measurements - Expressing Measurements: Numbers and the SI...

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MEASUREMENTS.wpd 5/12/10 Expressing Measurements: Numbers and the SI Units Expressing Measurements: Numbers and the SI Units If a student is to be successful in chemistry, or for that matter any science with quantitative aspects, the student must have a clear and complete knowledge and understanding of numbers. That is, how accurately do I know the number and what does the number represent? Measurements are the means by which scientists gather quantitative information. The beginning student generally recognizes the importance of making a careful measurement, but usually fails to recognize that great care must also be exercised in recording the measurement. Hence, it must be recognized that there are two basic parts to any measurement, the number and the label (or unit) associated with it. Both must be stated clearly when quoting a measurement. NUMBERS The numerical portion of a measurement relates the actual value obtained and the accuracy to which it is known. In your own work, you will always want to convey, exactly, the accuracy to which a value is known. This is true because, even if the exact degree of accuracy is not important to you in your work, it may be important to someone else who, at some later time, may try to use your results for a different purpose. There are two distinct classes of numbers. They are: C Exact numbers (known with absolute or infinite accuracy), and C Measured numbers (known with relative or finite accuracy). The exact numbers can be divided into two types. They are: C Counted numbers; as the "9" in 9 baseballs, or the "3" in 3 eggs, and C Defined numbers; as the "12" in 12 inches equal a foot, or the "16" in 16 ounces equal a pound. Exact Numbers There is no uncertainty in either type of exact number because it is not possible, for example, to have a fractional number of baseballs (that is, 9.1 baseballs is not possible) or anything other than exactly 16 ounces in a pound. Therefore, it must be kept in mind that the accuracy of any calculation involving exact numbers is limited only by the accuracy of those numbers which are not exact numbers, namely those numbers which are derived from measurements. One could argue that counting 9 000 000 baseballs makes that a counted number and therefore exact. However, in this case (because the number is so large), the odds are great that some error was made in the counting. Therefore, one must use some judgement in deciding the accuracy of any such number. Defined Numbers Conversion factors within one system of measurement are exact; they are defined numbers. (Conversion factors between different systems of measurement usually need to be compared by some measurement and, therefore, are approximate. However, exceptions do exist--when one system is defined in terms of the other system. E.g., 1 inch / 25.4 millimeters, and 1 calorie / 4.184 joules. Both of these conversions are now exact.) Therefore, it is easy to see why it is desirable to do all work within one system.
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2 Numbers like (an irrational number) and ð (a transcendental number) fall into the category of "defined numbers" even though you can't write the number out completely.
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This note was uploaded on 12/07/2011 for the course CHM 2045 taught by Professor Weinstein during the Fall '11 term at Santa Fe College.

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measurements - Expressing Measurements: Numbers and the SI...

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