretowrite
carefully,sothatthehorizontalbarisaimedatthemiddleofthex.
Here'sonemoreexampleofinterest.Whenenteredas2^3^4withoutparentheses,theTI85calculatorshows4096andtheTI89shows2.41785163923E24.(Thosearetheanswersto(2^3)^4and2^(3^4),respectively.)Thus,eventhecalculatorsmadebyonecompanydon'tallagreeontheirordersofoperations.Whenindoubt,useparentheses!ThankstoBillDodgeforthisexample.
Streamofconsciousnessequalitiesandimplications.(MythankstoH.G.Mushenheimforidentifyingthistypeoferrorandsuggestinganameforit.)Thisisanerrorintheintermediatestepsinstudents'computations.Itdoesn'toftenleadtoanerroneousfinalresultattheendofthatcomputation,butitistremendouslyirritatingtothemathematicianwhomustgradethestudent'spaper.Itmayalsoleadtoalossofpartialcredit,ifthestudentmakessomeothererrorinhisorhercomputationandthegraderisthenunabletodecipherthestudent'sworkbecauseofthisstreamofconsciousnesserror.
Toputitsimply:Somestudents(especiallycollegefreshmen)usetheequalssign(=)asasymbolfortheword"then"orthephrase"thenextstepis."Forinstance,whenaskedtofindthethirdderivativeofx4+7x25,somestudentswillwrite"x4+7x25=4x3+14x=12x2+14=24x."Ofcourse,thosefourexpressionsarenotactuallyequaltooneanother.
Aslightvariantofthiserrorconsistsofconnectingseveraldifferentequationswithequalsigns,wheretheintermediateequalssignsareintendedtoconvey"equivalentto"forexample,x=y3=x+3=y.Thisisveryconfusingandaltogetherwrong,becauseequalityistransitivei.e.,ifa=bandb=cthena=c,butxcertainlyisnotequaltox+3.Itwouldbebettertoreplacethatmiddleequalssignwithsomeothersymbol.Themostobvioussymbolforthispurposeis,whichmeans"isequivalentto,"butthatsymbolhasthedisadvantageoflookingtoomuchlikeanequalssign,andthuspossiblyleadingtothesameconfusion.Thus,abetterchoicewouldbeor,bothofwhichmean"ifandonlyif."Thus,Iwouldrewritetheexampleaboveasx=y3x+3=y.
Thereisalsoamore"advanced"formofthiserror.Somemoreadvancedstudents(e.g.,collegeseniors)usetheimplicationsymbol()asasymbolforthephrase"thenextstepis."Astringofstatementsofthe
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 13/31
form
ABCDshouldmeanthatAbyitselfimpliesB,andBbyitselfimpliesC,andCbyitselfimpliesDthatisthecoventionalinterpretationgivenbymathematicians.ButsomestudentsusesuchastringtomeanmerelythatifwestartfromA,thenthenextstepinourreasoningisB(usingnotonlyAbutotherinformationaswell)andthenthenextstepisC(perhapsusingbothAandB),etc.
Actually,thereisasymbolfor"thenextstepis."Itlookslikethis: Itisalsocalled"leadsto,"andintheLaTeXformattinglanguageitisgivenbythecode\leadsto.However,Ihaven'tseenitusedveryoften.
ErrorsinReasoning
Goingoveryourwork.Unfortunately,mosttextbooksdonotdevotealotofattentiontocheckingyourwork,andsometeachersalsoskipthistopic.Perhapsthereasonisthatthereisnowellorganizedbodyoftheoryonhowtocheckyourwork.Unfortunately,somestudentsendupwiththeimpressionthatitisnotnecessarytocheckyourworkjustwriteituponce,andhopethatit'scorrect.Butthat'snonsense.Allofusmakemistakessometimes.Inanysubject,ifyouwanttodogoodwork,youhavetoworkcarefully,andthenyouhavetocheckyourwork.InEnglish,thisiscalled"proofreading"incomputerscience,thisiscalled"debugging."
Moreover,inmathematics,checkingyourworkisanimportantpartofthelearningprocess.Sure,you'lllearnwhatyoudidwrongwhenyougetyourhomeworkpaperbackfromthegraderbutyou'lllearnthesubjectmuchbetterifyoutryveryhardtomakesurethatyouranswersarerightbeforeyouturninyourhomework.
It'simportanttocheckyourwork,but"goingoveryourwork"istheworstwaytodoit.Ihavetwistedsomewordshere,inordertomakeapoint.By"goingoveryourwork"Imeanreadingthroughthestepsthatyou'vejustdone,toseeiftheylookright.Thedrawbackofthatmethodisyou'requitelikelytomakethesamemistakeagainwhenyoureadthroughyoursteps!Thisisparticularlytrueofconceptualerrorse.g.,forgettingtocheckforextraneousroots(discussedlateronthiswebpage).
Youwouldbemuchmorelikelytocatchyourerrorif,instead,youcheckedyourworkbysomemethodthatisdifferentfromyouroriginalcomputation.Indeed,withthatapproach,theonlywayyourerrorcangoundetectedisifyoumaketwodifferenterrorsthatsomehow,justbyaremarkablecoincidence,managetocanceleachotheroute.g.,ifyouarriveatthesamewronganswerbytwodifferentincorrectmethods.Thathappensoccasionally,butveryseldom.
Inmanycases,yoursecondmethodcanbeeasier,becauseitcanmakeuseofthefactthatyoualreadyhaveananswer.Thistypeofcheckingisnot100%reliable,butitisveryhighlyreliable,anditmaytakeverylittletimeandeffort.
Hereisasimpleexample.Supposethatwewanttosolve3(x2)+7x=2(x+1)forx.Hereisacorrectsolution:
3(x2)+7x=2(x+1)3x6+7x=2x+2
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 14/31
3x+7x2x=2+68x=8x=1
Now,oneeasywaytocheckthisworkistoplugx=1intoeachsideoftheoriginalequation,andseeiftheresultscomeoutthesame.Ontheleftside,wehave3(x2)+7x=3(12)+7(1)=3(1)+7(1)=(3)+7=4.Ontherightside,wehave2(x+1)=2(1+1)=2(2)=4.Thosearethesame,sothecheckworks.It'seasierthantheoriginalcomputation,becauseintheoriginalcomputationwewerelookingforxinthecheck,wealreadyhaveacandidateforx.Nevertheless,thiscomputationwasbyadifferentmethodthanouroriginalcomputation,sotheanswerisprobablyright.
Differentkindsofproblemsrequiredifferentkindsofchecking.Forafewkindsofproblems,noothermethodofcheckingbesides"goingoveryourwork"willsuggestitselftoyou.Butformostproblems,somesecondmethodofcheckingwillbeevidentifyouthinkaboutitforamoment.
Ifyouabsolutelycan'tthinkofanyothermethod,hereisalastresorttechnique:Putthepaperawaysomewhere.Severalhourslater(ifyoucanaffordtowaitthatlong),dothesameproblemsoverbythesamemethod,ifneedbebutonanewsheetofpaper,withoutlookingatthefirstsheet.Thencomparetheanswers.Thereisstillsomechanceofmakingthesameerrortwice,butthismethodreducesthatchanceatleastalittle.Unfortunately,thistechniquedoublestheamountofworkyouhavetodo,andsoyoumaybereluctanttoemploythistechnique.Well,that'suptoyouit'syourdecision.Buthowbadlydoyouwanttomasterthematerialandgetthehighergrade?Howmuchimportancedoyouattachtotheintegrityofyourwork?
Onemethodthatmanystudentsusetochecktheirhomeworkisthis:beforeturninginyourpaper,compareitwithaclassmate'spaperseeifthetwoofyougotthesameanswers.I'lladmitthatthisdoessatisfymycriterion:Ifyougotthesameanswerforaproblem,thenthatanswerisprobablyright.Thisapproachhasbothadvantagesanddisadvantages.Onedisadvantageisthatitmayviolateyourteacher'srulesabouthomeworkbeinganindividualeffortperhapsyoushouldaskyourteacherwhathisorherrulesare.Anotherconcernis:howmuchdoyoulearnfromthecomparisonofthetwoanswers?Ifyoudiscusstheproblemwithyourclassmate,youmaylearnsomething.Withorwithoutaclassmate'sinvolvement,ifyouthinksomemoreaboutthedifferentsolutionstotheproblem,youmaylearnsomething.
Whenyoudofindthatyourtwoanswersdiffer,workverycarefullytodeterminewhichone(ifeither)iscorrect.Don'thurrythroughthiscruciallastpartoftheprocess.You'vealreadydemonstratedyourfallibilityonthistypeofproblem,sothereisextrareasontodoubttheaccuracyofanyfurtherworkonthisproblemcheckyourresultsseveraltimes.
Perhapstheerroroccurredthroughmerecarelessness,becauseyouweren'treallyinterestedintheworkandyouwereinahurrytofinishitandputitaside.Ifso,don'tcompoundthaterror.Younowmustpayforyourneglectyounowmustputinevenmoretimetomasterthematerialproperly!Theproblemwon'tjustgoawayorloseimportanceifyouignoreit.Mathematics,morethananyothersubject,isverticallystructured:eachconceptbuildsonmanyconceptsthatprecededit.Onceyouleaveatopicunmastered,itwillhauntyourepeatedlythroughoutmanyofthetopicsthatfollowit,inallofthemathcoursesthatfollowit.
Also,ifdiscoverthatyou'vemadeanerror,trytodiscoverwhattheerrorwas.Itmaybeatypeoferrorthatyouaremakingwithsomefrequency.Onceyouidentifyit,youmaybebetterabletowatchoutforitinthefuture.
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 15/31
Notnoticingthatsomestepsareirreversible.Ifyoudothesamethingtobothsidesofatrueequation,you'llgetanothertrueequation.Soifyouhaveanequationthatissatisfiedbysomeunknownnumberx,andyoudothesamethingtobothsidesoftheequation,thenthenewequationwillstillbesatisfiedbythesamenumberx.Thus,thenewequationwillhaveallthesolutionsxthattheoldequationhadbutitmightalsohavesomenewsolutions.
Someoperationsarereversiblei.e.,wehavethesamesetofsolutionsbeforeandaftertheoperation.Forinstance,
Theoperation"multiplybothsidesoftheequationby2"isreversible.Forexample,thesetofallnumbersxthatsatisfyx23x2=4isthesameasthesetofallnumbersxthatsatisfy2x26x4=8.Infact,toreversetheoperation,wejusthavetomultiplybothsidesofanequationby1/2.Theoperation"subtract7frombothsides"isreversible.Toreverseit,justadd7tobothsides.
Someoperationsarenotreversible,andsowemaygetnewsolutionswhenweperformsuchanoperation.Forinstance,
Theoperation"squarebothsides"isnotreversible.Forinstance,theequationx=3hasonlyonesolution,butwhenwesquarebothsides,wegetx2=9,whichhastwosolutions.Theoperation"multiplybothsidesbyx4"isnotreversible.Theresultingequationwillhaveforitssolutionsallofthesolutionsoftheoriginalequationplustheadditionalnewsolutionx=4.
Acommonlyusedmethodforsolvingequationsisthis:Constructasequenceofequations,goingfromoneequationtothenextbydoingthesamethingtobothsidesofanequation,choosingtheoperationstograduallysimplifytheequation,untilyougettheequationdowntosomethingobviouslike"x=5".Thismethodisnotbadfordiscovery,butasamethodofcertificationitisunreliable.Tomakeitreliable,youneedtoaddonemorerule:
ifanyofyourstepsareirreversible,thenyoumustcheckforextraneousrootswhenyougettotheendofthecomputation.
That'sbecause,attheendofyourcomputationalprocedure,you'llhavenotonlythesolution(s)totheoriginalproblem,butpossiblyalsosomeadditionalnumbersthatdonotsolvetheoriginalproblem.Howdoyoucheckforthem?Justplugeachofyouranswersintotheoriginalproblem,toseewhetheritworks.Manystudents,unfortuntely,omitthatlaststep.
Firstexample:
Thegivenproblemis 2=x.Tobeginsolvingthisproblem,add2tobothsides(areversiblestep)weobtain =x+2.Squarebothsidesanirreversiblestepweobtain
2x+12=x2+4x+4.Byaddingandsubtractingappropriateamountstobothsidesoftheequation(areversiblestep),weobtainx2+2x8=0.Nowsolvethatquadraticequationbyyourfavoritemethodbythequadraticformula,bycompletingthesquare,orbyfactoringbyinspection.Weobtain
x=2orx=4.
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 16/31
Unfortunately,manystudentsstopatthissteptheybelievethey'redonetheywrite{2,4}fortheiranswer.Acorrectsolutioncontinuesasfollows:Sinceatleastoneofthestepsinourprocedurewasirreversible,wemustcheckforextraneousroots.Checkeachofthenumbers2and4,toseeifitsatisfiestheequationoriginallygivenintheproblem.
Whenx=2,then 2=(4+12)1/22=42=x,sowehaveacorrectsolution.However,Whenx=4,then 2=(8+12)1/22=22x,sowehaveanincorrectsolution.
Thusthecorrectansweris{2}.
Secondexample.
Thegivenproblemis .Byaddingandsubtracting(reversible),
weobtain .Byfactoring(reversible),weobtain .Canceloutanx
(irreversible)weobtainx(x+3)=0.Thesolutionsappeartobex=0andx=3.Somestudentsunfortunatelystophere,butweshouldn'toneofourstepswasirreversible.Checkingrevealsthatx=0doesn'tworkinthegivenproblem,soit'sextraneous.Thecorrectanswerisjustx=3.
Ofcourse,evenasidefromtheissueofextraneousroots,anotherreasontocheckyouranswersistoavoidarithmeticerrors.Thisisaspecialcaseof"checkingyourwork,"mentionedelsewhereonthiswebpage.Weallmakecomputationalmistakeswecancatchmostofourcomputationalmistakeswithalittleextraeffort.
TheextraneousrootserrorwasbroughttomyattentionbyDr.RichardBeldin.ProfessorBeldintellsmethathegaveatestheavilylacedwithextraneousrootsproblems,andwarnedthestudentsthat
suchproblemswouldappearonthetest,andtheappearanceofanextraneousrootinananswerwouldcosthalfthecreditforaproblem,sothestudentsshouldcheckforextraneousroots.
ProfessorBeldinreportsthat,nevertheless,aboutathirdofthestudentsneglectedtocheck,onsomanyproblemsthattheylosttwolettergradesontheoverallthetestscore.
ProfessorStephenGlasbyreportsthisinterestingexampleofignoringirreversibility.Wewishtoprovesinx=(1cos2x).Webeginbysquaringbothsidesofthatequationweobtainsin2x=1cos2x.Rearrangetermstoobtainsin2x+cos2x=1.That'strue,soapparentlytheproofisdone.Butit'snot,becausesquaringbothsideswasirreversible.Infact,theequationsinx=(1cos2x)thatwe'vejust"proved"isn'ttrueforinstance,tryx=/2.
Confusingastatementwithitsconverse.Theimplication"AimpliesB"isnotthesameastheimplication"BimpliesA."Forinstance,
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 17/31
ifIwentswimmingatthebeachtoday,thenIgotwettoday
isatruestatement.But
ifIgotwettoday,thenIwentswimmingatthebeachtoday
doesn'thavetobetruemaybeIgotwetbytakingashowerorbathathome.Thedifferenceiseasytoseeinconcreteexampleslikethese,butitmaybehardertoseeintheabstractsettingofmathematics.
Sometechnicalterminologymightbehelpfulhere.Thesymbolmeans"implies."Thetwostatements"AB"and"BA"aresaidtobeconversesofeachother.Whatwe'vejustexplainedisthatanimplicationanditsconversegenerallyarenotequivalent.
Ishouldemphasizetheword"generally"inthelastparagraph.Inafewcasestheimplications"AB"and"BA"doturnouttobeequivalent.Forinstance,letp,q,rbethelengthsofthesidesofatriangle,withrbeingthelongestsidethen
p2+q2=r2ifandonlyifthetriangleisarighttriangle.
The"if"partofthatstatementisthewellknownPythagoreanTheoremthe"onlyif"partisitsconverse,whichalsohappenstobetruebutislesswellknown.
Somestudentsconfuseastatementwithitsconverse.Thismaystempartlyfromthefactthat,inmanynonmathematicalsituations,astatementisequivalenttoitsconverse,andsoineveryday"human"Englishweoftenusetheword"if"interchangeablywiththephrase"ifandonlyif".Forinstance,
I'llgotothevendingmachineandbuyasnackifIgethungry
soundsreasonable.ButmostpeoplewouldfigurethatifIdonotgethungry,thenIwon'tgobuyasnack.So,evidently,whatIreallymeantwas
I'llgotothevendingmachineandbuyasnackifandonlyifIgethungry.
Mostpeoplewouldjustsaytheshortersentence,andmeanthelongeroneit'sasortofverbalshortcut.Generallyyoucanfigureoutfromthecontextjustwhattherealmeaningis,andusuallyyoudon'teventhinkaboutitonaconsciouslevel.
Tomakemattersmoreconfusing,mathematiciansarehumanstoo.Incertaincontexts,evenmathematiciansuse"if"whentheyreallymean"ifandonlyif."Youhavetofigurethisoutfromthecontext,andthatmaybehardtodoifyou'renewtothelanguageofmathematics,andnotafluentspeaker.Chiefly,mathematiciansusetheverbalshortcutwhenthey'regivingdefinitions,andthenyouhaveahint:thewordbeingdefinedusuallyisinitalicsorboldface.Forinstance,hereisthedefinitionofcontinuityofarealvaluedfunctionf:
fiscontinuousifforeachrealnumberpandeachpositivenumberthereexistsapositivenumber(whichmaydependonpand)suchthat,foreachrealnumberq,if|pq|
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 18/31
Conversesalsoshouldnotbeconfusedwithcontrapositives.Thosetwowordssoundsimilarbuttheymeanverydifferentthings.Thecontrapositiveoftheimplication"AB"istheimplication"(~B)(~A)",where~means"not."Thosetwostatementsareequivalent.Forinstance,
ifIwentswimmingatthebeachtoday,thenIgotwettoday
hasexactlythesamemeaningasthemorecomplicatedsoundingstatement
ifIdidn'tgetwettoday,thenIdidn'tgoswimmingatthebeachtoday.
Sometimeswereplaceastatementwithitscontrapositive,becauseitmaybeeasiertoprove,evenifitismorecomplicatedtostate.(ThankstoValeryMishkinforbringingthisclassoferrorstomyattention.)
Workingbackward.Thisisanunreliablemethodofproofused,unfortunately,bymanystudents.Westartwiththestatementthatwewanttoprove,andgraduallyreplaceitwithconsequences,untilwearriveatastatementthatisobviouslytrue(suchas1=1).Fromthatsomestudentsconcludethattheoriginalstatementistrue.Theyoverlookthefactthatsomeoftheirstepsmightbeirreversible.
Hereisanexampleofasuccessfulandcorrectuseof"workingbackward":weareaskedtoprovethatthecuberootof3isgreaterthanthesquarerootof2.Wewritethesesteps:
Startbyassumingthethingthatwe'retryingtoprove:31/3>21/2.Raisebothsidestothepower6thatyields(31/3)6>(21/2)6.Simplifybothsides,usingtheruleofexponentsthatsays(ab)c=a(bc).Thusweobtain32>23.Evaluate.Thatyields9>8,whichisclearlytrue.
Somestudentswouldbelievethatwehavenowproved31/3>21/2.Butthat'snotaproofyoushouldneverbeginaproofbyassumingtheverythingthatyou'retryingtoprove.Inthisexample,however,allthestepshappentobereversible,sothosestepscanbemadeintoaproof.Wejusthavetorewritethestepsintheirproperorder:
9>8isobviouslytrue.Rewritethatas32>23.Rewritethatas(31/3)6>(21/2)6,byusingtheruleofexponentsthatsays(ab)c=a(bc).Nowtakethesixthrootonbothsides.Thatleaves31/3>21/2.
Workingbackwardcanbeagoodmethodfordiscoveringproofs,thoughithastobeusedwithcaution,asdiscussedbelow.Butitisanunacceptablemethodforpresentingproofsafteryouhavediscoveredthem.Studentsmustdistinguishbetweendiscovery(whichcanbehaphazard,informal,illogical)andpresentation(whichmustberigorous).Thereasoningusedinworkingbackwardisareversalofthereasoningneededforpresentationoftheproofbutthatmeansreplacingeachimplication"AB"withitsconverse,"BA".Aswepointedoutafewparagraphsago,thosetwoimplicationsaresometimesnotequivalent.
Insomecases,theimplicationisreversiblei.e.,somereversibleoperation(likemultiplyingbothsidesofanequationby2,orraisingbothsidesofaninequalitytothesixthpowerwhenbothsideswerealreadypositive)transformsstatementAintostatementB.Perhapsthestudentshavegottenintothehabitofexpectingallimplicationstobereversible,becauseearlyintheireducationtheywereexposedtomany
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 19/31
reversibletransformationsaddingthree,multiplyingbyahalf,etc.Butinfact,mostimplicationsofmathematicalstatementsarenotreversible,andso"workingbackward"isalmostneveracceptableasamethodofpresentingaproof.
Workingbackwardcanbeusedfordiscoveringaproof(and,infact,sometimesitistheonlydiscoverymethodavailable),butitmustbeusedwithappropriatecaution.Ateachstepinthediscoveryprocess,youstartfromsomestatementA,andyoucreatearelatedstatementBitmaybethecasethattheimplicationAimpliesBisobvious.ButyouhavetothinkaboutwhetherBimpliesA.IfyoucanfindaconvincingdemonstrationthatBimpliesA,thenyoucanproceed.Ifyoucan'tfindademonstrationofBimpliesA,thenyoumightaswelldiscardstatementB,becauseitisofnouseatalltoyoulookforsomeotherstatementtouseinstead.
Beginnersoftenmakemistakeswhentheyuse"workingbackward,"becausetheydon'tnoticethatsomestepisirreversible.Forinstance,thestatementx> isnottrueforallrealnumbersx.Butifwedidn'tknowthat,wemightcomeupwiththisproof:
Startwithwhatwewanttoprove:x> .Since meansthenonnegativesquarerootofx21,weknowthatitisnonnegative.Sincexisevenlarger,itisalsononnegative.Squaringbothsidesofaninequalityisareversiblestepifbothsidesarenonnegative.Thusweobtainx2>x21.Subtractx2frombothsides0>1.Andthat'sclearlytrue,regardlessofwhatxis."Thereforex> forallrealnumbersx."
Butthatconclusioniswrong.Therightsideoftheinequalityisundefinedwhenx=0.5.Andwhenx=2,thenbothsidesoftheinequalityaredefined,buttheinequalityisfalse.Seeifyoucanfindwherethereasoningwentawry.
Wellthen,ifreasoningbackwardisnotacceptableasapresentationofaproof,whatisacceptable?Adirectproofisacceptable.Atheoremhascertainhypotheses(assumptions)andcertainconclusions.Inadirectproof,youstartwiththehypotheses,andyougenerateconsequencesi.e.,youstartmakingsentences,whereeachsentenceiseitherahypothesisofthetheorem,anaxiom(ifyou'reusinganabstracttheory),oraresultdeducedfromsomeearliertheoremusingsentencesyou'vealreadygeneratedintheproof.Theymustbeinorderi.e.,ifonesentenceAisusedtodeduceanothersentenceB,thensentenceAshouldappearbeforesentenceB.Thegoalistoeventuallygenerate,amongtheconsequences,theconclusionofthedesiredtheoreom.
Somevariantsonthisarepossible,butonlyiftheexplanatorylanguageisusedverycarefullysuchvariantsarenotrecommendedforbeginners.Thevariantsinvolvephraseslike"itsufficestoshowthat...".Thesephrasesarelikeforeshadowinginastory,orlikedirectionsignsonahighway.Theyintentionallyappearoutofchronologicalorder,tomaketheintendedroutemoreunderstandable.Butinsomesensetheyarenotreallypartoftheofficialprooftheyarejustcommentariesontheside,tomaketheofficialproofeasiertounderstand.Whenyoupassasignthatsays"100milestoNashville,"you'renotactuallyinNashvilleyet.
Perhapsthebiggestfailureintheproofsofbeginnersisaseverelackofwords.Abeginnerwillwritedownanequationthatshouldbeaccompaniedbyeitherthephrase"wehavenowshown"orthephrase"weintendtoshow",toclarifyjustwhereweareintheproof.Butthebeginnerwritesneitherphrase,andthereaderisexpectedtoguesswhichitis.Thisislikeanovelinwhichtherearemanyflashbacksand
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 20/31
alsomuchforeshadowing,butalltheverbsareinpresenttensethereadermusttrytofigureoutalogicalorderinwhichtheeventsactuallyoccur.
OneeasymethodthatIhavebegunrecommendingtostudentsisthis:Putaquestionmarkoveranyrelationship(equalssign,greaterthansign,etc.)thatrepresentsanassertionthatyouwanttoprove,buthavenotyetproved.Anequalssignwithoutaquestionmarkwillthenbeunderstoodtorepresentanequationthatyouhavealreadyproved.Lateryoucanputacheckmarknexttotheequationswhosedoubthasbeenremoved.Thismethodmayhelpthestudentwritingthework,butunfortunatelyitdoesnotgreatlyhelptheteacherorgraderwhoisreadingtheworktheorderofstepsisstillobscured.
Anothercommonstyleofproofistheindirectproof,alsoknownasproofbycontradiction.Inthisproof,westartwiththehypothesesofthedesiredtheorembutwemayalsoadd,asadditionalhypothesis,thestatementthat"thedesiredtheorem'sconclusionisfalse."Inotherwords,wereallywanttoproveAB,sowestartbyassumingbothAand~B(where~means"not").Wethenstartgeneratingconsequences,andwetrytogenerateacontradictionamongourconsequences.Whenwedoso,thisestablishesthatABmusthavebeentrueafterall.Thiskindofproofishardertoread,butitisactuallyeasiertodiscoverandtowrite:wehavemorehypotheses(notonlyA,butalso~B),soitiseasiertogenerateconsequences.Irecommendthatbeginnersavoidindirectproofsaslongaspossiblebutifyoucontinuewithyourmatheducation,youwilleventuallyrunintosomeabstracttheoremsinhighermaththatcanonlybeprovedbyindirectproof.
Difficultieswithquantifiers.Quantifiersarethephrases"thereexists"and"forevery."Manystudentsevenbeginninggraduatestudentsinmathematics!havelittleornounderstandingoftheuseofquantifiers.Forinstance,whichofthesestatementsistrueandwhichisfalse,usingthestandardrealnumbersystem?
Foreachpositivenumberathereexistsapositivenumberbsuchthatbislessthana.
Thereexistsapositivenumberbsuchthatforeachpositivenumberawehaveblessthana.
Difficultywithquantifiersmaybecommon,butI'mnotsurewhatcausesthedifficulty.Perhapsitisjustthatmathematicalsentencesaregrammaticallymorecomplicatedthannonmathematicalones.Forinstance,arealvaluedfunctionfdefinedonthereallineiscontinuousif
foreachpointpandforeachnumberepsilongreaterthanzero,thereexistsanumberdeltagreaterthanzerosuchthat,foreachpointq,ifthedistancefromptoqislessthandelta,thenthedistancefromf(p)tof(q)islessthanepsilon.
Thissentenceinvolvesseveralnestedclauses,basedonseveralquantifiers:
foreachpointp...foreachnumberepsilon...thereexistsdeltasuchthat...foreachpointq...
Nonmathematicalgrammargenerallydoesn'tinvolvesomanynestedclausesandsuchcrucialattentiontotheorderofthewords.
Ithinkthatmanystudentswouldbenefitfromthinkingofquantifiersasindicatorsofacompetitionbetweentwoadversaries,asinacourtoflaw.Forinstance,whenIassertthatthefunctionfis
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 21/31
continuous,Iamassertingthat
nomatterwhatpointpandwhatpositivenumberepsilonyouspecify,Icanthenspecifyacorrespondingpositivenumberdelta,suchthat,nomatterwhatpointqyouthenspecify,ifyoudemonstratethatyourqhasdistancefromyourplessthanmydelta,thenIcandemonstratethattheresultingf(p)andf(q)areseparatedbyadistancelessthanyourepsilon.
Ofcourse,itmustbeunderstoodthatthetwoadversariesinmathematicsareemotionallyandmorallyneutral.Inacourtoflaw(atleast,asdepictedontelevision),itisoftenthecasethatonesideisthe"goodguys"andtheothersideisthe"badguys,"butinprinciplethelawissupposedtobeaneutralwayofseekingthetruthmathematicalreasoningistoo.
Somestudentsmayhaveaneasiertimeavoidingerrorswithquantifiersiftheyactuallyusesymbolsinsteadofwords.Thismaymakethedifferencesinthequantifiersmorevisuallyprominent.Thesymbolstouseare
universalquantifier "forall"(or"foreach")
existentialquantifier "thereexists"(or"thereexistsatleastone")
Withthosesymbols,myearliertwostatementsaboutrealnumberscanbewritten,respectively,as
( a>0)( b>0)(b0)( a>0)(b0)( >0)( q)(if|pq|
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 22/31
( p)~( >0)( >0)( q)(if|pq|0)( q)(if|pq|0)~( q)(if|pq|0)( q)~(if|pq|0)( q)(|pq|
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 23/31
DaveRusinhasputtogethersomenotesonthewidevarietyoferrorsonecanmakebynotunderstandingone'scalculator.Bytheway,I'lltakethisopportunitytomentionthatDaveRusinhasputtogetherasuperwebsite,MathematicalAtlas:AgatewaytoMathematics,whichoffersdefinitions,introductions,andlinkstoallsortsoftopicsinmath.
UnwarrantedGeneralizations
Aformulaornotationmayworkproperlyinonecontext,butsomestudentstrytoapplyitinawidercontext,whereitmaynotworkproperlyatall.RobinChapmanalsocallsthistypeoferror"crassformalism."Hereisoneexamplethathehasmentioned:
Everypositivenumberhastwosquareroots:onepositive,theothernegative.Thenotationbgenerallyisonlyusedwhenbisanonnegativerealnumberitmeans"thenonnegativesquarerootofb,"andnotjust"thesquarerootofb."Thenotationbprobablyshouldnotbeusedatallinthecontextofcomplexnumbers.Everynonzerocomplexnumberbhastwosquareroots,butingeneralthereisnonaturalwaytosaywhichoneshouldbeassociatedwiththeexpressionb.Theformula iscorrectwhen
aandbarepositiverealnumbers,butitleadstoerrorswhengeneralizedindiscriminatelytootherkindsofnumbers.Beginnersintheuseofcomplexnumbersarepronetoerrorssuchas
.Infact,thegreatmathematicianLeonhardEuler
publishedacomputationsimilartothisinabookin1770,whenthetheoryofcomplexnumberswasstillyoung.
Hereisanotherexample,frommyownteachingexperience:Whatisthederivativeofxx?Ifyouaskthisduringthefirstweekofcalculus,acorrectansweris"wehaven'tcoveredthatyet."Butmanystudentswillveryconfidentlytellyouthattheanswerisxxx1.Someofthemmayevensimplifythatexpressionitreducestoxxandafewstudentswillevenremark:"Say,that'sinterestingxxisitsownderivative!"Ofcourse,allthesestudentsarewrong.Thecorrectanswer,coveredafteraboutasemesterofcalculus,is (xx)=xx(1+lnx).
Thedifficultyisthat,inhighschoolorshortlyaftertheyarriveatcollege,thestudentshavelearnedthat
(xk)=kxk1
ThatformulaisactuallyWRONG,butinaverysubtleway.Thecorrectformulais
(xk)=kxk1(forallxwheretherightsideisdefined),ifkisanyconstant.
Theequationisunchanged,butit'snowaccompaniedbysomewordstellinguswhentheequationisapplicable.I'vethrownintheparenthetical"forallxwheretherightsideisdefined,"inordertoavoiddiscussingthecomplicationsthatarisewhenx0.ButthepartthatIreallywanttodiscusshereisthe
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 24/31
otherparti.e.,thephrase"ifkisanyconstant."
Tomostteachers,thatadditionalphrasedoesn'tseemimportant,becauseintheteacher'smind"x"usuallymeansavariableand"k"usuallymeansaconstant.Thelettersxandkareusedindifferentwayshere,alittlelikethedifferencebetweenboundandfreevariablesinlogic:Fixanyconstantkthentheequationstatesarelationshipbetweentwofunctionsofthevariablex.Sothelanguagesuggeststousthatxisprobablynotsupposedtoequalk.
Butthemathteacherisalreadyfluentinthislanguage,whereasmathematicsisaforeignlanguagetomoststudents.Tomoststudents,thedistinctionbetweenthetwoboxedformulasisonewhichdoesn'tseemimportantatfirst,becausetheonlyexamplesshowntothestudentatfirstarethoseinwhichkactuallyisaconstant.Whybothertomentionthatkmustbeaconstant,whentherearenootherconceivablemeaningsfork?Sothestudentmemorizesthefirst(incorrect)formula,ratherthanthesecond(correct)formula.
EverymathematicalformulashouldbeaccompaniedbyafewwordsofEnglish(oryournaturallanguage,whateveritis).ThewordsinEnglishtellwhentheformulacanorcan'tbeapplied.Butfrequentlyweneglectthewords,becausetheyseemtobeclearfromthecontext.Whenthecontextchanges,thewordsthatwe'veomittedmaybecomecrucial.
Studentshavedifficultywiththis.HereisanexperimentthatIhavetriedafewtimes:Atthebeginningofthesemester,Itellthestudentsthatthecorrectanswerto (xx)isnotxx,butratherxx(1+lnx),and
Itellthemthatthisproblemwillbeontheirfinalexamattheendofthesemester.Irepeatthesestatementsonceortwiceduringthesemester,andIrepeatthemagainattheveryendofthesemester,justbeforeclassesend.Nevertheless,alargepercentage(sometimesathird)ofmystudentsstillgettheproblemwrongonthefinalexam!Theiroriginal,incorrectlearningpersistsdespitemyefforts.
Ihaveacoupleoftheoriesaboutwhythishappens:(i)Formoststudents,mathematicsisaforeignlanguage,andthestudentfocuseshisorherattentiononthepartwhichseemsmostforeigni.e.,theformulas.Thewordshavetheappearanceofsomethingfamiliar("oh,that'sjustEnglish,andIalreadyknowEnglish"),andsothestudentdoesn'tpayalotofattentiontothewords.(ii)Undergraduatestudentstendtofocusonmechanicalcomputationstheyarenotyetmathematicallymatureenoughtobeabletothinkeasilyabouttheoreticalandabstractideas.
Asortoffootnote:Hereisacommonerroramongreadersofthiswebpage.Severalpeoplehavewrittentometoask,shouldn'tthatformulasay"ifkisanyconstantexcept0",or"ifkisanyconstantexcept1",orsomethinglikethat?Theythinksomespecialnoteneedstobemadeaboutthelogarithmcase.Actually,myformulaiscorrectasitstandsi.e.,foreveryconstantrealnumberkbutifyouwanttotellthewholestory,you'dhavetoappendsomeadditionalformula(s).Whenk=0,myformulajustsaysthederivativeof1is0x1that'struebutnotveryenlightening.Myformuladoesn'tmention,butalsodoesn'tcontradict,thefactthatthederivativeofln(x)is1x1.Youcanalwayssaymoreaboutanysubject,butIjustwantedtocontrasttheformulasxkandxxassimplyaspossible....Andofcourse,forsimplicity'ssake,Ihaven'tmentionedthecomplicationsyourunintowhenxiszerooranegativenumberI'monlyconsideringthosevaluesofxforwhichxkandxxareeasytodefine.
OtherCommonCalculusErrors
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 25/31
Jumpingtoconclusionsaboutinfinity.Someproblemsinvolvinginfinitycanbesolvedusing"theelementaryarithmeticofinfinity".Somestudentsjumptotheconclusionthatallproblemsinvolvinginfinitycanbesolvedbythissortof"elementaryarithmetic,"andsotheyguessallsortsofincorrectanswers(mainly0orinfinity)tosuchproblems.
Hereisanexampleofthe"elementaryarithmetic":Ifweusetheequationcautiously,wecansay(informally)that1/=0thoughperhapsitwouldbelessmisleadingtowriteinstead1/0.(MythankstoHansAbergforthissuggestionandforseveralothersuggestionsonthiswebpage.)Whatthisrulereallymeansisthatifyoutakeamediumsizednumberanddivideitbyanenormousnumber,yougetanumberverycloseto0.Forinstance,withoutdoinganyrealwork,wecanusethisruletoconcludeataglancethat
Thus,theproblem1/hastheanswer0.Theproblemdoesnothaveananswerinanyanalogousfashionwemightsaythatisundefined.Thisdoesnotmeanthat"Undefined"istheanswertoanyproblemoftheform.Whatitmeans,rather,isthateachprobleminvolvingrequiresaseparateanalysisdifferentproblemsofthistypehavedifferentanswers.Forinstance,
Thosefirsttwoproblemsarefairlyobviousthelastproblemtakesmoresophisticatedanalysis.Justguessingwouldnotgetyouananswerof1/2.(Ifyoudon'tunderstandwhatisgoingoninthelastproblem,trygraphingthefunctions andxononedisplayscreenonyourgraphingcalculator.Thatmayprovidealotofinsight,thoughit'snotaproof.)
Inasimilarfashion, donothavequickandeasyanswerstheytoorequiremorespecialized
andsophisticatedanalyses.
HereisacommonerrormentionedbyStuartPrice:Somestudentsseemtothinkthatlimn(1+(1/n))n=1.Theirreasoningisthis:"Whenn,then1+(1/n)1.Nowcomputelimn1n=1."Ofcourse,thisreasoningisjustabittoosimplistic.Youhavetodealwithbothofthen'sintheexpression(1+(1/n))natthesametimei.e.,theybothgotoinfinitysimultaneouslyyoucan'tfigurethatonegoestoinfinityandthentheothergoestoinfinity.Andinfact,ifyoulettheotheronegotoinfinityfirst,you'dgetadifferentanswer:limn(1+0.0000001)n=.Soevidentlytheanswerliessomewherebetween1and.Thatdoesn'ttellusmuchmypointhereisthateasymethodsdonotworkonthisproblem.Thecorrectanswerisanumberthatisnear2.718.(It'sanimportantconstant,knowntomathematiciansas"e".)There'snowayyoucouldgetthatbyaneasymethod.
Thatremindsmeofarelatedquestionthatseemstobothermanystudents:Whatis00?
Thereasonthataquestionarisesatallisbecausexyisdiscontinuousat(0,0).Indeed,wehavex0=1forallx>0,andwehave0y=0forally>0.Andlimx,y0+xydoesn'texist,becausethatexpressionmeansthelimitofxyasthepoint(x,y)approaches(0,0)alongallpathswherexyisdefined.
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 26/31
Nevertheless,many(most?)mathematicianswilldefine00=1,justforconvenience,becausethatmakesthemostformulaswork(andthentheywillnoteexceptionsforformulasthatrequireadifferentdefinition).
Forinstance,ifwe'reworkingwithpolynomialsorpowerseries,
p(x)=a0x0+a1x1+a2x2+a3x3+...+anxn+...
perhapsthemostcommonplacefor00toarisethenit'sconvenienttohave00=1,sincea0x0needstobeequaltoa0.TheBinomialTheoremwouldbemorecomplicatedtowriteifwedefined00anyotherway.
Problemswithseries.SeanRaleighreportsthatthemostcommonserieserrorhehasseenisthis:Ifa1,a2,a3,...isasequenceconvergingto0,thenmanystudentsconclude(erroneously)thattheseriesa1+a2+a3+...mustbeconvergent(i.e.,mustadduptoafinitenumber).Perhapstheyholdthatbeliefbecauseitistrueformostoftheexamplesthattheyhaveseen.Mostcounterexamplesaretooadvancedtobeincludedinanelementarytextbook.Ofcourse,everycalculusbookgivesthesimpleexampleoftheharmonicseries:
1+(1/2)+(1/3)+(1/4)+...=butonesingleexampleofdivergencedoesnotseemtooutweighinthestudents'mindsthemanyexamplesofconvergencethattheyhaveseen.
Lossormisuseofconstantsofintegration.Theindefiniteintegralofafunctioninvolvesan"arbitraryconstant",andthiscausesconfusionformanystudents,becausethenotationdoesn'tconveytheconceptverywell.Anexpressionsuchas"3x2+5x+C"reallyissupposedtorepresentaninfinitecollectionoffunctionsitrepresentsallofthefunctions
3x2+5x+7,3x2+5x+19,3x2+5x3.19,etc.
plusmorefunctionsofthesamesort.Oneofthedifficulties,also,isthatthesameletter"C"iscustomarilyusedforallsucharbitraryconstantsbutonecomputationmayinvolveseveraldifferentarbitraryconstants.ItwouldbemoreaccuratetoputsubscriptsontheC's,todifferentiateoneofthemfromanotheri.e.,writeC1,C2,C3,etc.andIoftendothatinmylectures.
Hereisanexample.TheformulaforIntegrationByParts,initsbriefestform,isudv=uvvduthatcanbeunderstoodmoreeasilyas
u(x)v'(x)dx=u(x)v(x)u'(x)v(x)dx.
Now,thatformulaiscorrect,butitcaneasilybemishandledandcanleadtoerrors.Hereisoneparticularlyamusingerror:Plugu(x)=1/xandv(x)=xintotheformulaabove.Weget
(1/x)(1)dx=(1/x)(x)(1/x2)(x)dx
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 27/31
whichsimplifiesto
(1/x)dx=1+(1/x)dx.
Now,regardlessofwhatyouthinkisthevalueof(1/x)dx,youjusthavetosubtractthatamountfrombothsidesoftheprecedingequation,toobtain0=1.Wait,howcanthatbe????Well,ifwe'reverycareful,werealizethatthetwo(1/x)dx'sonthetwosidesofthelastequationarenotactuallythesame.Whatthatlastequationreallysaysis
[ln|x|+C1]=1+[ln|x|+C2].Thatisatrueequation,ifwechoosetheconstantsC1andC2appropriatelyi.e.,ifwechoosethemsothatC1C2=1.Thus,thetwoconstantsarenotindependentofeachothertheyarenotcompletely
"arbitrary".Perhapsamoreaccurateexplanationisthis:Thetwoexpressions[ln|x|+C1]and1+[ln|x|+C2]donotactuallyrepresentindividualfunctionsrather,eachofthoseexpressionsrepresentsasetoffunctions.
Theexpression[ln|x|+C1]representsthesetofallthefunctionsofxthatcanbeobtainedbystartingwiththefunctionln|x|andthenaddingaconstant.
Theexpression1+[ln|x|+C2]representsthesetofallthefunctionsofxthatcanbeobtainedbystartingwiththefunctionln|x|,thenaddingaconstant,andthenadding1.
Thosetwodescriptionsmaysounddifferent,butifyouthinkaboutit,you'llseethatthosedescriptionsareneverthelessspecifyingthesameset.MythankstoAntonioFerraioli("Ferra")forthis0=1paradoxanditsexplanation.
Somestudentsmanagetomakethiskindoferrorevenwithdefiniteintegrals.Theystartfromtheformula(1/x)dx=1+(1/x)dx,whichiscorrectbutthenwhenthey"switchtodefiniteintegrals",theygettheformulaab(1/x)dx=1+ab(1/x)dx,whichisnotcorrect.Ifyoureallywantto"switchtodefiniteintegrals",youneedtothinkofthatconstant1asaspecialsortoffunction.Whenyouswitchtodefiniteintegrals,anyfunctionp(x)getsreplacedbyp(b)p(a).Inparticular,theconstantfunction1isthefunctiongivenbyp(x)=1forallx.Sop(b)p(a)becomes11,or0.
Somestudentsmayunderstandthisbetterifwedothewholethingwithdefiniteintegrals,rightfromthestart.Let'susetheformula
abu(x)v(x)dx=u(b)v(b)u(a)v(a)abu(x)v(x)dx.
Notethatthisformulahasonemoretermthanmypreviousboxedformulawhenweconvertu(x)v(x)tothedefiniteintegralversion,wereplaceitwithu(b)v(b)u(a)v(a).Nowpluginu(x)=1/xandv(x)=x.Weget
ab(1/x)(1)dx=(1/b)(b)(1/a)(a)ab(1/x2)(x)dx
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 28/31
which(assuming0isnotintheinterval[a,b])simplifiesto
[ln|b|ln|a|]=11[ln|b|+ln|a|]whichistruei.e.,thereisnocontradictionhere.
SomestudentsmaybepuzzledbythedifferencesbetweenthetwoversionsoftheIntegrationbyPartsformula(inboxes,inthelastfewparagraphs).Iwilldescribeinalittlemoredetailhowyougetfromthedefiniteintegralformula(inthelastbox)totheindefiniteintegralformula(inthefirstboxinthissection).Thinkofaasaconstantandbasavariable,andyou'llgetsomethinglikethis:
[u(x)v(x)dx+C1]=[u(x)v(x)C2][u(x)v(x)dx+C3].Notethattheu(b)v(b)termgetsreplacedbyu(x)v(x),andtheu(a)v(a)term"disappears"becauseitisconstant.Finally,wecan"absorb"thearbitraryconstantsintotheindefiniteintegralsi.e.,wedon'tneedtowriteC1,C2,C3,becauseanyindefiniteintegralisonlydetermineduptoaddingorsubtractingaconstantanyway.Thus,wearriveatthebrieferformulau(x)v(x)dx=u(x)v(x)u(x)v(x)dx.
Handlingconstantsofintegrationgetsevenmorecomplicatedinthefirstcourseondifferentialequations,andthereareevenmorekindsoferrorspossible.Iwon'ttrytolistallofthemhere,buthereisthesimplestandmostcommonerrorthatI'veseen:Incalculus,somestudentsgettheideathatyoucanjustomitthe"+C"inyourintermediatecomputations,andthentackitonattheendofyouranswer,ifyouknowwhichkindsofproblemsrequireanarbitraryconstant.Thatwillusuallyworkincalculus,butitdoesn'tworkindifferentialequations,becauseindifferentialequationsthe"C"canshowupanywherenotnecessarilyasa"+C"attheendoftheanswer.
Here'sasimpleexample:Let'ssolvethedifferentialequationxy+7=y(whereymeansdy/dx).Onewaytosolveitisbythefollowingsteps:
Rewritetheproblemasy(1/x)y=7/x,toshowthatitislinear.
Theintegratingfactoristhenexp[(1/x)dx]=1/x.Multiplybothsidesofthedifferentialequationbytheintegratingfactor,toobtainanexactdifferentialequation:(y/x)=(1/x)y(1/x2)y=7/x2.
Integratebothsides.Thusy/x=(7/x)+C.
Solvefory.Thusy=7+Cx.
That'sthecorrectanswer.Butifwehadtakentheattitude"don'tbotherwithC,justtackitonwhenyou'redone,"insteadofthelasttwostepswe'dhavewritten:
"Integratebothsides.Thusy/x=7/x."
"Solvefory.Thusy=7."
"Tackonthe"+C".Thusy=7+C."
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 29/31
That'swrong,whetheryousimplifyitornot.
Lossofdifferentials.Thisshowsupbothindifferentiationandinintegration.The"lossofdifferentials"ismuchlikethe"lossofinvisibleparentheses"discussedearlierinthisdocumentitisatypeofsloppywritinginintermediatestepswhichleadstoactualerrorsinthefinalanswer.
Whenstudentsfirstbegintolearntodifferentiate,theyarealwaysdifferentiatingwithrespecttothesamevariable,andsotheyseenoreasontomentionthatvariable.Thus,indifferentiatingthefunctiony=f(x)=7x3+5x,theymaycorrectlywrite
ortheymayincorrectlywrite"dy=21x2+5."Theomissionofthe"dx"fromthislastequationmakesnorealdifferenceinthestudent'smind,andthisslovenlyomissionmaybecomeahabit.Butitwillcausedifficultieslaterinthecourse.Infact,Iamstartingtothinkthatwecouldavoidalotofdifficultyifwediscouragebeginningcalculusstudentsfromusingthenotationsf(x)orDy.Ifwerequirethemtousethenotationdy/dx,andpenalizethemforwritingitasdy,wemightsavethemalotofheadacheslater.
Thedifficulty,ofcourse,showsupwhenwearriveattheChainRule.Suddenly,thequestionisnolonger"Whatisthederivativeofy",butrather,"Whatisthederivativeofywithrespecttox?withrespecttou?Howarethosetwoderivativesrelated?"Thestudentwhodoesnotmakeahabitofdistinguishingbetweendy/dxanddy/duinwritingmayalsohavedifficultydistinguishingbetweenthemconceptually,andthuswillhavedifficultyunderstandingtheChainRule.
Thisalsoleadstodifficultieswiththe"usubstitutions"rule,whichisjusttheChainRuleturnedintoaruleaboutintegrals.Forinstance:
Whatcausestheseerrors?
Forthefirstthreeproblems,thestudentisattemptingtousetheformula(1/u)du=ln|u|+C(whichisacorrectformula,butnotdirectlyapplicable).However,thestudenthaslearneditincorrectlyas"(1/u)=ln|u|+C."Substituteu=1+x2oru=x3oru=cosxintothatformulatogetthefirstthreeerroneousanswersinthetableabove.Theexpressions(1/u)duand(1/u)dxhaveverydifferent
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 30/31
meanings,butyou'relikelytoconfusethemifyouwritethembothas(1/u).
Forthelastprobleminthetableabove,thestudentisattemptingtousetheformulau2du=(1/3)u3+C,whichisacorrectformula,butnotrelevanttothepresentproblem.Thestudenthasprobabymemorizedthatformulaintheincorrectformu2=(1/3)u3+C.Theexpressionsu2duandu2dxhaveverydifferentmeanings,butyou'relikelytoconfusethemifyouwritethembothasu2.
Anothercorrectwaytowritetheruleaboutlogarithmsis .Sincethis
expresseseverythingintermsofthevariablex,itmaymakeerrorslesslikely.Admittedly,itisacomplicatedlookingformula,butitispreferabletoawrongformula.Thefirst,third,andfourthproblemsintheprecedingtableallrequiremorecomplicatedmethodsjustusinglogarithmswon'tsolvetheproblemsforyou.Theproblemofintegratingx3actuallyrequiresalesscomplicatedmethodi.e.,withoutlogarithms.
Weshouldprohibitstudentsfromwritinganintegralsignwithoutamatchingdifferential.Justasany"("mustbematchedwitha")",sotooanyintegralsignmustbematchedwitha"dx"or"du"or"dt"orwhatever.Theexpression isunbalanced,andshouldbeprohibited.Ifwe'reconsideringa
substitutionofu=1+x2,then(1/u)duisverydifferentfrom(1/u)dx,andsotheexpression(1/u)isambiguousandmeaningless.Ifyouwrite(1/u)inoneofyourintermediatesteps,youmayforgetwhetheritrepresents(1/u)duor(1/u)dx,andyoumayinadvertentlyswitchfromonetotheotherthusreplacingonemathematicalquantitywithanothertowhichitisnotequal.
Bytheway,somestudentsgetconfusedaboutwhether(1/u)dushouldbeln|u|+Corln(u)+C.Hereisananswer.(1/u)duisalwaysequaltoln|u|+C,butsometimesthatanswercanbesimplifiedandsometimesitcan't.Inmath,wegenerallyprefertowriteouranswersinsimplestform(andwesometimesinsistonit).Inthosesituationswhereweknowthatuwillonlytakepositivevalues(e.g.,whenu=1+x2,orwhenthedomainisrestrictedsothatucan'tbenegative),then(1/u)dushouldbewrittenasln(u)+C.Inthosesituationswherewedon'tknowwhetheruwillbepositive,weshouldwritetheanswerasln|u|+C.(Butsometimesweomittheabsolutevaluesignoutofsheerlaziness,justifyingthiswiththeexcusethatwecanmakethedomainsmaller.)
Theselossofdifferentialserrorsindifferentiationandinintegrationcanbecaughteasilybyabitof"dimensionalanalysis"(discussedearlier).Todothat,itisusefultothinkintermsof"infinitesimals"i.e.,numbersthatare"infinitelysmall"butstillnotzero.NewtonandLeibnizhadinfinitesimalsinmindwhentheyinventedcalculus300yearsago,buttheydidn'tknowhowtoexplaininfinitesimalsrigorously.Infinitesimalsbecameunfashionableacenturyortwolater,whenrigorousepsilondeltaproofswereinvented.Ifweusetherealnumbersystemthatmostmathematiciansusenowadays,therearenoinfinitesimalsexcept0.Butin1960alogiciannamedAbrahamRobinsoninventedanotherkindofrealnumbersystemthatincludesnonzeroinfinitesimalshefoundawaytobackuptheNewtonLeibnizintuitionwithrigorousproofs.
WiththeNewtonLeibnizRobinsonviewpoint,thinkofdxanddyasinfinitesimals.Now,dy/dxisaquotientoftwoinfinitelysmallnumbers,soitcouldbeamediumsizednumber.Thusanequationsuchasdy/dx=6x2couldmakesense.Anequationsuchasdy=6x2cannotpossiblybecorrecttheleftside
6/1/2015 CommonErrorsinCollegeMath
http://www.math.vanderbilt.edu/~schectex/commerrs/ 31/31
isinfinitelysmall,andtherightsideismediumsized.
Thesummationsignmeansaddtogetherfinitelyorcountablymanythingsforinstance,
butgenerallyisnotusedforaddinguncountablymanythings.
(Occasionallyitissoused:Thesumofanarbitrarycollectionofnonnegativerealnumbersisthesupofthesumsoffinitelymanymembersofthatcollection.Butalltheinterestingactionishappeningonacountableset.Itcanbeprovedthatifmorethancountablymanyofthosenumbersbeingaddedarenonzero,thesummustbeinfinity.Also,theremaybeother,moreesotericusesforthesymbol.Butthiswebpageisintendedforundergraduates.)
However,insomesenseweaddtogetheruncountablymanythingswhenweuseanintegral.Anequationsuchas3x2dx=x3+Csaysthatweaddtogetheruncountablymanyinfinitesimals,andwegetamediumsizednumber.Anequationsuchas3x2=x3+Ccouldn'tpossiblyberightitsaysweaddtogetheruncountablymanymediumsizednumbersandgetamediumsizednumber.
Arelateddifficultyisintryingtounderstandwhat"differentials"are.Mostrecentcalculusbookshaveafewpagesonthistopic,shortlybeforeoraftertheChainRule.Iamverysorrythattheauthorsofcalculusbookshavechosentocoverthistopicatthispointinthebook.Ithinktheyaremakingabigmistakeindoingso.WhenIteachcalculus,Iskipthatsection,withtheintentionofcoveringitinalatersemester.Hereiswhy:
Wheny=f(x),thendy=f(x)dxisreallyafunctionoftwovariablesitisafunctionofbothxanddx.Butinmanycalculustextbooks,thatfactisnotconfronteddirectlyitissweptundertherugandhidden.Severalhundredpageslaterinmostcalculustextbooks,weareintroducedtofunctionsoftwovariables,andgivenadecentnotationfortheme.g.,wemayhavez=h(u,v).Atthispointthestudentmaybegintounderstandfunctionsoftwovariables,andwehavepartialderivativesetc.Butbeforethispoint,wearenotgivenanygoodnotationsforafunctionoftwovariables.Ourbeginningmathstudentshavedifficultyenoughwithabstractionsevenwhentheyareprovidedwithdecentnotationhowcanweexpectthemtothinkabstractlywithoutthenotation?Thus,whenIteachcalculus,Idescribe"dx"and"dy"as"piecesofthenotationdy/dx,"withnoindependentmeaningsoftheirown.Ithinkthatthisapproachismuchkindertothebeginningstudents.
Thiswebpagewasselectedasthe"coolmathwebpageoftheweek",fortheweekofMay22,2002,byKaBoL.