calculations with significant figures. since all measurements contain an estimated digit, … …...
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Calculations with Significant Figures
Calculations with Significant Figures• Since ALL measurements contain an estimated
digit, …
… then ALL measurements contain some error (or uncertainty).
• When measurements are used in calculations, the results have even more error (or uncertainty) than the original measurements themselves.
• In order to control for this increasing error, rules for rounding the results of mathematically derived values have been established.
Calculations with Significant Figuresa) The Addition and Subtraction Rule:
When adding or subtracting measurements, the final answer is rounded so that the last significant figure in the answer is in the same number place as the last significant figure of the least sensitive of the original measurements.
10 cm 10. cm 10.0 cm 10.00 cm
least sensitivemeasurement
most sensitivemeasurement
Calculations with Significant FiguresEx. (1) 3.1000g + 2.67 g + 1.954 g =
_______________ _______________(calculator answer) (final rounded answer)
Where is the last significant figure?
Ex. (2) 7.83 mol – 1.5 mol =
_______________ _______________ (calculator answer) (final rounded answer)
Where is the last significant figure?
7.724 g 7.72 g
6.33 mol 6.3 mol
Calculations with Significant FiguresEx. (3) 427 cal + 173.6 cal =
_______________ _______________(calculator answer) (final rounded answer)
Where is the last significant figure?
Ex. (4) 8.2 mL – 1.2 mL =
_______________ _______________ (calculator answer) (final rounded answer)
Where is the last significant figure?
600.6 cal 601 cal
7 mL 7.0 mL
Calculations with Significant FiguresEx. (5) 5.4oC + 20.6oC =
_______________ _______________(calculator answer) (final rounded answer)
Where is the last significant figure?
Ex. (6) 8300 mm – 1380 mm =
_______________ _______________ (calculator answer) (final rounded answer)
Where is the last significant figure?
26oC 26.0oC
6920 mm 6900 mm
Calculations with Significant Figuresb) The Multiplication and Division Rule:
When multiplying or dividing a measurement by other measurements, the final answer is rounded so that the number of significant figures in the answer is equal to the fewest number of significant figures in the original measurements.
10 cm 10. cm 10.0 cm 10.00 cm
fewest number ofsignificant figures
greatest number ofsignificant figures
Calculations with Significant FiguresEx. (1) 458.800 cal 4.830 g =
_______________ _______________(calculator answer) (final rounded answer)
How many total significant figures are there?
Ex. (2) 3.14 (2.15 cm)2 7.00 cm =
_______________ _______________ (calculator answer) (final rounded answer)
How many total significant figures are there?
94.98964803 cal/g 94.99 cal/g
101.60255 cm3 102 cm3
Calculations with Significant FiguresEx. (3) 2.50 mm 2.8 mm =
_______________ _______________(calculator answer) (final rounded answer)
How many total significant figures are there?
Ex. (4) 5.100 kJ 273 K =
_______________ _______________ (calculator answer) (final rounded answer)
How many total significant figures are there?
7 mm2 7.0 mm2
0.018681319 kJ/K 0.0187 kJ/K
Calculations with Significant Figures• WARNING!
When you multiply or divide a measurement by a true NUMBER, round the answer to the same number place of the last significant figure in the original measurement.
Ex. (5) 3.019 g 30 =
_______________ _______________(calculator answer) (final rounded answer)
true number
90.57 g 90.570 g
Calculations with Significant Figures• WARNING!
When you multiply or divide a measurement by a true NUMBER, round the answer to the same number place of the last significant figure in the original measurement.
Ex. (6) 2.13 cm 2 =
_______________ _______________(calculator answer) (final rounded answer)
true number
1.065 cm 1.07 cm