significant figures dealing with uncertainty in measurements
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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 BETWEEN NON-ZERO DIGITS are SIGNIFICANT parts of a measurement.
• Examples 5005 120301
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• 3. Zeros BOTH TO THE RIGHT OF a non-zero digit AND a WRITTEN DECIMAL are significant.
• Examples 124.000 5.000
• 4. Zeros that SERVE ONLY AS PLACEHOLDERS are NOT SIGNIFICANT.
• Examples 0.000003432 0.0021111
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• 5. Zeros to THE RIGHT OF A NON-ZERO DIGIT BUT to the LEFT OF AN UNDERSTOOD DECIMAL are NOT SIGNIFICANT…..they can be the RESULT OF ROUNDING OFF!
• If a BAR is placed ABOVE A ZERO it makes ALL digits OVER TO AND INCLUDING THE ZERO WITH THE BAR SIGNIFICANT.
_
• Example 3400 1250000
• NOTE – If the number is in SCIENTIFIC NOTATION only consider the COEFFICIENT when determining SFs.
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Practice Problems
• Determine how many figures are significant in each of these measurements:
• 1. 375 2. 89.000
• 3. -0.00032 4. 4300
• 5. 12.0900 6. 0.00003200
• 7. 900001 8. 2.34 x 104
• 9. -0.000212000 10. 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