Common Errors in College Math

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



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.



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



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,



(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:



[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:



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).



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



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:



$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



"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



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



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



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.



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.



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,



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|



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



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



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



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|



( p)~( >0)( >0)( q)(if|pq|0)( q)(if|pq|0)~( q)(if|pq|0)( q)~(if|pq|0)( q)(|pq|



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



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



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.



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



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



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."



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



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



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.

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Common Errors in College Math

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