Significant Figures in Combined Expressions 1) Calculate in order of precedence: ()→→→ PEMDAS 2) Apply the x and ± rules separately to each step. → Do not round off intermediate numbers! Only count their N(s.f.) (2.34 + 5.6 + 3) x 1.90 = 21 (24.00 +6) x (-0.24 + 0.5) = 1294 × 1.90 =20.186 =\'s 2 s.f. 's2s.f. 3 s.f. 2s.f. (25.41 0.075) x (0.0078 +3.04 -0.109) ÷ (30.1) 2 X • Exact numbers have N(s.f.) =00 7(chairs) x π- 2.34 = 19.65 21.99115... - 2.34 =19.611 s.f. 0.0\s ÷ == = check only inexact, measured numbers for the lowest N(s.f.) & the highest decimal place of all last s.f's →0.0\s (100(pens)+3.2) ÷ 1.68= ÷ -202.0 x (10.6 + 1.01 60(pages)) ÷ (30(pins) - 8) = -202.0 x [Decimal] = [NSN] To convert decimal → NSN: Same Number o Ways ...now the figure 1) Shift decimal point to follow the first non-0 digit is in [1;10) 2) Compensate with a x10" term to keep the value the same: x10+ left [to undo decimal point shift by N places] right>>> x10-N Multiply NSN back out to decimal to check the sign & number of N. 3) Check trailing zeros in the [1;10) figure: drop if they become Ø, zeros of scale in the decimal. Count N(s.f.): must stay the same! 57000 = 5.1 x10 0.0020 = 2.0x10-200 = 4) To fix general scientific notation into NSN, update x 10" term: 46697.0 x 103 = 4.66970x103 x 104 = 4.66970 x10 0.00930 x 10-2 = -730.4 x 101 = To convert NSN decimal, multiply out the NSN product. → 1.34 x 10-2 = 0.0134 7.50 x 103 =1500 3.04 x 103 = Show zeros of scale as needed. Count N(s.f.): must stay the same! SFN Arithmetic in NSN calculator rounded decimal NSN 4.30×10-1.1.761×10-2=0.0015123 0.00151 3 s.f. 1.51x10 20. 4×10-2=418.848 400 1 s.f. 4x102 • 20. 4×10-2 1.91x10-3 = 2.01x102 4x 10-2 = 5.004x105 140 2.13x10¹ . = 1.91e-3 1.44×10-2x 5(pens) 0.0219 8.20x10-2 = 1.16x103 - 4.762x103=-3b02 -3000 0.01 sx103 0.00sx103 10s => → 12s =_s 590 7.1x10-29.103x 10² = - _S S - s.f. s.f. s.f. s.f. s.f. ⇒ _S 1.03×101 + 2.1x104 + 68 = 203.898 >>0.1_s 167.2 _s =125598 2s.f._s s.f. : N(§.f.) i count N(s.f.) in each 3 area A = 1.1 length² 1 m volume V = 1.1.1 length³ speed v = //t length/time acceleration a =S/2 Aspeed/time force F = m.a mass acceleration pressure P = F/A force/area N/m² mx3=m² 1mmxmx M = m³ M/S m/s² = m/s/s 2 = N = Pa =-kg energy Ek = 1/2m-v2 1/2 mass.speed2 kg.2 = Nm = J 2 molarity c = n/V moles/volume mol/ density P = m/V /= 5 x 103 m = 5 mA(square) = /x/ = 5 km x 5 km =30km² m = 0.371 g V = 0.03 mL P.V= n.R.T ? ⇒ p= = 0.371 q = 10a=1 g/mL m V 0.03 mL ⇒n= PV= RT n = mL 0.082057 atm.mol-1-1.273 3.04.5.6 (0.082057-273) 3.04 at 5.62 mol = mol 1.89×103 min = ? h = 1.89×103 min x h = 31.5-3.15x10 h 60min 50 lb = kg = 50 lb x 0.048 L= ? qt = 0.048 L x kg qt 6700 oz = ? ton = 6700 oz x •Can multiply together as many conversion factors as you need: 11x04536x 1 ton=0.1899 116 103 kg -> ton 10% 2.0 qt = 2 mL = 2.0 qt x-gal x- Lx mL qt 0.0340 mi? dm=0.0340 mi x gal Must write conversion factors with units first- then the numbers: == L mL km x mi mx dm = km m dm 11 cm = 2 ft = 11 cm x 2.54 cm 1 i 12 = 1 X ... Check the result in o.m.: ft ...X 1 ft ~ 30 cm 12 in. _ft • May switch in place ones of old units for their values in new units: 0.75 d x?s= 0.75-24 60.60s S 1 d = 24 h d = 24 h 1 h = 60 min h = 60 min 1 min = 60 s onvert s) power, plan for the plain units... then raise the plain conversion factors you know to that power: 1.39×104 μm² = 2 cm² = 1.39×104 \cm² 1 μm = 10 m 1 cm = 102 m 2×102 1492 X = 10412 (1 μm)2 = (10m)2 (1 cm)2 = (102 m)2 1 um² = 102m² 1 cm² = 10 m² -> cm² = 0.000139x 950 ft2 = ? m² = 950 pt2 x A in.2 2.54 cm² 10-4 m² 1 ft = 12 in. 950×144×2,542 (1 ft)2(12 in.)2 1 ft2=144 in.2 ft in. X1X10-4 1 in. = 2.54 cm =1 = 1 cm² m² 1 cm = 102 m (1 in.)2 = (2.54 cm)2 (1 cm)2 = (102 m)2 1 in.2 = 2.54 2 cm² 1 cm² = 10 m² 30000 dL = ? ft3 = 30000 dtx 10xx Im 1 ft = 30.48 cm (1 ft)3 = (30.48 cm)³ = X ft3 mx cm³ 103 mL 30.48 cm³ 1dL 103 == 30000 x10/(1 x 10-3)/(3048°) 1 ft3 = 30.483 cm³ 1 mL = 1 cm³ • To convert units ratios: unit/unit, (unit)-1, write ir g fractions: 430 L/km = ? L/m = 430 x 1 km = 0.43 ± 30m Km ID3m m → Conversion factors work on units in denominators just as well! 30 mol/qt=? mol/pint = 30 molx qt x qt gal 2.90x105 h = ? Hz = 2.90×105 1x gal pint \x\min= = mol pint mol/pint 1 K bomin 60 s S 1 Hz= 1 s²= -> Hz 120 ft/s ? m/h = X = 120 ft x 60s bomin 30.48cm 102 m= X × S \ min m= h h \ ft 1 cm m/h 64.0 mm³/kg = ? mL/lb =b4x(1x10-3)3/ (1×10-2)3x04536 64.0 mm3x (10-3)-m-x (1 cm)- x 1 mL x 0.4530 kg = kg 1 mL = 1 cm³ (1 mm) (10-2)-m- \ cm³ \ lb t mL lb mL/lb Significant Figures in Combined Expressions 1) Calculate in order of precedence: ()→→→ PEMDAS 2) Apply the x and ± rules separately to each step. → Do not round off intermediate numbers! Only count their N(s.f.) (2.34 + 5.6 + 3) × 1.90 = 21 (24.00 +6) x (-0.24 + 0.5)= 1294 × 1.90 =20.186 's2s.f. 3 s.f. 2s.f. ⇒ (25.41 0.075) x (0.0078 +3.04 -0.109) ÷ (30.1)² X • Exact numbers have N(s.f.) =00 7(chairs) x π- 2.34 = 19.65 21.99115... - 2.34 = 19,611 s.f. Q.Q1s ÷ = = check only inexact, measured numbers for the lowest N(s.f.) & the highest decimal place of all last s.f's → 0.01s (100 (pens)+3.2) ÷ 1.68= ÷ -202.0 x (10.6 + 1.01 60(pages)) ÷ (30(pins) - 8) = -202.0 x