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Significant Figures I. Concept and Definition Suppose a ruler is used to measure the length of an object as shown in the figure below. When using a measuring device you should always use the measuring device to the limit of its precision. This means that you should record all numbers that are known with certainty and then include one estimated digit. For the ruler shown above, the smallest increments are in tenths of a centimeter, so we should be able to tell the length of the string with certainty to a tenth of a centimeter. We would then estimate one digit beyond the tenths and our final measurement should be recorded to a hundredth of a centimeter. Some of you may say its length is 6.75 cm. Some may say it is 6.74 cm or 6.76 cm. We are quite certain that the length is somewhere between 6.7 cm and 6.8 cm. The third (last) digit is our estimated digit and there is an uncertainty of at least 0.01cm. It is unreasonable to report a value like 6.75342183 cm since we are not even sure about the third digit. The last six digits are meaningless. Suppose we take the value 6.75 cm. In this reported measurement, the first two digits are definitely significant. The third digit is also significant but has some uncertainty associated with it. It is our best estimate of where it is between 6.7 and 6.8 cm. Therefore, there are three significant figures in the measured quantity reported above. Similarly, all measured quantities are generally reported in such a way that the last digit is uncertain. All digits in a measurement including the uncertain one are called significant figures. II. Guidelines for determining the number of significant figures. 1. All nonzero digits are significant. 2. Zeros between nonzero digits are significant. 3. Zeros to the right of the decimal point at the end of a measurement are significant. 4. Zeros used solely for spacing the decimal point are not significant. This is the only time that a number in a measurement is not significant. Examples Measured Value # of Significant Figures 2.456 4 1003.2 5 1.03000 6 0.0000402 3 (the zeros before the 4 are spacers for the decimal point and are not significant) 230000 2-6 In the last example above, it is not clear how many significant figures there are. Suppose there are three significant figures, the number represents 230,000±1,000. If there are two significant figures, the number represents 230,000±10,000. To overcome this ambiguity as well as for ease of manipulation, such numbers should always be written in exponential (scientific) notation: 2.30 x 105 (3 significant figures). Very large and very small numbers are usually expressed in exponential notation: 0.00000001230 = 1.230 x 10-8 (four significant figures) 3000000000. = 3.0 x 109 (two significant figures) Note: Only those digits before the exponent are used to express the number of significant figures. Do not add the exponential term to the number of significant figures. Exact numbers are considered to have an infinite number of significant figures. For example, if you said, "A is twice (or two times) as large as B", the number 2 would be exact. Or, if you said "There are 4 quarts in a gallon", the number 4 would be exact. Exact numbers usually involve counted values or definitions. Problem 1: State the number of significant figures in each of the following numbers: a) 74906 b) 0.00459 c) 274 d) 3.74 x 106 e) 4.730 x 106 f) 500 g) 78 million h) 190.0022 Problem 2: Express in scientific notation with proper attention to significant figures: a) 1,358 to 3 significant figures b) 1004 c) 0.000831 d) 36,000,000 to 4 significant figures III. Making calculations with significant figures RULE #1 - In combined math operations that consist of only addition and subtraction, the answer must be rounded off to contain only as many decimal places as are in the value with the least number of decimal places. In the following examples the answer rounded to the correct number of significant figures is shown in parentheses. 16.234 - 4.120 + 0.433 ---------12.547 (12.547) 258.4 - 4.25 + 0.226 ----------254.376 (254.4) 7.30 - 1.451 + 0.016 ---------5.865 (5.87) RULE #2 - In combined math operations that consist of only multiplication and division, the answer must be rounded off to contain only as many significant figures as are in the value with the least number of significant digits. In the following example the answer rounded to the correct number of significant figures is shown in parentheses.. 2.423 ------1.24 x 1.9620 = 3.83381 (3.83) RULE #3 - In combined math operations that consist of multiplication/division and addition/subtraction it is conventional to round your answer after each step according to the rule that applies. In the following example the answer rounded to the correct number of significant figures is shown after each step. 2.423 ------1.24 = 1.9540322258 (1.95) + .37 = 2.32403 (2.32)