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Chem 160- Ch # 2l. Numbers from measurements. Measurements • Experiments are performed. • Numerical values or data are obtained from these measurements. Exact Numbers • Exact numbers have an infinite number of significant figures. • Exact numbers occur in simple counting operations 12345 • Defined numbers are exact. 12 inches 100 centimeters = 1 foot = 1 meter Form of a Measurement numerical value 70.0 kilograms = 154 pounds unit Significant Figures • The number of digits that are known plus one estimated digit are considered significant in a measured quantity known 5.16143estimated Significant Figures on Reading a Thermometer The temperature Temperature is oC is expressed 21.2 estimated to be oC. The last 2 is to 3 significant 21.2 figures. uncertain. The temperature Temperature is oC is expressed 22.0 estimated to be to 3 osignificant 22.0 C. The last 0 is figures. uncertain. The temperature Temperature is oC isto 22.11 expressed estimated be oC. The last 1 to 4 significant 22.11 figures. is uncertain. Significant Figures All nonzero numbers are significant. 461 Significant Figures All nonzero numbers are significant. 461 Significant Figures All nonzero numbers are significant. 3 Significant Figures 461 Significant Figures A zero is significant when it is between nonzero digits. 3 Significant Figures 401 Significant Figures A zero is significant when it is between nonzero digits. 5 Significant Figures 93 . 006 Significant Figures A zero is significant at the end of a number that includes a decimal point. 5 Significant Figures 55 . 000 Significant Figures A zero is significant at the end of a number that includes a decimal point. 5 Significant Figures 2 . 1930 Significant Figures A zero is not significant when it is before the first nonzero digit. 1 Significant Figure 0 . 006 Significant Figures A zero is not significant when it is before the first nonzero digit. 3 Significant Figures 0 . 709 Significant Figures A zero is not significant when it is at the end of a number without a decimal point. 1 Significant Figure 50000 Rounding off Numbers Rounding Off Numbers • Often when calculations are performed extra digits are present in the results. • It is necessary to drop these extra digits so as to express the answer to the correct number of significant figures. • When digits are dropped the value of the last digit retained is determined by a process known as rounding off numbers. Rounding Off Numbers Rule 1. When the first digit after those you want to retain is 4 or less, that digit and all others to its right are dropped. The last digit retained is not changed. 4 or less 80.873 Rounding Off Numbers Rule 2. When the first digit after those you want to retain is 5 or greater, that digit and all others to its right are dropped. The last digit retained is increased by 1. drop 5 or these greater figures increase by 1 6 5.459672 Significant Figures in Calculations Multiplication or Division In multiplication or division, the answer must contain the same number of significant figures as in the measurement that has the least number of significant figures. 2.3 has two significant figures. (190.6)(2.3) = 438.38 190.6 has four significant figures. Answer given by calculator. The answer should have two significant figures because 2.3 is the number with the fewest significant figures. Round off this digit to four. Drop these three digits. 438.38 The correct answer is 440 or 4.4 x 102 Addition or Subtraction The results of an addition or a subtraction must be expressed to the same precision as the least precise measurement. The result must be rounded to the same number of decimal places as the value with the fewest decimal places. Add 125.17, 129 and 52.2 Least precise number. Answer given by calculator. Round off to the Correct answer. nearest unit. 306.37 125.17 129. 52.2 306.37 Scientific Notation of Numbers • Very large and very small numbers are often encountered in science. 602200000000000000000000 0.00000000000000000000625 • Very large and very small numbers like these are awkward and difficult to work with. A method for representing these numbers in a simpler form is called scientific notation. 23 602200000000000000000000 6.022 x 10 -21 0.00000000000000000000625 6.25 x 10 Scientific Notation • Write a number as a power of 10 • Move the decimal point in the original number so that it is located after the first nonzero digit. • Follow the new number by a multiplication sign and 10 with an exponent (power). • The exponent is equal to the number of places that the decimal point was shifted. Write 6419 in scientific notation. decimal after first nonzero digit power of 10 1 2 3 6.419 641.9x10 64.19x10 6419. 6419 x 10 Write 0.000654 in scientific notation. decimal after first nonzero digit 6.54 x 0.000654 0.00654 0.0654 0.654 -4 -2 -1 -3 10 power of 10