significant figures dealing with uncertainty in measurements

Significant Figures

Dealing with uncertainty in measurements.

What values are shown below?

• 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.

• 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.

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?

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

• 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

• 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.

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• Example 3400 1250000

• NOTE – If the number is in SCIENTIFIC NOTATION only consider the COEFFICIENT when determining SFs. 

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_

Mathematical Operations with Significant Figures

• 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.

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!

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

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.

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