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5.2 Uncertainty in Measurement and Significant Figures

5.2 Uncertainty in Measurement and Significant Figures. Period 5 Group# 4. Uncertainty in Measurement. A measurement always has some degree of uncertainty. Certain numbers are always the same and accurate. Uncertainty depends on the tool used for measuring.

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5.2 Uncertainty in Measurement and Significant Figures

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  1. 5.2 Uncertainty in Measurement and Significant Figures Period 5Group# 4

  2. Uncertainty in Measurement • A measurement always has some degree of uncertainty. • Certain numbers are always the same and accurate. • Uncertainty depends on the tool used for measuring. • Why is the last digit called an uncertain number? • Solution: The last digit is usually estimated and can vary.

  3. Uncertainty in Measurement • http://cyberbridge.mcb.harvard.edu/images/math2_1.png • http://cyberbridge.mcb.harvard.edu/images/math2_2.png • In the first ruler, it does not include the centimeters, so measuring the leaf’s length would have to be estimated causing it to be inaccurate. • In the second ruler, it includes the centimeters, so measuring the leaf’s length is more accurate than using the first ruler.

  4. Significant Figures • Significant Figures - The certain digits and the first uncertain digit of a measurement. • Any measurement that has an estimate is uncertain. • Rules • Leading zeroes - are never significant • Captive zeroes - are always significant • Trailing zeroes - are sometimes significant

  5. Significant Figures http://online.redwoods.cc.ca.us/instruct/milo/1/sld042.jpg • Why do we use significant figures? Solution: Significant figures allow us to signify the degree of certainty for a measurement. The uncertainty in the last number of a measurement is usually either +1 or -1. For example: 3.56 could have been 3.54 or 3.57.

  6. Rules for Rounding 1. If the digits to be removed is a. less than 5, the preceding digit stays the same. b. equal to or greater than 5, the preceding digit is increased by 1. 2. Carry extra digits through to the final result and then round off.

  7. Rules for Rounding • Rounding up example: Round 0.0876 to the nearest tenth. • Round down example: Round o.246 to the nearest tenth. • Answer: 0.1 because 8 is above 5 so you round up 0 up to 1. • Answer: 0.2 because 4 is below 0 so it stays the same.

  8. Rules for Multiplying and Dividing Significant Figures • Rules: The number of significant figures used when multiplying is equal to the factor with the least significant figures. • Ex: 8.315 / 298 = 0.0279027  2.79 x 10-2 There are three significant figures in this case because 298 has three, which has less sig. figs than the other factor, 8.315.

  9. Rules for Multiplying and Dividing Significant Figures • Explanation: 1.6 is the limiting term in this case which only has 2 sig. figs, so the answer will end up with 2 sig. figs. • Explanation: 45.2 is the limiting term in this case which has 3 sig. figs, so the answer will end up with 2 sig. figs. http://www.astro.washington.edu/courses/labs/clearinghouse/labs/Scimeth/images/multiply.gif http://www.astro.washington.edu/courses/labs/clearinghouse/labs/Scimeth/images/division.gif

  10. Rules for Addition and Subtraction Significant Figures • Rules: The limiting term is the one with the smallest number of decimal places for addition and subtraction which determines the number of decimal places that are sig. figs for the result. • 0.72 - limiting term There are 2 significant+0.0429 figures because there are 0.7629  0.76 2 decimal places in 0.72.

  11. Rules for Addition and Subtraction Significant Figures • Explanation: 2.02 is the limiting term because it has the least decimal places, so the result will be ending with 2 decimal places, 8.04. • Explanation: 1.0236 is the limiting term because it has the least decimal places, so the result will end with 4 decimal places, 0.0509. http://www.astro.washington.edu/courses/labs/clearinghouse/labs/Scimeth/images/addition.gif http://www.astro.washington.edu/courses/labs/clearinghouse/labs/Scimeth/images/subtract.gif

  12. Quiz • Why are the first few digits called certain numbers? • Why do we use significant figures? • Round 4.2786 x 10^3 to the nearest tens. • Solve (2.87 x 10^-2)(8.79x10^3) with the correct number of sig figs • Explain the limiting term for adding and subtracting.

  13. Quiz Answers • Answer 1: The first digits are always the same regardless of who makes the measurement • Answer 2: Significant figures allow us to signify the degree of certainty for a measurement • Answer 3: 4278.6  4279 • Answer 4: (0.0287)(8970) = 252.273  252 • Answer 5: The limiting term is the smallest number with the least digits past the decimal.

  14. Reference • http://cyberbridge.mcb.harvard.edu/images/math2_1.png • http://cyberbridge.mcb.harvard.edu/images/math2_2.png • http://online.redwoods.cc.ca.us/instruct/milo/1/sld042.jpg • http://www.astro.washington.edu/courses/labs/clearinghouse/labs/Scimeth/images/multiply.gif • http://www.astro.washington.edu/courses/labs/clearinghouse/labs/Scimeth/images/division.gif • http://www.astro.washington.edu/courses/labs/clearinghouse/labs/Scimeth/images/addition.gif • http://www.astro.washington.edu/courses/labs/clearinghouse/labs/Scimeth/images/subtract.gif

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