MATH AND METHODS

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

2. Accuracy Versus Precision • What is the difference between accuracy and precision? • Precision: is a measure of how closely an 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

3. An archery target illustrates the difference between accuracy and precision. (Not in notes)

4. 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

5. Example • 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%

6. 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

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

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

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

11. 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 • There are a standard set of rules for figuring out whether or not zeros are significant. • This is something you have probably never been introduced to before but it will play a role in nearly everything we do all semester so pay close attention and memorize these rules

12. Rules for zeros: • All zeros count except placeholder zeros • These are the ones that disappear when you write the number in scientific notation. • Zeros between nonzero digits are always significant • E.g. 1005 kg (4 sig. fig) and 1.03 (3 sig. fig) • Zeros at the beginning of a number are never significant • E.g. 0.02 (1 sig. fig) and 0.0026 (2 sig. fig) • Zeros at the end of a number are significant if the number contains a decimal point • 0.0200g (3 sig. fig), 3.0 cm (2 sig. fig), 5000 (1 sig. fig)

13. Practice • How many significant figures are in • 400.0 g • 4000 g • 4004 g • 0.004 g Answers: • 4 • 1 • 4 • 1

14. Scientific notation • Scientific notation has two purposes: • Showing a very large or very small number • Showing only the significant digits is 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

15. 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 = _________________

16. 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

17. 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

18. 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 4 3 4 4 4 6

19. 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 3 3 3 3 5 3 4

20. 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.

21. 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)

22. Practice • Add the following measurements: (don’t forget conversions) • 2.6g + 3.47g + 7.678g • 13.7 g • 30.0 mL – 2.35 mL • 27.7 mL • 5.678 cm + 3.76 cm • 9.44 cm

23. 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)

24. Practice • 24m x 13.6m x 3.24m • 1100 m3 • 47g ÷ 32.34 mL • 1.5 g/mL • 40m ÷ 4.3 sec • 9 m/s