arc considered optional. They can be o

Srarong /h~ uIKmimyAssumed ""ur/(JilllyWhen giving the result of il meilsu r ~menl, it is im port'lnl to st,'le Ihe u ti.mated uncertainty in the mea>uremenL For e~il mplc, the widlh of;l bo;lrd mighlbe written ;IS 8.8 ± O. t em. The ± 0,1 em ("plus or minus 0.1 em") repro.urement, so thilt the ildu;ll widlh most likelylies belween 8.7 and 8.9em, The percent uncertainly is simply the r;Jrio of theunecrtilinty to the measured v;Jlue, multiplied by tOO. For e~ilmple, if the me,lSurementis 8.8 ,lnd the unccrtairl\y aboul 0.1 cm, the pe re~rl\ uncert;Jinty is0.1- X 100% "" 1%8.'where '" means ~is roughly equal to."Of len the uncertainty in a measured val ue is nOI spi'cificd explicitly. In suchcases, Ihe uncertainty is generally assumed to be one or a few unils in Ihe laSIdigit specified. For example. if a length is given as 8.8cm. the uncertainty isassumed 10 be about 0.1 em or O.2cm.lt is importanl in this case that you do nOIwrile 8.SOem, for this implies an uncertainly on Ihe order of 0.01 em; il assumesIhal the lenglh is probably between 8.7gem and 8.81 em. when actually yo ubelieve it is between 8.7 and 8.9cm.CONCEPTUAL EXAMPLE 1-1 Is the diamond yours? A friend asks toborrow your precious diamond for a day to show her family. You Jre a bilworried. so you carefully have your diamond weighed on a s.cale which reads8.17 grams. The scale's accu racy is claimed to be ± 0.05 gram. The next day youweigh the relurned diamond again, getting 8.09 grams. Is this your diamond?RESPONSE The s.cale readings arc measuremenlS and do not necessarily givethe "true" value of Ihe mass. Each meJSuremenl could have been high or lowby up 10 0.05 gram or so. The actual mass of you r diamond lies most likelybetween 8.12 grams Jnd 8.22 grams. The aClual mass of Ihe returned diamondis most likely between 8.04 grams and 8.14 grams. These IWO ranges overlap, sothere is nOI a strong reason to doubt that Ihe returned diamond is yours, atleast based on the s.cJIe reJdings._ PROBLEM SOLVING.'>'lImber of signifieo," figllres in fi/Ullres,,11 should be SlIme os leas/significant inpou ,,,,/,,e6 CHAPTER 1Significant FiguresThe number of reliably known digi\.s in a number is cillled the number of~ ignifi ca nt figurl"S. Thus there ilr~ four signifiCilnl figures in the number 23.21 cmilnd two in the lIum!J.t, r 0.062 cm (the 'l.eros in the Iilller ,lie merely plilee holdersthill show whe re the decimill point goes), The number of signifiC;lnt figures m;Jynot alw;Jys be ck;lr. Take, for e' ilm ple, the lIumber SO. Arc Ihere one or twosignificanl figures? If we say it is ilIJ01I1 80km betweell two cities, Ihere is onlyone significant figure (the 8) since Ihe ze ro is merely a place holder. If it isHilelly 80 km within an accuracy of 1 or 2 km, then the SO has IWO significanlfigures. ' If il is precisely RO km, to within ± 0.1 km, Ihen we write RO.O km.When making mea\urement~ or when doing calculations, you should avoidthe lemptation 10 kee p more digits in Ihe final answer than is juslified. Forexample, to calculate the area of a rectangle 11.3cm hy 6.8cm, the result ofmultiplication would he 76.84cml. But this answer is clearly not accurale 100.01 cm 1 , since (using the outer limits of Ihe as.~umed uncertainty for eachmeasurement) the result could be between 11.2 em x 6.7 em = 75.04 cm 1 andII.4cm x 6.9cm = 78.66cm 1 . At best. \\"e can quote the answer as 77C111 1 ,which im plies an uncertainly of about 1 or 2cm 1 . The olher IWO digits (i n thenumber 76.84cm 1 ) must be dropped sillce they arc not significant. As a roughgeneral rule (i.e., in the absence of a detailed cOllsideration of uncertainties), wecan say that Ihe final resufl of a ",ulliplictlliOIl or divisioll sllOfiM have only asmally digilS as Ihe number ,villl lire leasl number of significant figures used ill lirecalmlillioll. [n our example, 6.8cm has the least number of significant figures.namely t\\"o. Thus Ihe resolt 76.84 cm 1 needs to be ro unded off to 77 cm 1 .' If tho 80 has Iv.'(> ,ignofocant figu,o,"."mc pcw tc r refor 10 ""ito il 8!J. v.'ilh • docomal roin l. Thi, i,not u,u.lly done. '" 'he num!>