5.2 Uncertainty in Measurement and Significant Figures

Period 5Group# 4

• 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.

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.

• Significant Figures - The certain digits and the first uncertain digit of a measurement.

• Any measurement that has an estimate is uncertain.

• Rulesa. Leading zeroes -

are never significant

b. Captive zeroes - are always significant

c. Trailing zeroes - are sometimes significant

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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.

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.

• 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.

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.

• 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.

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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.

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.

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1. Why are the first few digits called certain numbers?

2. Why do we use significant figures?

3. Round 4.2786 x 10^3 to the nearest tens.

4. Solve (2.87 x 10^-2)(8.79x10^3) with the correct number of sig figs

5. Explain the limiting term for adding and subtracting.

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 4279Answer 4: (0.0287)(8970) = 252.273 252Answer 5: The limiting term is the smallest

number with the least digits past the decimal.

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

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