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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. _ • 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 • 4 sf • • 45.6 0.002454 = 18581.90709 = 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) 63.66 (2 sf) + 102.44 (2 sf) 191.50 = 191.5 15.000 – 2.3791 = 12.6209 (3 sf) (4 sf) = 12.621