1. Slide 1 of 48 Measurements and Their Uncertainty 3.1 3.1

2. © Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 2 of 48 3.1 Using and Expressing Measurements A measurement is a quantity that has both a number and a unit. Measurements are fundamental to the experimental sciences. For that reason, it is important to be able to make measurements and to decide whether a measurement is correct.

3. © Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 3 of 48 3.1 Using and Expressing Measurements In scientific notation, a given number is written as the product of two numbers: a coefficient and 10 raised to a power. The number of stars in a galaxy is an example of an estimate that should be expressed in scientific notation.

4. © Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 4 of 48 Scientific Notation Proper Scientific Notation Form: M x 10 N 1 < M <10 N is an integer

5. © Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 5 of 48 Calculator Use with Scientific Notation When using a Texas Instrument Calculator, look for the EE key. On a graphing calculator it will be a second function. For 4.4 x 10 25 you will see 4.4 E 25 on the display. The calculator understands E as meaning x10 This method will give you more reliable results than using the carrot (^). Second Function EE is Second Function on comma key

6. © Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 6 of 48 Multiply and Divide with Scientific Notation 3.6 x 10 12 cm x 4.7 x 10 15 cm = 16.920 x 10 27 cm 2 = 1.7 x 10 28 cm 2 5.3 x 10 7 m 3 / 3.5 x 10 2 m = 1.51428 x 10 5 m 2 = 1.5 x 10 5 m 2 To multiply – Multiply the numbers Add the exponents cm x cm = cm 2 To divide – Divide the numbers Subtract the exponents m 3 / m = m 2

7. © Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 7 of 48 Addition and Subtraction with Scientific Notation 12.5 x 10 3 cm 13.7 x 10 3 cm +3.4 x 10 3 cm 29.6 x 10 3 cm Addition/Subtraction – Add/Subtract the numbers All numbers must be to the same power of 10 Keep the same power of 10 Keep the same units

8. © Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 8 of 48 3.1 Accuracy, Precision, and Error Accuracy and Precision Accuracy is a measure of how close a measurement comes to the actual or true value of whatever is measured. Precision is a measure of how close a series of measurements are to one another.

9. © Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 9 of 48 3.1 Accuracy, Precision, and Error To evaluate the accuracy of a measurement, the measured value must be compared to the correct value. To evaluate the precision of a measurement, you must compare the values of two or more repeated measurements.

10. © Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 10 of 48 3.1 Accuracy, Precision, and Error

11. © Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 11 of 48 3.1 Accuracy, Precision, and Error Determining Error The accepted value is the correct value based on reliable references. The experimental value is the value measured in the lab. The absolute value of the difference between the experimental value and the accepted value is called the error. Error = |Experimental – Accepted|* *This is different than the formula in the textbook.

12. © Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 12 of 48 3.1 Accuracy, Precision, and Error

13. © Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 13 of 48 Percent Error Calculation

14. © Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 14 of 48 3.1 Accuracy, Precision, and Error Just because a measuring device works, you cannot assume it is accurate. The scale below has not been properly zeroed, so the reading obtained for the person’s weight is inaccurate.

15. © Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 15 of 48 Significant Figures in Measurements Suppose you estimate a weight that is between 2.4 lb and 2.5 lb to be 2.46 lb. The first two digits (2 and 4) are known. The last digit (6) is an estimate and involves some uncertainty. All three digits convey useful information, however, and are called significant figures. 3.1

16. © Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 16 of 48 Significant Figures The significant figures in a measurement include all of the digits that are known, plus a last digit that is estimated.

17. © Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 17 of 48 Significant Figures in Measurements 3.1

18. © Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 18 of 48 Significant Figures in Measurements 3.1

19. © Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 19 of 48 Significant Figures 1.All non-zero digits (1 to 9) are always significant. (Ex. 245 has 3 SF) 2.Zeros sandwiched between non-zero digits or other significant zeros are always significant. (Ex. 101 has 3 SF – also note example in #4 below) 3.Zeros to the left of all non-zero digits are never significant. (Ex. 0.0025 has 2 SF and 0000001 has 1 SF)

20. © Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 20 of 48 Significant Figures 4.Zeros to the right of all non-zero digits must be to the right of the decimal to be significant. (Ex. 250.0 has 4 SF and 0.00370 has 3 SF) 5.Counting numbers (numbers with no estimation) have infinite significant figures. (Ex. There are ____ students in this classroom)

21. © Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 21 of 48 Significant Figures 6.Conversion factors also have infinite significant figures. (Ex. 100 cm/1m or 60 sec/1 min.)

22. © Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 22 of 48 Please note in the next slide that the type of measuring device has an impact on the number of significant figures. (i.e. the markings on the device determine the number of known digits. (There is always 1 estimated digit)

23. © Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 23 of 48 Significant Figures in Measurements 3.1

24. © Copyright Pearson Prentice Hall Slide 24 of 48

25. © Copyright Pearson Prentice Hall Slide 25 of 48 Practice Problems Problem Solving 3.2 Solve Problem 2 with the help of an interactive guided tutorial. for Conceptual Problem 3.1

26. Slide 26 of 48 © Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Significant Figures in Calculations In general, a calculated answer cannot be more precise than the least precise measurement from which it was calculated. The calculated value must be rounded to make it consistent with the measurements from which it was calculated. 3.1

27. © Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 27 of 48 3.1 Significant Figures in Calculations Rounding To round a number, you must first decide how many significant figures your answer should have. The answer depends on the given measurements and on the mathematical process used to arrive at the answer.

28. © Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 28 of 48 Rounding 1.If the number after the last significant digit is less than 5, the number remains unchanged. 2.If the number after the last significant digit is 5 or greater, the last significant digit is increased by 1. Ex. 75.223 m to 3 SF = 75.2 m 75.277 m to 3 SF = 75.3 m

29. © Copyright Pearson Prentice Hall Slide 29 of 48 Practice Problems Problem Solving 3.3 Solve Problem 3 with the help of an interactive guided tutorial. for Sample Problem 3.1

30. © Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 30 of 48 3.1 Significant Figures in Calculations Addition and Subtraction The answer to an addition or subtraction calculation should be rounded to the same number of decimal places (not digits) as the measurement with the least number of decimal places.

31. © Copyright Pearson Prentice Hall SAMPLE PROBLEM Slide 31 of 48 3.2

32. © Copyright Pearson Prentice Hall SAMPLE PROBLEM Slide 32 of 48 3.2

33. © Copyright Pearson Prentice Hall Slide 33 of 48 Practice Problems for Sample Problem 3.2 Problem Solving 3.6 Solve Problem 6 with the help of an interactive guided tutorial.

34. © Copyright Pearson Prentice Hall Measurements and Their Uncertainty > Slide 34 of 48 3.1 Significant Figures in Calculations Multiplication and Division In calculations involving multiplication and division, you need to round the answer to the same number of significant figures as the measurement with the least number of significant figures. The position of the decimal point has nothing to do with the rounding process when multiplying and dividing measurements.

35. © Copyright Pearson Prentice Hall SAMPLE PROBLEM Slide 35 of 48 3.3

36. © Copyright Pearson Prentice Hall Slide 36 of 48 Practice Problems for Sample Problem 3.3 Problem Solving 3.8 Solve Problem 8 with the help of an interactive guided tutorial.

37. © Copyright Pearson Prentice Hall Slide 37 of 48 Section Quiz -or- Continue to: Launch: Assess students’ understanding of the concepts in Section 3.1. Section Assessment

38. © Copyright Pearson Prentice Hall Slide 38 of 48 1. In which of the following expressions is the number on the left NOT equal to the number on the right? a.0.00456  10 –8 = 4.56  10 –11 b.454  10 –8 = 4.54  10 –6 c.842.6  10 4 = 8.426  10 6 d.0.00452  10 6 = 4.52  10 9 3.1 Section Quiz

39. © Copyright Pearson Prentice Hall Slide 39 of 48 2. Which set of measurements of a 2.00-g standard is the most precise? a.2.00 g, 2.01 g, 1.98 g b.2.10 g, 2.00 g, 2.20 g c.2.02 g, 2.03 g, 2.04 g d.1.50 g, 2.00 g, 2.50 g 3.1 Section Quiz

40. © Copyright Pearson Prentice Hall Slide 40 of 48 3. A student reports the volume of a liquid as 0.0130 L. How many significant figures are in this measurement? a.2 b.3 c.4 d.5 3.1 Section Quiz

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