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Math Review Significant figures When a number is used to describe a measured quantity such as weight, length, volume, or temperature, the number indicates not only the magnitude of the measurement but also its precision, i.e. the reproducibility of the measurement. For example, if the length of a wire is reported as 72 cm, it means this measurement was made only to the nearest cm; we do not know the length of the wire more precisely than 72 ± 1 cm. In other words, we know only that the exact length of the wire is between 71 and 73 cm. The 7 is a certain digit but the 2 is a doubtful digit. If the length of the wire is instead reported as 72.13 cm, it indicates that the measurement was made with equipment capable of measuring to the nearest 0.01 cm. In this instance, we know that the length of the wire is 72.13 ± 0.01 cm and that its exact length is between 72.12 and 72.14 cm. Here, the 7, 2, and 1 are known with certainty and the 3 is doubtful. Digits that indicate the precision of a measurement are called significant figures. The signifi-cant figures (sig figs) of a number include all the certain digits and the first doubtful digit. The first measurement described above, 72 cm, contains two sig figs; the second, 72.13 cm, contains four. Suppose an object is weighed on a balance having 0.0001 g precision, and the weight is 1.2300 g. This number has five sig figs. It would be incorrect to drop the final zeros (reporting the weight as 1.23 g) because this would imply the weight was found using a balance having only 0.01 g precision. Counting Significant Figures Due to the relationship between sig figs and precision of measurement, we must be able to determine the number of sig figs contained in any given number. All digits except zero are significant. The significance of a zero depends on its position in the number relative to other digits and the decimal point. The following rules apply to zeros as sig figs. 1. A zero is significant if it is both followed by and preceded by a digit other than zero: 10.203 has five sig figs 1.002 has four sig figs 12.023 has five sig figs 2. A zero necessary to locate the decimal point (preceded only by zero) is not significant: 0.123 has three sig figs 0.0123 has three sig figs 0.0001 has one sig fig 0.1023 has four sig figs 10.0123 has six sig figs2 3. The final zeros of a number are significant if they lie to the right of the decimal point: 1.000 has four sig figs 140.00 has five sig figs 0.010 has two sig figs 0.100 has three sig figs 4. Final zeros to the left of the decimal point may or may not be significant. In the expression 1200 g, we do not know whether one, both, or neither of the zeros is/are significant. If an object is weighed on a balance that weighs to the nearest g, both zeros are significant. If the object is weighed to the nearest 10 g, only the first zero is significant, etc. Any ambiguity surrounding the significance of final zeros preceding the decimal point can often be removed by expressing the measurement in some larger unit. For example, if we express the weight 1200 g in kilograms, the number of sig figs becomes apparent: 1.200 kg has four sig figs 1.20 kg has three sig figs 1.2 kg has two sig figs When conversion to larger units is impossible, ambiguity can be removed by expressing the num-ber in exponential terms. In the expression 120,000 atoms, the number of sig figs is questionable. When the measurement is written as 1.20 × 105 atoms, the number of sig figs (three) is clear. Significant Figures in Calculations Results of calculations with numbers that represent experimentally measured (inexact) quantities are themselves inexact. Results should imply no greater precision than the least precise measure-ment in the calculation. Results must often be rounded to indicate the proper number of sig figs. Rounding. When one or more digits must be dropped from a calculated result to get the proper number of sig figs, the following rules apply: 1. When the first digit to be dropped <5, the last digit retained remains unchanged. 2. When the first digit to be dropped >5, the last digit retained is increased by 1. 3. When the first digit to be dropped is 5, the last digit retained remains unchanged if it is even and is increased by 1 if it is odd. These rules are illustrated below, in which the numbers are rounded off to three sig figs: 1.6723 rounds off to 1.67 1.677 rounds off to 1.68 1.6652 rounds off to 1.66 1.6752 rounds off to 1.683 Multiplication and Division. As a general rule, the result of a calculation involving multipli-cation or division should contain the same number of sig figs as the factor with the smallest number of sig figs in the calculation. For example, in 2.137 × 5.62 = 5.13 2.3425 the least precisely known factor is 5.62, with three sig figs. The result thus contains three sig figs. Further examples appear below: 4.3 × 10-13 × 6.022 × 1023 = 2.6 × 1011 26.98 = 0.2023 133.34 5.0 × 0.18 × 6.022 × 1023 = 1.4 × 1023 4.00 4.785 × 1000.0 = 0.294 271.5 60.0 Addition and Subtraction. The rule for determining the number of sig figs different in addition and subtraction from multiplication and division. When numbers are added or subtracted, units must all be the same; i.e. we cannot add pounds and inches. Therefore, the least precise number is not necessarily the number with the fewest sig figs. It is the number with the fewest digits to the right of the decimal point. For example, assume we have determined the weights of a number of objects and wish to add them. The individual weights and precisions are: 1.02 g ± 0.01 g 107.3 g ± 0.1 g 14.273 g ± 0.001 g 0.12 g ± 0.01 g 122.713 g = 122.7 g ± 0.1 g The least precise weight is 107.3 g (though it is not the number with the fewest sig figs). Units and Conversion of Units The physical sciences are based on precise measurement. A measurement is defined as the ratio of the magnitude (how much) of any quantity to a standard value (unit). To record a person's4 height as 64 is meaningless. A unit must also be indicated, hence you must record it as 64 inches (magnitude + units). The three fundamental units of mechanical measurement are length, mass, and time. Many other units are derived, since they are written as some combination of the fundamental units.


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DePaul HON 225 - Math Review

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