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BELL RINGER – Complete on a sheet of paper and TURN IN before working on notes! A student needed to calibrate a graduated cylinder [a device to measure liquids]. She collected the following data: Trail #1 99.98 mL Trial #2 100.02 mL Trial #3 99.99 mL The accepted value of the cylinder’s volume is 100.00 mL. What is the PERCENT ERROR of her measurements?

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Average = 99.98 + 100.02 + 99.99 = 99.99 3 Error = 99.99 – 100.00 = -.01 Percent Error = 0.1 x 100 = 0.01 % error 100.00

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Significant Figures Dealing with uncertainty in measurements.

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What values are shown below?

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Why is it difficult to be certain about some of the measurements you make? –All measurements have SOME DEGREE OF UNCERTAINTY due to limits associated with the measuring device. –Generally, uncertainty begins with the LAST DIGIT of the measurement.

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In a measurement, ALL THE DIGITS KNOWN FOR CERTAIN plus the first ESTIMATED DIGIT are known as the SIGNIFICANT FIGURES of the measurement. It is generally accepted that when a measurement is given, ALL NON-ZERO DIGITS are considered SIGNIFICANT. For example 175.4 grams Digits known for certain. First estimated digit.

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The Problem with Zero While all NON-ZERO DIGITS are considered significant, ZEROS present a particular problem. –Zeros can be measurements –Zeros can be place holders How do you decide whether or not a zero is significant?

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Rules for Significant Figures 1. ALL NON-ZERO digits are considered significant. Examples 125.45 5648 1.1211 2. Zeros IN THE MIDDLE OF NUMBERS are SIGNIFICANT parts of a measurement. Examples 5005120301

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3. Zeros AT THE BEGINNING OF A NUMBER are not significant. Examples 0.000003432 0.0021111 4. Zeros AT THE END OF A NUMBER are only significant IF THE FOLLOW A DECIMAL or a BAR is placed over a zero… when this occurs, ALL digits up to and including the zero with the bar are significant. _ Example 45.23000 1.000 505.32000 4750000

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NOTE – If the number is in SCIENTIFIC NOTATION only consider the COEFFICIENT when determining Significant Figures. Example 4.965 x 10 16

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Practice Problems Determine how many figures are significant in each of these measurements: 1. 3752. 89.000 3. -0.000324. 4300 5. 12.09006. 0.00003200 7. 9000018. 2.34 x 10 4 9. -0.00021200010. 4002000 _

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Mathematical Operations with Significant Figures

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When completing math calculation, the final answer must be reported rounded to the appropriate number of significant figures. The answer is rounded according to the LAST mathematical operation completed.

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Rules 1. Complete calculations following the order of operations. 2. If the FINAL step is MULTIPLICATION or DIVISION: –A. Look at each value given in the problem and find the one with the LEAST number of significant figures. –B. Round the FINAL ANSWER to the same number of significant figures. –DO NOT ROUND UNTIL THE FINAL STEP!

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Mult/Div Examples 4.59 X 1.22 = 5.5998 = 5.5998 = 5.60 3 sf 3sf 3sf 3sf 3 sf 45.6 = 18581.90709 4 sf 0.002454 = 18587.90709 3sf = 18600 3sf

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ADD/SUBTRACT Complete calculations following order of operations. If the FINAL step is addition or subtraction: –A. Only consider digits to the RIGHT of the decimal. –B. Determine the fewest SF to the right of the decimal. –C. Round final answer to this number of SF.

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ADD/SUBTRACT EXAMPLES 25.4 (1 sf) 15.000 – 2.3791 = 12.6209 63.66 (2 sf) (3 sf) (4 sf) = 12.621 + 102.44 (2 sf) 191.50 = 191.5

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