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The Importance of measurement Scientific Notation

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Scientific notation 36,000 is 3.6 x 10 4 36,000 is 3.6 x 10 4 The exponent indicates how many times the coefficient 3.6 must be multiplied by 10 to equal the number 36,000. Large numbers have positive exponents

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Small numbers (less than one) have negative exponents. 0.0081 is 8.1 x 10 -3, the exponent -3 indicates that the coefficient 8.1 must be divided by 10 three times to equal 0.0081

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Multiplication and Division To multiply numbers written in scientific notation, multiply the coefficients and add the exponents (3.0 x 10 4 ) x (2.0 x 10 2 )= 6.0 x 10 6

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To divide numbers written in scientific notation, first divide the coefficient, then subtract the exponent in the dominator from the exponent in the numerator 3.0 x 10 4 / 2.0 x 10 2 = 1.5 x 10 2

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Sample Problems Write each measurement in scientific notation a.222 meters 2.22 x 10 2 meters 2.22 x 10 2 meters b.728500 meters 7.285 x 10 5 meters 7.285 x 10 5 meters c.0.00054 meters 5.4 x 10 -4 meters 5.4 x 10 -4 meters

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Multiplication and Division a.(3.0 x 10 5 ) x (2.6 x 10 3 )= 7.8 x 10 8 7.8 x 10 8 b.(2.8 x 10 3 ) / (1.4 x 10 -2 )= 2.0 x 10 5 2.0 x 10 5

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Uncertainty in Measurements Accuracy, Precision and Error - You must be able to make reliable measurements in the lab. Ideally, measurements are both correct and reproducible.

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Accuracy- is a measure of how close a measurement comes to the actual or true value of the object measured

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Precision- is a measure of how close a series of measurements are to one another Example- Which set of measurements is more precise? a.2g, 3g, 4g b.2.1g, 2.2g, 2.3g b. is correct because the measurements are closer together

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Error in Measurements Error in Measurements Percent error can be used to evaluate the accuracy of a measurement, it must be compared to the correct value. % error = experimental value - accepted value x100 accepted value

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Significant Figures in a measurement include all of the digits that are known, plus a last digit that is estimated. Measurements must always be recorded to the correct number of significant digits. Calculated answers depend upon the number of significant figures in the values used in the calculation

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Making measurements with sig figs

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Determining Significant Digits Nonzero digits: all are considered to be significant Example: 3279g has 4 sig figs Leading zeros: none are significant. They are considered to be place holders and not part of the measurement Example: 0.0045 has 2 sig figs (only the 4 and the 5)

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Captive zeros: zeros between two nonzero digits They are considered to be significant. Example: 5.007 has 4 sig figs

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Trailing zeros: zeros at the end of a measurement They are counted only if the number contains a decimal point Examples: 100 has 1 sig fig 100. has 3 sig figs 100.0 has 4 sig figs 0.0100 has 3 sig figs

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Scientific notation: all numbers listed in the coefficient are considered to be significant. Examples: 1.7 x 10 -4 has 2 sig figs 1.30 x 10 -2 has 3 sig figs

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Exact numbers: have unlimited significant digits Examples: 4 chairs (determined by counting) 1 inch = 2.54 cm (determined by definition)

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How many sig figs are in each measurement? 1. 123 meters 3 2. 0.123 meters 3 3. 40,506 meters 5 4. 9.8000 x 10 3 meters 5 5. 30.0 meters 3

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More practice 6. 22 meter sticks unlimited sig figs (count sig figs for measured values only) unlimited sig figs (count sig figs for measured values only) 7. 0.07080 meters 4 sig figs 4 sig figs 8. 98,000 meters 2 sig figs 2 sig figs

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Sig figs in calculations Rounding: In general, a calculated answer cannot be more precise than the least precise measurement from which it was calculated. Once you know the number of significant digits your answer should have, round to that many digits, counting from the left.

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Rounding Round each measurement to the number of sig figs shown in parentheses. 1. 314.721 meters (round to 4 sig figs) 315.0 meters 315.0 meters 2. 0.001775 meters (round to 2 sig figs) 0.0018 meters 0.0018 meters 3. 8792 meters (round to 2 sig figs) 8800 meters 8800 meters

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Rules for multiplication and division: When multiplying or dividing with measurements, round the answer to the same number of sig figs as the measurement with the least number of sig figs.

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Rules for multiplication and division: Example: 7.55 meters x 0.34 meters = 2.567 meters. = 2.567 meters. Round the answer to 2.6 meters (2 sig figs) Round the answer to 2.6 meters (2 sig figs) (The position of the decimal point has nothing to do with the rounding process when multiplying and dividing measurements.)

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Rules for addition and subtraction: The answer to an addition or subtraction calculation should be rounded to the same number of decimal places (not digits) as the measurement with the least number of decimal places.

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Rules for addition and subtraction Example: 7.055 meters + 0.35 meters = 7.405 meters. = 7.405 meters. Round the answer to 7.41 meters (2 places to the right of the decimal)

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Sample Problems 1. 7.55 meters x 0.34 meters = 2.6 m 2 2.6 m 2 2. 2.10 meters x 0.70 meters = 1.5 m 2 1.5 m 2 3. 2.4526 meters / 8.4 seconds=.29 m/s.29 m/s 4. 0.365 meters / 18.25 seconds = 0.0200 m/s 0.0200 m/s

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More problems 5. 12.52 meters + 349.0 meters+ 8.24 meters = 369.8 m 369.8 m 6. 74.626 meters- 28.34 meters= 46.29 m 46.29 m 7.80.0 meters + 0.0002 meters = 80.0 m

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