Significant figures - Wikipedia

For example, if a length measurement gives 114.8 mm while the smallest interval between marks on the ruler used in the measurement is 1 mm, then the first three ... Significantfigures FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Anydigitofanumberwithinitsmeasurementresolution,asopposedtospuriousdigits ForthebookbyIanStewart,seeSignificantFigures(book). "Firstdigit"redirectshere.Forthebodypart,seeFirstdigit(anatomy). Thisarticleneedsadditionalcitationsforverification.Relevantdiscussionmaybefoundonthetalkpage.Pleasehelpimprovethisarticlebyaddingcitationstoreliablesources.Unsourcedmaterialmaybechallengedandremoved.Findsources: "Significantfigures" – news ·newspapers ·books ·scholar ·JSTOR(July2013)(Learnhowandwhentoremovethistemplatemessage) Fitapproximation Concepts OrdersofapproximationScaleanalysis ·BigOnotationCurvefitting ·FalseprecisionSignificantfigures Otherfundamentals Approximation ·GeneralizationerrorTaylorpolynomialScientificmodelling vte Significantfigures(alsoknownasthesignificantdigits,precisionorresolution)ofanumberinpositionalnotationaredigitsinthenumberthatarereliableandabsolutelynecessarytoindicatethequantityofsomething. Ifanumberexpressingtheresultofameasurement(e.g.,length,pressure,volume,ormass)hasmoredigitsthanthenumberofdigitsallowedbythemeasurementresolution,thenonlyasmanydigitsasallowedbythemeasurementresolutionarereliable,andsoonlythesecanbesignificantfigures. Forexample,ifalengthmeasurementgives114.8 mmwhilethesmallestintervalbetweenmarksontherulerusedinthemeasurementis1 mm,thenthefirstthreedigits(1,1,and4,showing114 mm)arecertainandsotheyaresignificantfigures.Digitswhichareuncertainbutreliablearealsoconsideredsignificantfigures.Inthisexample,thelastdigit(8,whichadds0.8 mm)isalsoconsideredasignificantfigureeventhoughthereisuncertaintyinit.[1] Anotherexampleisavolumemeasurementof2.98 Lwithanuncertaintyof± 0.05 L.Theactualvolumeissomewherebetween2.93 Land3.03 L.Evenwhensomeofthedigitsarenotcertain,aslongastheyarereliable,theyareconsideredsignificantbecausetheyindicatetheactualvolumewithintheacceptabledegreeofuncertainty.Inthisexampletheactualvolumemightbe2.94 Lormightinsteadbe3.02 L.Andsoallthreearesignificantfigures.[2] Thefollowingdigitsarenotsignificantfigures.[3] Allleadingzeros.Forexample,013 kghastwosignificantfigures,1and3,andtheleadingzeroisnotsignificantsinceitisnotnecessarytoindicatethemass;013 kg=13 kgso0isnotnecessary.Inthecaseof0.056 mtherearetwoinsignificantleadingzerossince0.056 m=56 mmandsotheleadingzerosarenotabsolutelynecessarytoindicatethelength. Trailingzeroswhentheyaremerelyplaceholders.Forexample,thetrailingzerosin1500 masalengthmeasurementarenotsignificantiftheyarejustplaceholdersforonesandtensplacesasthemeasurementresolutionis100 m.Inthiscase,1500 mmeansthelengthtomeasureiscloseto1500 mratherthansayingthatthelengthisexactly1500 m. Spuriousdigits,introducedbycalculationsresultinginanumberwithagreaterprecisionthantheprecisionoftheuseddatainthecalculations,orinameasurementreportedtoagreaterprecisionthanthemeasurementresolution. Ofthesignificantfiguresinanumber,themostsignificantisthedigitwiththehighestexponentvalue(simplytheleft-mostsignificantfigure),andtheleastsignificantisthedigitwiththelowestexponentvalue(simplytheright-mostsignificantfigure).Forexample,inthenumber"123",the"1"isthemostsignificantfigureasitcountshundreds(102),and"3"istheleastsignificantfigureasitcountsones(100). Significancearithmeticisasetofapproximaterulesforroughlymaintainingsignificancethroughoutacomputation.Themoresophisticatedscientificrulesareknownaspropagationofuncertainty. Numbersareoftenroundedtoavoidreportinginsignificantfigures.Forexample,itwouldcreatefalseprecisiontoexpressameasurementas12.34525 kgifthescalewasonlymeasuredtothenearestgram.Inthiscase,thesignificantfiguresarethefirst5digitsfromtheleft-mostdigit(1,2,3,4,and5),andthenumberneedstoberoundedtothesignificantfiguressothatitwillbe12.345 kgasthereliablevalue.Numberscanalsoberoundedmerelyforsimplicityratherthantoindicateaprecisionofmeasurement,forexample,inordertomakethenumbersfastertopronounceinnewsbroadcasts. Radix10(base-10,decimalnumbers)isassumedinthefollowing. Contents 1Identifyingsignificantfigures 1.1Rulestoidentifysignificantfiguresinanumber 1.2Waystodenotesignificantfiguresinanintegerwithtrailingzeros 2Roundingtosignificantfigures 3Writinguncertaintyandimplieduncertainty 3.1Significantfiguresinwritinguncertainty 3.2Implieduncertainty 4Arithmetic 4.1Multiplicationanddivision 4.1.1Exception 4.2Additionandsubtraction 4.3Logarithmandantilogarithm 4.4Transcendentalfunctions 4.5Roundonlyonthefinalcalculationresult 5Estimatinganextradigit 6Estimationinstatistic 7Relationshiptoaccuracyandprecisioninmeasurement 8Incomputing 9Seealso 10References 11Externallinks Identifyingsignificantfigures[edit] Thissectionneedsadditionalcitationsforverification.Pleasehelpimprovethisarticlebyaddingcitationstoreliablesources.Unsourcedmaterialmaybechallengedandremoved.(May2021)(Learnhowandwhentoremovethistemplatemessage) Rulestoidentifysignificantfiguresinanumber[edit] Digitsinlightbluearesignificantfigures;thoseinblackarenot. Notethatidentifyingthesignificantfiguresinanumberrequiresknowingwhichdigitsarereliable(e.g.,byknowingthemeasurementorreportingresolutionwithwhichthenumberisobtainedorprocessed)sinceonlyreliabledigitscanbesignificant;e.g.,3and4in0.00234 garenotsignificantifthemeasurablesmallestweightis0.001 g.[4] Non-zerodigitswithinthegivenmeasurementorreportingresolutionaresignificant. 91hastwosignificantfigures(9and1)iftheyaremeasurement-alloweddigits. 123.45hasfivesignificantdigits(1,2,3,4and5)iftheyarewithinthemeasurementresolution.Iftheresolutionis0.1,thenthelastdigit5isnotsignificant. Zerosbetweentwosignificantnon-zerodigitsaresignificant(significanttrappedzeros). 101.12003consistsofeightsignificantfiguresiftheresolutionisto0.00001. 125.340006hassevensignificantfiguresiftheresolutionisto0.0001:1,2,5,3,4,0,and0. Zerostotheleftofthefirstnon-zerodigit(leadingzeros)arenotsignificant. Ifalengthmeasurementgives0.052 km,then0.052 km=52 mso5and2areonlysignificant;theleadingzerosappearordisappear,dependingonwhichunitisused,sotheyarenotabsolutelynecessarytoindicatethemeasurementscale. 0.00034has4significantzerosiftheresolutionis0.001.(3and4arebeyondtheresolutionsoarenotsignificant.) Zerostotherightofthelastnon-zerodigit(trailingzeros)inanumberwiththedecimalpointaresignificantiftheyarewithinthemeasurementorreportingresolution. 1.200hasfoursignificantfigures(1,2,0,and0)iftheyareallowedbythemeasurementresolution. 0.0980hasthreesignificantdigits(9,8,andthelastzero)iftheyarewithinthemeasurementresolution. 120.000consistsofsignificantfiguresexceptforthelastzeroIftheresolutionisto0.01. Trailingzerosinanintegermayormaynotbesignificant,dependingonthemeasurementorreportingresolution. 45,600has3,4or5significantfiguresdependingonhowthelastzerosareused.Forexample,ifthelengthofaroadisreportedas45600 mwithoutinformationaboutthereportingormeasurementresolution,thenitisnotcleariftheroadlengthispreciselymeasuredas45600 morifitisaroughestimate.Ifitistheroughestimation,thenonlythefirstthreenon-zerodigitsaresignificantsincethetrailingzerosareneitherreliablenornecessary;45600 mcanbeexpressedas45.6 kmoras4.56×104 minscientificnotation,andneitherexpressionrequiresthetrailingzeros. Anexactnumberhasaninfinitenumberofsignificantfigures. Ifthenumberofapplesinabagis4(exactnumber),thenthisnumberis4.0000...(withinfinitetrailingzerostotherightofthedecimalpoint).Asaresult,4doesnotimpactthenumberofsignificantfiguresordigitsintheresultofcalculationswithit. Amathematicalorphysicalconstanthassignificantfigurestoitsknowndigits. π,astheratioofthecircumferencetothediameterofacircle,is3.14159265358979323...knownto50trilliondigits[5]calculatedasof2020-01-29,andthatcalculated'π'approximationhasthatmanysignificantdigits,whileinpracticalapplicationsfarfewerareused(andπitselfhasinfinitesignificantdigits,asallirrationalnumbersdo).Often3.14isusedinnumericalcalculations,i.e.3significantdecimaldigits,with7correctbinarydigits(whilethemoreaccurate22/7isalsoused,eventhoughitalsoonlyamountstothesame3significantcorrectdecimaldigits,ithas10correctbinarydigits),whichisagoodenoughapproximationformanypracticaluses.Mostcalculators,andcomputerprograms,canhandle3.141592653589793,16decimaldigits,thatiscommonlyusedincomputersandusedbyNASAfor"JPL'shighestaccuracycalculations,whichareforinterplanetarynavigation".[6]For"thelargestsizethereis:thevisibleuniverse[..]youwouldneed39or40decimalplaces."[6] ThePlanckconstantis h = 6.62607015 × 10 − 34 J ⋅ s {\displaystyleh=6.62607015\times10^{-34}J\cdots} andisdefinedasanexactvaluesothatitismoreproperlydefinedas h = 6.62607015 ( 0 ) × 10 − 34 J ⋅ s {\displaystyleh=6.62607015(0)\times10^{-34}J\cdots} .[7] Waystodenotesignificantfiguresinanintegerwithtrailingzeros[edit] Thesignificanceoftrailingzerosinanumbernotcontainingadecimalpointcanbeambiguous.Forexample,itmaynotalwaysbeclearifthenumber1300isprecisetothenearestunit(justhappenscoincidentallytobeanexactmultipleofahundred)orifitisonlyshowntothenearesthundredsduetoroundingoruncertainty.Manyconventionsexisttoaddressthisissue.However,thesearenotuniversallyusedandwouldonlybeeffectiveifthereaderisfamiliarwiththeconvention: Anoverline,sometimesalsocalledanoverbar,orlessaccurately,avinculum,maybeplacedoverthelastsignificantfigure;anytrailingzerosfollowingthisareinsignificant.Forexample,1300hasthreesignificantfigures(andhenceindicatesthatthenumberisprecisetothenearestten). Lessoften,usingacloselyrelatedconvention,thelastsignificantfigureofanumbermaybeunderlined;forexample,"1300"hastwosignificantfigures. Adecimalpointmaybeplacedafterthenumber;forexample"1300."indicatesspecificallythattrailingzerosaremeanttobesignificant.[8] Astheconventionsabovearenotingeneraluse,thefollowingmorewidelyrecognizedoptionsareavailableforindicatingthesignificanceofnumberwithtrailingzeros: Eliminateambiguousornon-significantzerosbychangingtheunitprefixinanumberwithaunitofmeasurement.Forexample,theprecisionofmeasurementspecifiedas1300 gisambiguous,whileifstatedas1.30 kgitisnot.Likewise0.0123 Lcanberewrittenas12.3 mL Eliminateambiguousornon-significantzerosbyusingScientificNotation:Forexample,1300withthreesignificantfiguresbecomes1.30×103.Likewise0.0123canberewrittenas1.23×10−2.Thepartoftherepresentationthatcontainsthesignificantfigures(1.30or1.23)isknownasthesignificandormantissa.Thedigitsinthebaseandexponent(103or10−2)areconsideredexactnumberssoforthesedigits,significantfiguresareirrelevant. Explicitlystatethenumberofsignificantfigures(theabbreviations.f.issometimesused):Forexample"20 000to2 s.f."or"20 000(2 sf)". Statetheexpectedvariability(precision)explicitlywithaplus–minussign,asin20 000 ± 1%.Thisalsoallowsspecifyingarangeofprecisionin-betweenpowersoften. Roundingtosignificantfigures[edit] Roundingtosignificantfiguresisamoregeneral-purposetechniquethanroundingtondigits,sinceithandlesnumbersofdifferentscalesinauniformway.Forexample,thepopulationofacitymightonlybeknowntothenearestthousandandbestatedas52,000,whilethepopulationofacountrymightonlybeknowntothenearestmillionandbestatedas52,000,000.Theformermightbeinerrorbyhundreds,andthelattermightbeinerrorbyhundredsofthousands,butbothhavetwosignificantfigures(5and2).Thisreflectsthefactthatthesignificanceoftheerroristhesameinbothcases,relativetothesizeofthequantitybeingmeasured. Toroundanumbertonsignificantfigures:[9][10] Ifthen+1digitisgreaterthan5oris5followedbyothernon-zerodigits,add1tothendigit.Forexample,ifwewanttoround1.2459to3significantfigures,thenthisstepresultsin1.25. Ifthen+1digitis5notfollowedbyotherdigitsorfollowedbyonlyzeros,thenroundingrequiresatie-breakingrule.Forexample,toround1.25to2significantfigures: Roundhalfawayfromzero(alsoknownas"5/4")[citationneeded]roundsupto1.3.Thisisthedefaultroundingmethodimpliedinmanydisciplines[citationneeded]iftherequiredroundingmethodisnotspecified. Roundhalftoeven,whichroundstothenearestevennumber.Withthismethod,1.25isroundeddownto1.2.Ifthismethodappliesto1.35,thenitisroundedupto1.4.Thisisthemethodpreferredbymanyscientificdisciplines,because,forexample,itavoidsskewingtheaveragevalueofalonglistofvaluesupwards. Foranintegerinrounding,replacethedigitsafterthendigitwithzeros.Forexample,if1254isroundedto2significantfigures,then5and4arereplacedto0sothatitwillbe1300.Foranumberwiththedecimalpointinrounding,removethedigitsafterthendigit.Forexample,if14.895isroundedto3significantfigures,thenthedigitsafter8areremovedsothatitwillbe14.9. Infinancialcalculations,anumberisoftenroundedtoagivennumberofplaces.Forexample,totwoplacesafterthedecimalseparatorformanyworldcurrencies.Thisisdonebecausegreaterprecisionisimmaterial,andusuallyitisnotpossibletosettleadebtoflessthanthesmallestcurrencyunit. InUKpersonaltaxreturns,incomeisroundeddowntothenearestpound,whilsttaxpaidiscalculatedtothenearestpenny. Asanillustration,thedecimalquantity12.345canbeexpressedwithvariousnumbersofsignificantfiguresordecimalplaces.Ifinsufficientprecisionisavailablethenthenumberisroundedinsomemannertofittheavailableprecision.Thefollowingtableshowstheresultsforvarioustotalprecisionattworoundingways(N/AstandsforNotApplicable). Precision Roundedtosignificantfigures Roundedtodecimalplaces 6 12.3450 12.345000 5 12.345 12.34500 4 12.34or12.35 12.3450 3 12.3 12.345 2 12 12.34or12.35 1 10 12.3 0 N/A 12 Anotherexamplefor0.012345.(Rememberthattheleadingzerosarenotsignificant.) Precision Roundedtosignificantfigures Roundedtodecimalplaces 7 0.01234500 0.0123450 6 0.0123450 0.012345 5 0.012345 0.01234or0.01235 4 0.01234or0.01235 0.0123 3 0.0123 0.012 2 0.012 0.01 1 0.01 0.0 0 N/A 0 Therepresentationofanon-zeronumberxtoaprecisionofpsignificantdigitshasanumericalvaluethatisgivenbytheformula:[citationneeded] 10 n ⋅ round ⁡ ( x 10 n ) {\displaystyle10^{n}\cdot\operatorname{round}\left({\frac{x}{10^{n}}}\right)} where n = ⌊ log 10 ⁡ ( | x | ) ⌋ + 1 − p {\displaystylen=\lfloor\log_{10}(|x|)\rfloor+1-p} whichmayneedtobewrittenwithaspecificmarkingasdetailedabovetospecifythenumberofsignificanttrailingzeros. Writinguncertaintyandimplieduncertainty[edit] Significantfiguresinwritinguncertainty[edit] Itisrecommendedforameasurementresulttoincludethemeasurementuncertaintysuchas x b e s t ± σ x {\displaystylex_{best}\pm\sigma_{x}} ,wherexbestandσxarethebestestimateanduncertaintyinthemeasurementrespectively.[11]xbestcanbetheaverageofmeasuredvaluesandσxcanbethestandarddeviationoramultipleofthemeasurementdeviation.Therulestowrite x b e s t ± σ x {\displaystylex_{best}\pm\sigma_{x}} are:[12] σxhasonlyoneortwosignificantfiguresasmorepreciseuncertaintyhasnomeaning. 1.79±0.06(correct),1.79 ±0.96(correct),1.79 ±1.96(incorrect). Thedigitpositionsofthelastsignificantfiguresinxbestandσxarethesame,otherwisetheconsistencyislost.Forexample,in1.79 ±0.067(incorrect),itdoesnotmakesensetohavemoreaccurateuncertaintythanthebestestimate.1.79 ±0.9(incorrect)alsodoesnotmakesensesincetheroundingguidelineforadditionandsubtractionbelowtellsthattheedgesofthetruevaluerangeare2.7and0.9,thatarelessaccuratethanthebestestimate. 1.79±0.06(correct),1.79 ±0.96(correct),1.79 ±0.067(incorrect),1.79 ±0.9(incorrect). Implieduncertainty[edit] Inchemistry(andmayalsobeforotherscientificbranches),uncertaintymaybeimpliedbythelastsignificantfigureifitisnotexplicitlyexpressed.[2]Theimplieduncertaintyis±thehalfoftheminimumscaleatthelastsignificantfigureposition.Forexample,ifthevolumeofwaterinabottleisreportedas3.78 Lwithoutmentioninguncertainty,then± 0.005 Lmeasurementuncertaintymaybeimplied.If2.97±0.07 kg,sotheactualweightissomewherein2.90to3.04 kg,ismeasuredanditisdesiredtoreportitwithasinglenumber,then3.0 kgisthebestnumbertoreportsinceitsimplieduncertainty± 0.05 kgtellstheweightrangeof2.95to3.05 kgthatisclosetothemeasurementrange.If2.97 ± 0.09 kg,then3.0 kgisstillthebestsince,if3 kgisreportedthenitsimplieduncertainty± 0.5tellstherangeof2.5to3.5 kgthatistoowideincomparisonwiththemeasurementrange. Ifthereisaneedtowritetheimplieduncertaintyofanumber,thenitcanbewrittenas x ± σ x {\displaystylex\pm\sigma_{x}} withstatingitastheimplieduncertainty(topreventreadersfromrecognizingitasthemeasurementuncertainty),wherexandσxarethenumberwithanextrazerodigit(tofollowtherulestowriteuncertaintyabove)andtheimplieduncertaintyofitrespectively.Forexample,6 kgwiththeimplieduncertainty± 0.5 kgcanbestatedas6.0 ± 0.5 kg. Arithmetic[edit] Mainarticle:Significancearithmetic Astherearerulestodeterminethesignificantfiguresindirectlymeasuredquantities,therearealsoguidelines(notrules)todeterminethesignificantfiguresinquantitiescalculatedfromthesemeasuredquantities. Significantfiguresinmeasuredquantitiesaremostimportantinthedeterminationofsignificantfiguresincalculatedquantitieswiththem.Amathematicalorphysicalconstant(e.g.,πintheformulafortheareaofacirclewithradiusrasπr2)hasnoeffectonthedeterminationofthesignificantfiguresintheresultofacalculationwithitifitsknowndigitsareequaltoormorethanthesignificantfiguresinthemeasuredquantitiesusedinthecalculation.Anexactnumbersuchas½intheformulaforthekineticenergyofamassmwithvelocityvas½mv2hasnobearingonthesignificantfiguresinthecalculatedkineticenergysinceitsnumberofsignificantfiguresisinfinite(0.500000...). Theguidelinesdescribedbelowareintendedtoavoidacalculationresultmoreprecisethanthemeasuredquantities,butitdoesnotensuretheresultedimplieduncertaintycloseenoughtothemeasureduncertainties.Thisproblemcanbeseeninunitconversion.Iftheguidelinesgivetheimplieduncertaintytoofarfromthemeasuredones,thenitmaybeneededtodecidesignificantdigitsthatgivecomparableuncertainty. Multiplicationanddivision[edit] Forquantitiescreatedfrommeasuredquantitiesviamultiplicationanddivision,thecalculatedresultshouldhaveasmanysignificantfiguresastheleastnumberofsignificantfiguresamongthemeasuredquantitiesusedinthecalculation.[13]Forexample, 1.234×2=2.468≈2 1.234×2.0=2.468≈2.5 0.01234×2=0.02468≈0.02 withone,two,andonesignificantfiguresrespectively.(2hereisassumednotanexactnumber.)Forthefirstexample,thefirstmultiplicationfactorhasfoursignificantfiguresandthesecondhasonesignificantfigure.Thefactorwiththefewestorleastsignificantfiguresisthesecondonewithonlyone,sothefinalcalculatedresultshouldalsohaveonesignificantfigure. Exception[edit] Forunitconversion,theimplieduncertaintyoftheresultcanbeunsatisfactorilyhigherthanthatinthepreviousunitifthisroundingguidelineisfollowed;Forexample,8inchhastheimplieduncertaintyof±0.5 inch=± 1.27 cm.Ifitisconvertedtothecentimetrescaleandtheroundingguidelineformultiplicationanddivisionisfollowed,then20.32 cm≈20 cmwiththeimplieduncertaintyof± 5 cm.Ifthisimplieduncertaintyisconsideredastoounderestimated,thenmorepropersignificantdigitsintheunitconversionresultmaybe20.32 cm≈20. cmwiththeimplieduncertaintyof± 0.5 cm. Anotherexceptionofapplyingtheaboveroundingguidelineistomultiplyanumberbyaninteger,suchas1.234×9.Iftheaboveguidelineisfollowed,thentheresultisroundedas1.234×9.000....=11.106≈11.11.However,thismultiplicationisessentiallyadding1.234toitself9timessuchas1.234+1.234+...+1.234sotheroundingguidelineforadditionandsubtractiondescribedbelowismoreproperroundingapproach.[14]Asaresult,thefinalansweris1.234+1.234+...+1.234=11.106=11.106(onesignificantdigitincrease). Additionandsubtraction[edit] Forquantitiescreatedfrommeasuredquantitiesviaadditionandsubtraction,thelastsignificantfigureposition(e.g.,hundreds,tens,ones,tenths,hundredths,andsoforth)inthecalculatedresultshouldbethesameastheleftmostorlargestdigitpositionamongthelastsignificantfiguresofthemeasuredquantitiesinthecalculation.Forexample, 1.234+2=3.234≈3 1.234+2.0=3.234≈3.2 0.01234+2=2.01234≈2 withthelastsignificantfiguresintheonesplace,tenthsplace,andonesplacerespectively.(2hereisassumednotanexactnumber.)Forthefirstexample,thefirsttermhasitslastsignificantfigureinthethousandthsplaceandthesecondtermhasitslastsignificantfigureintheonesplace.Theleftmostorlargestdigitpositionamongthelastsignificantfiguresofthesetermsistheonesplace,sothecalculatedresultshouldalsohaveitslastsignificantfigureintheonesplace. Theruletocalculatesignificantfiguresformultiplicationanddivisionarenotthesameastheruleforadditionandsubtraction.Formultiplicationanddivision,onlythetotalnumberofsignificantfiguresineachofthefactorsinthecalculationmatters;thedigitpositionofthelastsignificantfigureineachfactorisirrelevant.Foradditionandsubtraction,onlythedigitpositionofthelastsignificantfigureineachofthetermsinthecalculationmatters;thetotalnumberofsignificantfiguresineachtermisirrelevant.[citationneeded]However,greateraccuracywilloftenbeobtainedifsomenon-significantdigitsaremaintainedinintermediateresultswhichareusedinsubsequentcalculations.[citationneeded] Logarithmandantilogarithm[edit] Thebase-10logarithmofanormalizednumber(i.e.,a×10bwith1≤a<10andbasaninteger),isroundedsuchthatitsdecimalpart(calledmantissa)hasasmanysignificantfiguresasthesignificantfiguresinthenormalizednumber. log10(3.000×104)=log10(104)+log10(3.000)=4.000000...(exactnumbersoinfinitesignificantdigits)+0.4771212547...=4.4771212547≈4.4771. Whentakingtheantilogarithmofanormalizednumber,theresultisroundedtohaveasmanysignificantfiguresasthesignificantfiguresinthedecimalpartofthenumbertobeantiloged. 104.4771=29998.5318119...=30000=3.000×104. Transcendentalfunctions[edit] Ifatranscendentalfunction f ( x ) {\displaystylef(x)} (e.g.,theexponentialfunction,thelogarithm,andthetrigonometricfunctions)isdifferentiableatitsdomainelementx,thenitsnumberofsignificantfigures(denotedas"significantfiguresof f ( x ) {\displaystylef(x)} ")isapproximatelyrelatedwiththenumberofsignificantfiguresinx(denotedas"significantfiguresofx")bytheformula ( s i g n i f i c a n t   f i g u r e s   o f   f ( x ) ) ≈ ( s i g n i f i c a n t   f i g u r e s   o f   x ) − log 10 ⁡ ( | d f ( x ) d x x f ( x ) | ) {\displaystyle{\rm{(significant~figures~of~f(x))}}\approx{\rm{(significant~figures~of~x)}}-\log_{10}\left(\left\vert{{\frac{df(x)}{dx}}{\frac{x}{f(x)}}}\right\vert\right)} , where | d f ( x ) d x x f ( x ) | {\displaystyle\left\vert{{\frac{df(x)}{dx}}{\frac{x}{f(x)}}}\right\vert} istheconditionnumber.Seethesignificancearithmeticarticletofinditsderivation. Roundonlyonthefinalcalculationresult[edit] Whenperformingmultiplestagecalculations,donotroundintermediatestagecalculationresults;keepasmanydigitsasispractical(atleastonemoredigitthantheroundingruleallowsperstage)untiltheendofallthecalculationstoavoidcumulativeroundingerrorswhiletrackingorrecordingthesignificantfiguresineachintermediateresult.Then,roundthefinalresult,forexample,tothefewestnumberofsignificantfigures(formultiplicationordivision)orleftmostlastsignificantdigitposition(foradditionorsubtraction)amongtheinputsinthefinalcalculation.[15] (2.3494+1.345)×1.2=3.6944×1.2=4.43328≈4.4. (2.3494×1.345)+1.2=3.159943+1.2=4.359943≈4.4. Estimatinganextradigit[edit] Whenusingaruler,initiallyusethesmallestmarkasthefirstestimateddigit.Forexample,ifaruler'ssmallestmarkis0.1 cm,and4.5 cmisread,thenitis4.5(±0.1 cm)or4.4 cmto4.6 cmastothesmallestmarkinterval.However,inpracticeameasurementcanusuallybeestimatedbyeyetocloserthantheintervalbetweentheruler'ssmallestmark,e.g.intheabovecaseitmightbeestimatedasbetween4.51 cmand4.53 cm. Itisalsopossiblethattheoveralllengthofarulermaynotbeaccuratetothedegreeofthesmallestmark,andthemarksmaybeimperfectlyspacedwithineachunit.Howeverassuminganormalgoodqualityruler,itshouldbepossibletoestimatetenthsbetweenthenearesttwomarkstoachieveanextradecimalplaceofaccuracy.[16]Failingtodothisaddstheerrorinreadingtherulertoanyerrorinthecalibrationoftheruler.[17] Estimationinstatistic[edit] Mainarticle:Estimation Whenestimatingtheproportionofindividualscarryingsomeparticularcharacteristicinapopulation,fromarandomsampleofthatpopulation,thenumberofsignificantfiguresshouldnotexceedthemaximumprecisionallowedbythatsamplesize. Relationshiptoaccuracyandprecisioninmeasurement[edit] Mainarticle:Accuracyandprecision Traditionally,invarioustechnicalfields,"accuracy"referstotheclosenessofagivenmeasurementtoitstruevalue;"precision"referstothestabilityofthatmeasurementwhenrepeatedmanytimes.Hopingtoreflectthewayinwhichtheterm"accuracy"isactuallyusedinthescientificcommunity,thereisarecentstandard,ISO5725,whichkeepsthesamedefinitionofprecisionbutdefinestheterm"trueness"astheclosenessofagivenmeasurementtoitstruevalueandusestheterm"accuracy"asthecombinationoftruenessandprecision.(Seetheaccuracyandprecisionarticleforafulldiscussion.)Ineithercase,thenumberofsignificantfiguresroughlycorrespondstoprecision,nottoaccuracyorthenewerconceptoftrueness. Incomputing[edit] Mainarticle:Floating-pointarithmetic Computerrepresentationsoffloating-pointnumbersuseaformofroundingtosignificantfigures(whileusuallynotkeepingtrackofhowmany),ingeneralwithbinarynumbers.Thenumberofcorrectsignificantfiguresiscloselyrelatedtothenotionofrelativeerror(whichhastheadvantageofbeingamoreaccuratemeasureofprecision,andisindependentoftheradix,alsoknownasthebase,ofthenumbersystemused). Seealso[edit] Benford'slaw(first-digitlaw) Engineeringnotation Errorbar Falseprecision IEEE754(IEEEfloating-pointstandard) Intervalarithmetic Kahansummationalgorithm Precision(computerscience) Round-offerror References[edit] ^"SignificantFigures-WritingNumberstoReflectPrecision".Chemistry-Libretexts.2019-09-04. ^abLower,Stephen(2021-03-31)."SignificantFiguresandRounding".Chemistry-LibreTexts. ^ChemistryintheCommunity;Kendall-Hunt:Dubuque,IA1988 ^Givingaprecisedefinitionforthenumberofcorrectsignificantdigitsissurprisinglysubtle,seeHigham,Nicholas(2002).AccuracyandStabilityofNumericalAlgorithms(PDF)(2nd ed.).SIAM.pp. 3–5. ^Mostaccuratevalueofpi ^ab"HowManyDecimalsofPiDoWeReallyNeed?-EduNews".NASA/JPLEdu.Retrieved2021-10-25. ^"Resolutionsofthe26thCGPM"(PDF).BIPM.2018-11-16.Archivedfromtheoriginal(PDF)on2018-11-19.Retrieved2018-11-20. ^Myers,R.Thomas;Oldham,KeithB.;Tocci,Salvatore(2000).Chemistry.Austin,Texas:HoltRinehartWinston.p. 59.ISBN 0-03-052002-9. ^Engelbrecht,Nancy;et al.(1990)."RoundingDecimalNumberstoaDesignatedPrecision"(PDF).Washington,D.C.:U.S.DepartmentofEducation. ^NumericalMathematicsandComputing,byCheneyandKincaid. ^Luna,Eduardo."UncertaintiesandSignificantFigures"(PDF).DeAnzaCollege. ^"SignificantFigures".PurdueUniversity-DepartmentofPhysicsandAstronomy. ^"SignificantFigureRules".PennStateUniversity. ^"UncertaintyinMeasurement-SignificantFigures".Chemistry-LibreTexts.2017-06-16. ^deOliveiraSannibale,Virgínio(2001)."MeasurementsandSignificantFigures(Draft)"(PDF).FreshmanPhysicsLaboratory.CaliforniaInstituteofTechnology,PhysicsMathematicsAndAstronomyDivision.Archivedfromtheoriginal(PDF)on2013-06-18. ^ExperimentalElectricalTesting.Newark,NJ:WestonElectricalInstrumentsCo.1914.p. 9.Retrieved2019-01-14.ExperimentalElectricalTesting.. ^"Measurements".slc.umd.umich.edu.UniversityofMichigan.Retrieved2017-07-03. 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