Lesson 2 – Sci. Notation, Accuracy, and Significant Figures MATH AND METHODS Lesson 2 – Sci. Notation, Accuracy, and Significant Figures

Accuracy Versus Precision What is the difference between accuracy and precision? Precision: is a measure of how closely individual measurements agree with one another Can be precise but inaccurate Accuracy: refers to how closely individual measurements agree with the correct, or “true” value

An archery target illustrates the difference between accuracy and precision.

Calculating Accuracy (Percent Error) Percent error allows you to compare your answer to the actual answer to see how accurate you were. The “actual answer” is referred as the “accepted value.” “Your answer” is referred to as the “experimental value” % error = | Accepted value – Experimental value | x 100 Accepted value **Notice the absolute value, percent error will never be negative

Example % error = | 1.00g/mL – 1.18 g/mL| x 100 1.00 g/mL The density of water is known to be 1.00 g/mL. You measure the mass and volume of a water sample and calculate its density to be 1.18 g/mL. What is your percent error? % error = | 1.00g/mL – 1.18 g/mL| x 100 1.00 g/mL % error = 18%

Scale Reading and Uncertainty Uncertainty: Limit of precision of the reading (based on ability to guess the final digit). Exists in measured quantities not in counted quantities Counted quantities are exact numbers Addition and substraction(no. of sig. figs) Division and Multiplication (no. of sig. figs)

Measurements between users What is the length of this arrow? Likely we have many different possible answers based on our own eyes.

Significant Figures Significant Figures Digits in a measurement that have meaning relative to the equipment being used 9

Significant Figures Digits with meaning Digits that can be known precisely plus a last digit that must be estimated. 10

How to determine which figures are significant in a given number All non-zero digits (1-9) are significant. The zeros in a number are not always significant, depending on their position in the number.

Significant Figures Pacific to Atlantic Rule Examples Pacific = Decimal Present Start from the Pacific (left hand side), every digit beginning with the first 1-9 integer is significant 20.0 = 3 sig digits 0.00320400 = 6 sig digits 1000. = 4 sig digits 12

Significant Figures Atlantic Rule to Pacific Examples Atlantic = Decimal Absent Start from the Atlantic (right hand side), every digit beginning with the first 1-9 integer is significant 100020 = 5 sig digits 1000 = 1 sig digits 13

Practice How many significant figures are in 400.0 g 4000 g 4004 g Answers: 4 1 14

More Practice Determine the number of significant figures in: 72.3 g 60.5 cm 6.20 m 0.0253 kg 4320 years 0.00040230 s 4.05 moles 4500. L Answers: 3 5 4

Review Questions Determine the number of significant figures in the following: 1005000 cm 1.005 g 0.000125 m 1000. km 0.02002 s 2002 mL 200.200 days Answers: 4 3 6

Scientific notation Scientific notation has two purposes: Showing a very large or very small number Showing only the significant digits in a measurement Scientific notation has three parts: a coefficient that is 1 or greater and less than 10, a base and a power: 6.34 x 10³ g Positive (or 0) exponent means the number is one or greater. Negative exponent means the number is less than one. A negative exponent does NOT mean the number is negative.

Scientific Notation Practice Convert to scientific notation: 89540 = _______________ 0.000345 = ______________ 0.0041 = _______________ 7890000 = _________________ 23000 = _________________ Convert to standard form: 6.72 x 10³ = ______________ 2.341 x 10­ˉ³ = ______________ 5.6 x 10² = _______________ 1.29 x 10º = ________________ 4.78 x 10ˉ² = _________________ 8.954 x 104 3.45 x 10-4 4.1 x 10-3 7.89 x 106 2.3 x 104 6720 0.002341 560 1.29 0.0478

Significant Figures when in Scientific Notation The number of significant figures in a measurement that is in scientific notation is simply the same number of digits that are in the coefficient of the measurement: 4.5 x 10³ has 2 significant figures 5.234 x 10² has 4 significant figures 9.65 x 10ˉ³ has 3 significant figures There is no need to convert to standard form before determining the number of significant figures.

What about when you add two measurements? When you add or subtract measurements, your answer must have the same number of digits to the right of the decimal point as the value with the fewest digits to the right of the decimal point. Ex 456.865g + 2g = 458.865g (do the calculation first) Since 2g has no digits right of the decimal, neither can your answer, which would be 459 (three sig figs)

Practice Add the following measurements: 2.6g + 3.47g + 7.678g 30.0 mL – 2.35 mL 27.7 mL 5.678 cm + 3.76 cm 9.44 cm

What about when you multiply/divide two measurements? When you multiply or divide measurements, your answer must have the same number of significant figures as the measurement with the fewest sig figs. This does not apply to counted values or unit conversions, they will not impact the number of significant figures Ex. Find the density of an object with a mass of 2.6g and a volume of 300 mL (Density=mass/volume) 2.6g/300mL = 0.009g/mL (one sig fig)

Practice 24m x 13.6m x 3.24m 47g ÷ 32.34 mL 40m ÷ 4.3 s 1100 m3