2.4: Significant Figures in Calculations - Chemistry LibreTexts

For example, the number 450 has two significant figures and would be written in scientific notation as 4.5 × 102, whereas 450.0 has four ... Skiptomaincontent RoundingMultiplicationandDivisionAdditionandSubtractionCalculationsInvolvingMultiplication/DivisionandAddition/SubtractionSummaryContributions&Attributions LearningObjectives Usesignificantfigurescorrectlyinarithmeticaloperations. Rounding Beforedealingwiththespecificsoftherulesfordeterminingthesignificantfiguresinacalculatedresult,weneedtobeabletoroundnumberscorrectly.Toroundanumber,firstdecidehowmanysignificantfiguresthenumbershouldhave.Onceyouknowthat,roundtothatmanydigits,startingfromtheleft.Ifthenumberimmediatelytotherightofthelastsignificantdigitislessthan5,itisdroppedandthevalueofthelastsignificantdigitremainsthesame.Ifthenumberimmediatelytotherightofthelastsignificantdigitisgreaterthanorequalto5,thelastsignificantdigitisincreasedby1. Considerthemeasurement\(207.518\:\text{m}\).Rightnow,themeasurementcontainssixsignificantfigures.Howwouldwesuccessivelyroundittofewerandfewersignificantfigures?FollowtheprocessasoutlinedinTable\(\PageIndex{1}\). NumberofSignificantFigures RoundedValue Reasoning Table\(\PageIndex{1}\):Roundingexamples 6 207.518 Alldigitsaresignificant 5 207.52 8roundsthe1upto2 4 207.5 2isdropped 3 208 5roundsthe7upto8 2 210 8isreplacedbya0androundsthe0upto1 1 200 1isreplacedbya0 Noticethatthemoreroundingthatisdone,thelessreliablethefigureis.Anapproximatevaluemaybesufficientforsomepurposes,butscientificworkrequiresamuchhigherlevelofdetail. Itisimportanttobeawareofsignificantfigureswhenyouaremathematicallymanipulatingnumbers.Forexample,dividing125by307onacalculatorgives0.4071661238…toaninfinitenumberofdigits.Butdothedigitsinthisanswerhaveanypracticalmeaning,especiallywhenyouarestartingwithnumbersthathaveonlythreesignificantfigureseach?Whenperformingmathematicaloperations,therearetworulesforlimitingthenumberofsignificantfiguresinananswer—oneruleisforadditionandsubtraction,andoneruleisformultiplicationanddivision. Inoperationsinvolvingsignificantfigures,theanswerisreportedinsuchawaythatitreflectsthereliabilityoftheleastpreciseoperation.Ananswerisnomoreprecisethantheleastprecisenumberusedtogettheanswer. MultiplicationandDivision Formultiplicationordivision,theruleistocountthenumberofsignificantfiguresineachnumberbeingmultipliedordividedandthenlimitthesignificantfiguresintheanswertothelowestcount.Anexampleisasfollows: Thefinalanswer,limitedtofoursignificantfigures,is4,094.Thefirstdigitdroppedis1,sowedonotroundup. Scientificnotationprovidesawayofcommunicatingsignificantfigureswithoutambiguity.Yousimplyincludeallthesignificantfiguresintheleadingnumber.Forexample,thenumber450hastwosignificantfiguresandwouldbewritteninscientificnotationas4.5×102,whereas450.0hasfoursignificantfiguresandwouldbewrittenas4.500×102.Inscientificnotation,allsignificantfiguresarelistedexplicitly. Example\(\PageIndex{1}\) Writetheanswerforeachexpressionusingscientificnotationwiththeappropriatenumberofsignificantfigures. 23.096×90.300 125×9.000 Solution a Explanation Answer Thecalculatoransweris2,085.5688,butweneedtoroundittofivesignificantfigures.Becausethefirstdigittobedropped(inthetenthsplace)isgreaterthan5,weroundupto2,085.6. \(2.0856\times10^3\) b Explanation Answer Thecalculatorgives1,125astheanswer,butwelimitittothreesignificantfigures. \(1.13\times10^3\) AdditionandSubtraction Howaresignificantfigureshandledincalculations?Itdependsonwhattypeofcalculationisbeingperformed.Ifthecalculationisanadditionorasubtraction,theruleisasfollows:limitthereportedanswertotherightmostcolumnthatallnumbershavesignificantfiguresincommon.Forexample,ifyouweretoadd1.2and4.71,wenotethatthefirstnumberstopsitssignificantfiguresinthetenthscolumn,whilethesecondnumberstopsitssignificantfiguresinthehundredthscolumn.Wethereforelimitouranswertothetenthscolumn. Wedropthelastdigit—the1—becauseitisnotsignificanttothefinalanswer. Thedroppingofpositionsinsumsanddifferencesbringsupthetopicofrounding.Althoughthereareseveralconventions,inthistextwewilladoptthefollowingrule:thefinalanswershouldberoundedupifthefirstdroppeddigitis5orgreater,androundeddownifthefirstdroppeddigitislessthan5. Example\(\PageIndex{2}\) 13.77+908.226 1,027+611+363.06 Solution a Explanation Answer Thecalculatoransweris921.996,butbecause13.77hasitsfarthest-rightsignificantfigureinthehundredthsplace,weneedtoroundthefinalanswertothehundredthsposition.Becausethefirstdigittobedropped(inthethousandthsplace)isgreaterthan5,weroundupto922.00 \(922.00=9.2200\times10^2\) b Explanation Answer Thecalculatorgives2,001.06astheanswer,butbecause611and1027hasitsfarthest-rightsignificantfigureintheonesplace,thefinalanswermustbelimitedtotheonesposition. \(2,001.06=2.001\times10^3\) Exercise\(\PageIndex{2}\) Writetheanswerforeachexpressionusingscientificnotationwiththeappropriatenumberofsignificantfigures. 217÷903 13.77+908.226+515 255.0−99 0.00666×321 Answera: \(0.240=2.40\times10^{-1}\) Answerb: \(1,437=1.437\times10^3\) Answerc: \(156=1.56\times10^2\) Answerd: \(2.14=2.14\times10^0\) Rememberthatcalculatorsdonotunderstandsignificantfigures.Youaretheonewhomustapplytherulesofsignificantfigurestoaresultfromyourcalculator. CalculationsInvolvingMultiplication/DivisionandAddition/Subtraction Inpractice,chemistsgenerallyworkwithacalculatorandcarryalldigitsforwardthroughsubsequentcalculations.Whenworkingonpaper,however,weoftenwanttominimizethenumberofdigitswehavetowriteout.Becausesuccessiveroundingcancompoundinaccuracies,intermediaterounding needstobehandledcorrectly.Whenworkingonpaper,alwaysroundanintermediateresultsoastoretainatleastonemoredigitthancanbejustifiedandcarrythisnumberintothenextstepinthecalculation.Thefinalansweristhenroundedtothecorrectnumberofsignificantfiguresattheveryend. Video\(\PageIndex{1}\):Significantfiguresinmixedoperations(https://www.youtube.com/watch?v=yBntMndXQWA). Video\(\PageIndex{2}\):https://www.youtube.com/watch?v=__csP0NtlGI Intheworkedexamplesinthistext,wewilloftenshowtheresultsofintermediatestepsinacalculation.Indoingso,wewillshowtheresultstoonlythecorrectnumberofsignificantfiguresallowedforthatstep,ineffecttreatingeachstepasaseparatecalculation.Thisprocedureisintendedtoreinforcetherulesfordeterminingthenumberofsignificantfigures,butinsomecasesitmaygiveafinalanswerthatdiffersinthelastdigitfromthatobtainedusingacalculator,wherealldigitsarecarriedthroughtothelaststep. Example\(\PageIndex{3}\) 2(1.008g)+15.99g 137.3s+2(35.45s) \({118.7g\over2}-35.5g\) Solution a. Explanation Answer 2(1.008g)+15.99g= Performmultiplicationfirst. 2(1.008g4sigfigs)=2.016g4sigfigs Thenumberwiththeleastnumberofsignificantfiguresis1.008g;thenumber2isanexactnumberandthereforehasaninfinitenumberofsignificantfigures. Then,performtheaddition. 2.016gthousandthsplace+15.99ghundredthsplace(leastprecise)=18.006g Roundthefinalanswer. Roundthefinalanswertothehundredthsplacesince15.99hasitsfarthestrightsignificantfigureinthehundredthsplace(leastprecise). 18.01g(roundingup) b. Explanation Answer 137.3s+2(35.45s)= Performmultiplicationfirst. 2(35.45s4sigfigs)=70.90s4sigfigs Thenumberwiththeleastnumberofsignificantfiguresis35.45;thenumber2isanexactnumberandthereforehasaninfinitenumberofsignificantfigures. Then,performtheaddition. 137.3stenthsplace(leastprecise)+70.90shundredthsplace=208.20s Roundthefinalanswer. Roundthefinalanswertothetenthsplacebasedon137.3s. 208.2s c. Explanation Answer \({118.7g\over2}-35.5g\)= Performdivisionfirst. \({118.7g\over2}\)4sigfigs=59.35g4sigfigs Thenumberwiththeleastnumberofsignificantfiguresis118.7g;thenumber2isanexactnumberandthereforehasaninfinitenumberofsignificantfigures. Performsubtractionnext. 59.35ghundredthsplace−35.5gtenthsplace(leastprecise)=23.85g Roundthefinalanswer. Roundthefinalanswertothetenthsplacebasedon35.5g. 23.9g(roundingup) Exercise\(\PageIndex{3}\) Completethecalculationsandreportyouranswersusingthecorrectnumberofsignificantfigures. 5(1.008s)-10.66s 99.0cm+2(5.56cm) Answera -5.62s Answerb 110.2cm Summary Rounding Ifthenumbertobedroppedisgreaterthanorequalto5,increasethenumbertoitsleftby1(e.g.2.9699roundedtothreesignificantfiguresis2.97). Ifthenumbertobedroppedislessthan5,thereisnochange (e.g.4.00443roundedtofoursignificantfiguresis4.004). Theruleinmultiplicationanddivisionisthatthefinalanswershouldhavethesamenumberofsignificantfiguresasthereareinthenumberwiththefewestsignificantfigures. Theruleinadditionandsubtractionisthattheanswerisgiventhesamenumberofdecimalplacesasthetermwiththefewestdecimalplaces. Contributions&Attributions Thispagewasconstructedfromcontentviathefollowingcontributor(s) andedited(topicallyorextensively)bytheLibreTextsdevelopmentteamtomeetplatformstyle,presentation,andquality: CK-12FoundationbySharonBewick,RichardParsons,ThereseForsythe, ShonnaRobinson,andJeanDupon. Marisa Alviar-Agnew (SacramentoCityCollege) HenryAgnew (UCDavis)