1.Tutorial on Uncertainty, Significant figures and Scientific notation .Prepared by Lawrence Kok http://lawrencekok.blogspot.com

2. Significant figuresUsed in measurementsDegree of precisionShow digits believed to be correct/certain + 1 estimated/uncertainAll reads 8080 80.0 80.00 80.000least preciseCertain 23.00Uncertain 523.005g more preciseNumber sf necessary to express a measurement Consistent with precision of measurement Precise equipment = Measurement more sf Last digit always an estimate/uncertainmeasurement 15.831g(15.831 0.001)g (5 sig figures)Deals with precision NOT accuracy!!!!!!!! Precise measurement doesnt mean, its accurate ( instrument may not be accurate) 3. Significant figuresUsed in measurementsDegree of precisionShow digits believed to be correct/certain + 1 estimated/uncertainAll reads 8080 80.0 80.00 80.000least preciseCertain 23.00Uncertain 5Zeros bet (significant) 4.109 = 4sf 902 = 3sf 5002.05 = 6sfmeasurement 15.831g23.005g more precise(15.831 0.001)g (5 sig figures)Rules for significant figuresAll non zero digit (significant) 31.24 = 4 sf 563 = 3 sf 23 = 2sfNumber sf necessary to express a measurement Consistent with precision of measurement Precise equipment = Measurement more sf Last digit always an estimate/uncertainZeros after decimal point (significant) 4.580 = 4 sf 9.30 = 3sf 86.90000 = 7sf 3.040 = 4sf 67.030 = 5sfZero right of decimal point and following a non zero digit (significant) 0.00500 = 3sf 0.02450 = 4sf 0.04050 = 4sf 0.50 = 2sfDeals with precision NOT accuracy!!!!!!!! Precise measurement doesnt mean, its accurate ( instrument may not be accurate)Zeros to left of digit (NOT significant) 0.0023 = 2sf 0.000342 = 3sf 0.00003 = 1sfZero without decimal (ambiguous) 80 = may have 1 or 2 sf 500 = may have 1 or 3 sfClick here and here for notes on sig figures 4. Significant figures 122Smallest division = 0.122Max = 21.63 2Certain 21.6Uncertainty = 1/10 of smallest division. = 1/10 of 0.1 = 1/10 x 0.1 = 0.013Certain = 21.64Uncertain = 21.62 0.015(21.62 0.01)Measurement = Certain digits + 1 uncertain digitMin = 21.61 Answer = 21.62 (4 sf) 21.6 (certain)2 (uncertain) 5. Significant figures 122Smallest division = 0.122Max = 21.63 2Certain 21.6Uncertainty = 1/10 of smallest division. = 1/10 of 0.1 = 1/10 x 0.1 = 0.013Certain = 21.64Uncertain = 21.62 0.015(21.62 0.01)Measurement = Certain digits + 1 uncertain digitMin = 21.61 Answer = 21.62 (4 sf) 21.6 (certain)1Smallest division = 12Uncertainty = 1/10 of smallest division. = 1/10 of 1 = 1/10 x 1 = 0.13Certain = 364Uncertain = 36.5 0.15Measurement = Certain digits + 1 uncertain digit2 (uncertain)Certain 36Max = 36.6(36.5 0.1) Min = 36.4 Answer = 36.5 (3 sf) 36. 5 (certain) (uncertain) 6. Significant figures 1Smallest division = 10Max = 47 2Certain 40Uncertainty = 1/10 of smallest division. = 1/10 of 10 = 1/10 x 10 = 13Certain = 404Uncertain = 46 15(46 1)Measurement = Certain digits + 1 uncertain digitMin = 45 Answer = 46 (2 sf) 4 (certain)6 (uncertain) 7. Significant figures 1Smallest division = 10Max = 47 2Certain 40Uncertainty = 1/10 of smallest division. = 1/10 of 10 = 1/10 x 10 = 13Certain = 404Uncertain = 46 15(46 1)Measurement = Certain digits + 1 uncertain digitMin = 45 Answer = 46 (2 sf) 4 (certain)1Certain 3.4Smallest division = 0.12Uncertainty = 1/10 of smallest division. = 1/10 of 0.1 = 1/10 x 0.1 = 0.013Certain = 3.44Uncertain = 3.410.015Measurement = Certain digits + 1 uncertain digit6 (uncertain)Max = 3.42(3.41 0.01) Min = 3.40 Answer = 3.41 (3sf) 3.4 (certain)1 (uncertain) 8. Significant figures 1Smallest division = 0.05Max = 0.480.1 20.2 0.3 0.4 0.5Certain 0.45Uncertainty = 1/10 of smallest division. = 1/10 of 0.05 = 1/10 x 0.05 = 0.005 (0.01)3Certain = 0.454Uncertain = 0.47 0.015(0.47 0.01)Measurement = Certain digits + 1 uncertain digitMin = 0.46 Answer = 0.47 (2 sf) 0.4 (certain)7 (uncertain) 9. Significant figures 1Smallest division = 0.05Max = 0.480.1 20.2 0.3Certain 0.45Uncertainty = 1/10 of smallest division. = 1/10 of 0.05 = 1/10 x 0.05 = 0.005 (0.01)3Certain = 0.454Uncertain = 0.47 0.0150.4(0.47 0.01)Measurement = Certain digits + 1 uncertain digitMin = 0.460.5Answer = 0.47 (2 sf) 0.4 (certain)7 (uncertain)Measurement 1Smallest division = 0.12Uncertainty = 1/10 of smallest division. = 1/10 of 0.1 = 1/10 x 0.1 = 0.013Certain = 5.74Uncertain = 5.72 0.01(5.72 0.01) Answer = 5.72 (3sf) 5.7 (certain)2 (uncertain)1Smallest division = 12Uncertainty = 1/10 of smallest division. = 1/10 of 1 = 1/10 x 1 = 0.13Certain = 34Uncertain = 3.0 0.1(3.0 0.1) Answer =3.0 (2 sf) 3 0 (certain) (uncertain) 10. Scientific notation How many significant figuresWritten as a = 1 to 9Scientific - notation = a 10bb = integerNumber too big/small 3 sf6,720,000,000Size sand= 6.72 109 4 sf0.0000000001254=1.254 10-10 3 sfSpeed of light300000000= 3.00 108 11. Scientific notation How many significant figuresWritten as a = 1 to 9Number too big/smallb = integer3 sfScientific - notation = a 10b6,720,000,000Size sand= 6.72 109 4 sf0.0000000001254=1.254 10-10 3 sfSpeed of light300000000How many significant figures4.66 x 10 6 4.660 x 10 64 sf4.6600 x 10 646600003 sf5 sfClick here practice scientific notationClick here practice scientific notation= 3.00 108 12. Scientific notation How many significant figuresWritten as a = 1 to 9Number too big/smallb = integer3 sfScientific - notation = a 10b6,720,000,000Size sand= 6.72 109 4 sf0.0000000001254=1.254 10-10 3 sfSpeed of light= 3.00 108300000000Scientific notation80 3 ways to write 80How many significant figures4.66 x466000010 63 sf4.660 x 10 65 sf80 8 x 101 (1sf) Digit 8 uncertain It can be 70 to 9080. 80. 8.0 x 101 (2sf) Digit 8 is certain It can be 79 to 8180.0 80.0 8.00 x 101 (3sf) Digit 80 is certain It can be 79.9 or 80.14 sf4.6600 x 10 68090 or 9 x 101 80 or 8 x 101 70 or 7 x 10181 or 8.1 x 101 80 or 8.0 x 101 79 or 7.9 x 10180.1 or 8.01 x 101 80.0 or 8.00 x 101 79.9 or 7.99 x 101 More preciseClick here practice scientific notationClick here practice scientific notation 13. Rules for sig figures addition /subtraction: Last digit retained is set by the first doubtful digit. Number decimal places be the same as least number of decimal places in any numbers being added/subtracted23.112233 1.3324 + 0.25 24.694633uncertain least number decimal placesround down4.7832 1.234 + 2.02 8.0372uncertain least number decimal placesround down1247 134.5 450 + 78 1909.5uncertainleast number decimal places1.0236 - 0.97268 0.050924.2 2.32 + 0.6157 7.13578.04 least number decimal places uncertainround downround up0.03uncertain least number decimal places68.7 - 68.42 0.280.0509 least number decimal places uncertain7.987 - 0.54 7.447uncertain least number decimal placesround upround downround up0.316.967.1 1.367 - 1.34 0.027191012.587 4.25 + 0.12 16.957uncertainround downround up24.69least number decimal placesuncertain least number decimal places2.300 x 103 + 4.59 x 103 6.890 x 103least number decimal places7.45 Convert to same exponent47.68 x 104 + 23.2 x 103476.8 x 103 + 23.2 x 103 500.0 x 103round up6.89 x 103500.0 x 103 5.000 x 105least number decimal places 14. Rules for sig figures addition /subtraction: Last digit retained is set by the first doubtful digit. Number decimal places be the same as least number of decimal places in any numbers being added/subtracted23.112233 1.3324 + 0.25 24.694633uncertain least number decimal placesround down4.7832 1.234 + 2.02 8.0372uncertain least number decimal placesround down1247 134.5 450 + 78 1909.5uncertainleast number decimal places1.0236 - 0.97268 0.050928.044.2 2.32 + 0.6157 7.1357least number decimal places uncertainround down0.03 3 x 10-2 (1sf)1.910 x 103 (4sf) uncertain least number decimal places5.09 x 10-2 (3sf)least number decimal places68.7 - 68.42 0.28uncertain7.987 - 0.54 7.447uncertain least number decimal placesround upround down0.316.967.1 x 100 (2sf)round up12.587 4.25 + 0.12 16.9570.0509round up7.1 1.367 - 1.34 0.02719108.04 x 100 (3sf)2.469 x 101 (4sf)uncertainround downround up24.69least number decimal places1.696 x 101 (4sf) uncertain least number decimal places2.300 x 103 + 4.59 x 103 6.890 x 1037.453 x 10-1 (1sf) least number decimal places7.45 x 100 (3sf)Convert to same exponent47.68 x 104 + 23.2 x 103476.8 x 103 + 23.2 x 103 500.0 x 103round up6.89 x 103 6.89 x 103 (3sf)500.0 x 103 5.000 x 105least number decimal places 15. Rules for sig figures - multiplication/division Answer contains no more significant figures than the least accurately known number.12.34 3.22 x 1.8 71.52264least sf (2sf)round up23.123123 x 1.3344 30.855495least sf (5sf)21.45 x 0.023 0.49335round downround down30.85572 16.235 0.217 x 5 17.614975least sf (1sf)round up4.52 6.3578 7.1093775least sf (3sf)923 20312 0.045441least sf (3sf)1300 x 57240 round down 744120004.60.00435 x 4.6 0.02001least sf (2sf)round down7.110.020least sf (2sf) Scientific notationleast sf (2sf)round up0.49round up202.8723 x I.6 4.59568least sf (2sf)6305 0.010 630500least sf (2sf)round down630000I.3*103 x 5.724*104 7.4412 x 107round down0.0454740000007.4 x 107 Click here for practice notes on sig figures 16. Rules for sig figures - multiplication/division Answer contains no more significant figures than the least accurately known number.12.34 3.22 x 1.8 71.5226423.123123 x 1.3344 30.855495least sf (2sf)round up21.45 x 0.023 0.49335least sf (5sf)round downround down30.85572 7.2 x 101 (2sf)3.0855 x 101 (5sf)16.235 0.217 x 5 17.614975least sf (1sf)round up4.52 6.3578 7.10937750.00435 x 4.6 0.02001923 20312 0.0454411300 x 57240 round down 74412000least sf (2sf)0.0207.11 x 100 (3sf) least sf (3sf)4.6 x 100 (2sf)round down7.112 x 101 (1sf)4.64.9 x 10-1 (2sf)least sf (3sf)2.0 x 10-2 (2sf) least sf (2sf) Scientific notationleast sf (2sf)round up0.49round up202.8723 x I.6 4.59568least sf (2sf)6305 0.010 630500least sf (2sf)round down630000 6.3 x 105 (2sf)I.3*103 x 5.724*104 7.4412 x 107round down0.0454740000007.4 x 1074.54 x 10-2 (3sf) Click here for practice notes on sig figures 17. Rules for sig figures Multiplication/Division/Addition/Subtraction Answer contains no more significant figures than the least accurately known number. Avoid rounding off error. 0.0000673 x 291 0.125 = ? least sf (3sf)0.0000673 x 291 0.01958430.0000673 x 291 0.0195843Intermediate step leave more sf DO NOT ROUND UP/DOWN intermediate steps Leave extra sf to avoid rounding off error0.0195843 0.125 0.15632ROUND UP/DOWN intermediate step0.0196 0.125 0.1568 round upround down0.156 1.56 x10-1 (3sf)Rounding off error0.15710-1(3sf)1.57 x 18. Rules for sig figures Multiplication/Division/Addition/Subtraction Answer contains no more significant figures than the least accurately known number. Avoid rounding off error. 0.0000673 x 291 0.125 = ? least sf (3sf)0.0000673 x 291 0.01958430.0000673 x 291 0.0195843Intermediate step leave more sf DO NOT ROUND UP/DOWN intermediate steps Leave extra sf to avoid rounding off error0.0195843 0.125 0.15632ROUND UP/DOWN intermediate step0.0196 0.125 0.1568 round upround down1.56 x10-1Rounding off error0.156(3sf)0.15710-1(3sf)1.57 x(21.5 + 21.53 + 22.548 ) x 8.45 = ? least number decimal places21.5 21.53 + 22.548 65.578Intermediate step leave more sf65.578 x 8.45 554.1341 round down554least number decimal places5.54 x 102 (3sf)DO NOT ROUND UP/DOWN Leave extra sf to avoid rounding off error least sf (3sf)ROUND UP/DOWN intermediate step least sf (3sf)21.5 21.53 22.548 65.578 65.5 x 8.45 553.47 round down553 5.53 x 102 (3sf) 19. Significant figures in measurement Recording measurement using significant figuresVolume, V = 4/3r3Radius, r = 2.15 cmRecording measurement using significant figuresCircumference, C = 2rRadius, r = 3.0 cmRecording measurement using significant figuresDisplacement, s = gt2Time, t = 2.25 sRecording measurement using significant figures G = (20 ) H = (16 ) Z = (106)Speed, s = (G + H) Z 20. Significant figures in measurement Recording measurement using significant figuresRadius, r = 2.15 cmVolume, V = 4/3r3V = 4/3 x x (2.15)3 = 4/3 x 3.14 x 2.15 x 2.15 x 2.15 = 41.60Recording measurement using significant figuresC = 2 x x (3.0) = 2 x 3.14 x 3.0 = 18.8495Recording measurement using significant figuresTime, t = 2.25 sRecording measurement using significant figures G = (20 ) H = (16 ) Z = (106)least sf (2sf)g and constant sf is not taken (not a measurement)Displacement, s = gt2s = 1/2 x 9.8 x (2.25)2 = 24.80625least sf (3sf)Speed, s = (G + H) Z 20 + 16 = 36 106 0.339least sf (3sf)2 and constant sf is not taken (not a measurement)Circumference, C = 2rRadius, r = 3.0 cm4/3 constant constant sf is not taken (not a measurement)least sf (2sf) 21. Significant figures in measurement Recording measurement using significant figuresRadius, r = 2.15 cmVolume, V = 4/3r3V = 4/3 x x (2.15)3 = 4/3 x 3.14 x 2.15 x 2.15 x 2.15 = 41.604/3 constant constant sf is not taken (not a measurement) least sf (3sf)round down41.6 Recording measurement using significant figuresRadius, r = 3.0 cm2 and constant sf is not taken (not a measurement)Circumference, C = 2rC = 2 x x (3.0) = 2 x 3.14 x 3.0 = 18.8495least sf (2sf)round up19 Recording measurement using significant figuresTime, t = 2.25 sg and constant sf is not taken (not a measurement)Displacement, s = gt2s = 1/2 x 9.8 x (2.25)2 = 24.80625least sf (3sf)round down24.8 Recording measurement using significant figures G = (20 ) H = (16 ) Z = (106)Speed, s = (G + H) Z 20 + 16 = 36 106 0.339least sf (2sf)round down0.34 22. Significant figures in measurement Recording measurement using significant figuresPeriod, T = 2 L gLength, I = 1.25 mRecording measurement using significant figuresArea, A = I x hLength, I = 4.52 cm Height, h = 2.0 cmRecording measurement using significant figuresConc, c = (2.00) M Volume, v = (2.0 )dm3Moles, n = Conc x Vol 23. Significant figures in measurement Recording measurement using significant figuresLength, I = 1.25 mRecording measurement using significant figuresLength, I = 4.52 cm Height, h = 2.0 cmRecording measurement using significant figuresConc, c = (2.00) M Volume, v = (2.0 )dm3Period, T = 2 L gleast sf (3sf)T = 2 x x (1.25/9.8) = 2 x 3.14 x 0.35714 = 2.24399Area, A = I x h4.52 x 2.0 9.04least sf (2sf)Moles, n = Conc x Vol2.00 x 2.0 4.00least sf (2sf)2, and g constant sf is not taken (not a measurement) 24. Significant figures in measurement Recording measurement using significant figuresLength, I = 1.25 mPeriod, T = 2 L gleast sf (3sf)T = 2 x x (1.25/9.8) = 2 x 3.14 x 0.35714 = 2.24399 round down2.24 Recording measurement using significant figuresLength, I = 4.52 cm Height, h = 2.0 cmArea, A = I x h4.52 x 2.0 9.04least sf (2sf)round down9.0 Recording measurement using significant figuresConc, c = (2.00) M Volume, v = (2.0 )dm3Moles, n = Conc x Vol2.00 x 2.0 4.00least sf (2sf)round down4.02, and g constant sf is not taken (not a measurement) 25. Significant figures in measurement Recording measurement using significant figuresDensity = Mass VolumeMass, m = 482.63g Volume, v = 258 cm3Recording measurement using significant figuresEnthalpy, H = m x c x TMass water, m = 2.00 g Temp, T = 2.0 CRecording measurement using significant figuresVolt, v = 2.0 V Current, I = 3.0A Time, t = 4.52st I2 Energy = 1/2 v 26. Significant figures in measurement Recording measurement using significant figuresDensity = Mass VolumeMass, m = 482.63g Volume, v = 258 cm3Recording measurement using significant figures482.63 258 1.870658Enthalpy, H = m x c x TMass water, m = 2.00 g Temp, T = 2.0 CxRecording measurement using significant figuresVolt, v = 2.0 V Current, I = 3.0A Time, t = 4.52sleast sf (3sf)2.00 4.18 2.0 16.72t I2 Energy = 1/2 v 4.52 3.0 x 3.0 40.68 1.414 28.769c (4.18) - constant sf is not taken (not a measurement)least sf (2sf)least sf (2sf) 27. Significant figures in measurement Recording measurement using significant figuresDensity = Mass VolumeMass, m = 482.63g Volume, v = 258 cm3482.63 258 1.870658least sf (3sf)round down1.87 Recording measurement using significant figuresEnthalpy, H = m x c x TMass water, m = 2.00 g Temp, T = 2.0 Cx2.00 4.18 2.0 16.72c (4.18) - constant sf is not taken (not a measurement)least sf (2sf)round upRecording measurement using significant figuresVolt, v = 2.0 V Current, I = 3.0A Time, t = 4.52s17 t I2 Energy = 1/2 v 4.52 3.0 x 3.0 40.68 1.414 28.769 round up29least sf (2sf) 28. Acknowledgements Thanks to source of pictures and video used in this presentation http://crescentok.com/staff/jaskew/isr/tigerchem/econfig/electron4.htm http://pureinfotech.com/wp-content/uploads/2012/09/periodicTable_20120926101018.png http://www.wikihow.com/Find-the-Circumference-and-Area-of-a-CircleThanks to Creative Commons for excellent contribution on licenses http://creativecommons.org/licenses/Prepared by Lawrence KokCheck out more video tutorials from my site and hope you enjoy this tutorial http://lawrencekok.blogspot.com