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Using and Expressing Measurements Learning Objectives Write numbers in scientific notation. Evaluate accuracy and precision. Explain why measurements must be reported to the correct number of significant figures. Two types of data that is gathered: Qualitative: descriptive data Quantitative: numerical data Scientific Notation Scientific notation is used to express very large and very small numbers. Width of human hair = 0.00007 m = (7 × 0.00001) m = 7 × 10–5 m Writing Numbers in Scientific Notation Scientific notation includes a coefficient and an exponent. 6,300,000 = 6.3 × 106 Coefficient: number between 1 and 10 Exponent: power of 10 0.0000074 = 7.4 × 10–6 In class practice: Write each of these numbers in scientific notation: 17 3 500.0 215 7,000,631 1.7 x 101 3x 100 5.000 x 102 2.15 x 102 7.000631 x 106 0.000000614 6.14 x 10-7 0.0037004 3.7004 x 10-3 0.00000038 0.01010 0.00000000001 3.8 x 10-7 1.010 x 10-2 1 x 10-11 Accuracy and Precision Accuracy and precision relate to the reliability of a measurement. Accuracy refers to how close experimental measurements are to the actual (book) value. Precision refers to how closely the data gathered is to each other. Good accuracy Poor accuracy Poor accuracy Good precision Good precision Poor precision YOU TRY IT. THREE STUDENTS PERFORMED AN EXPERIMENT SEVERAL TIMES AND OBTAINED THE FOLLOWING RESULTS. THE ACTUAL VALUE OF THE MASS WAS 9.00 G. WHICH OF THESE DATA SETS (IF ANY) SHOW BETTER RESULTS? Student 3 has the best results Student 1 Student 2 Student 3 9.28 g 9.27 g 9.50 g 9.48 g 9.00 g 9.01 g 9.05 g 9.10 g 9.52 g 9.50 g 8.98 g 9.00 g 9.30 g 9.25 g 9.49 g 9.51 g 9.03 g 9.04 g Low accuracy Low precision Low accuracy High precision High accuracy High precision Error and Percent Error Error is a description of how far a measurement is from its accepted value. Calculating Percent Error The boiling point of pure water is measured to be 99.1°C. Calculate the percent error. Calculating Percent Error, cont. The boiling point of pure water is measured to be 99.1°C. Calculate the percent error. Significant Figures Significant figures indicate the precision of a measurement. Rules for Determining Significant Figures 1. Every nonzero digit in a reported measurement is assumed to be significant and Zeros appearing between nonzero digits are significant. 2. Leftmost zeros appearing in front of nonzero digits are not significant. (Leading zeros are not significant.) 3. Zeros at the end of a number and to the right of a decimal point are always significant. 4. For values written in scientific notation, the digits in the coefficient are significant. 5. Counted numbers and exact conversion factors have an infinite number of significant figures. The rules are given to you on your reference material. We will learn how to apply and use those rules. ALWAYS refer to the rules. 1. Non-zero digits and zeros between non-zero digits are always significant. Number Number of Significant Figures 125 3 1142 4 2005 4 7 1000007 2. Leading zeros are not significant. Leading zeros are zeros that precede all nonzero digits. These zeros do not count as significant figures. 0.0025 2 sig figs 0.0321 3 sig figs 3. Zeros to the right of all non-zero digits are only significant if a decimal point is shown. These are called trailing zeros. Examples: 100 1 sig fig 100. 3 sig fig 4. For values written in scientific notation, the digits in the coefficient are significant. Examples: 1.00×102 2.0560 x 104 3 sig fig 5 sig fig 5. In a common logarithm, there are as many digits after the decimal point as there are significant figures in the original number. (log base 10) Examples: Log 75 is 1.87 which is two sig figs. WHOA!! Ok how? Let’s look at it this way: 74 is 7.4x101 Now consider the log of each part: the log of 101 is 1, an exact number The log of 7.4 is 0.87 -- with a proper 2 SF. Add those together, and you get log 74 = 1.87 -with 2 SF. Pacific Ocean Decimal Present: Start on the left of the number and don’t start counting sig figs until you get to the first non-zero number. Memory Helper Atlantic Ocean Decimal Absent. Start on the right side of the number and don’t start counting sig figs until you get to the first non-zero Number of Sig Figs Number of Sig Figs 0.02 1 142 3 0.020 2 0.073 2 501 3 1.071 4 501.0 4 10810 4 5.00 3 5000 1 Round each of the following to three significant digits. Rounded Number 88.473 8505 976450 699.5 123.98 0.00086321 88.5 8510 976000 700. 124 0.000863 Rounded Number 69.95 70.0 0.000056794 0.0000568 67.048 67.0 3.002 3.00 0.0300 0.0300 90100 90100