Exact Numbers and Conversion Statements
Date: 07/25/2005 at 21:26:28 From: Jason Subject: Exact Numbers and Conversion Statements I am currently taking chemistry, and something that keeps coming up is exact numbers, which are, by definition, numbers obtained by counting (as opposed to measuring) as in 8 apples. Also, an exact number could be part of a definition, like a conversion statement such as 1 inch = 2.54 cm. Is there a general rule for determining whether or not a conversion statement is exact? I believe that fairly recently, the inch was redefined to be EXACTLY 2.54 cm. Are there others that have been defined as exact? And are conversion statements from metric to English or vice versa the only conversion statements that might be inexact? I would really appreciate your help here, since knowing the exactness of a conversion factor plays a huge role in dimensional analysis and rounding an answer to the correct number of significant digits (because an exact number has no limits on significant digits, but an inexact number does). And on a more general note, what other types of numbers are classified as exact (e.g., infinite, infinite repeating, fractions, pi)? I know a lot of times it has to do with the context. An example of my dilemma is in the following problem: What is the mass of a troy ounce of gold in grams? 1 troy ounce = 20 pennyweight (exact) 1 pennyweight = 24 grains (exact) 1 grain = 0.0648 gram (NOT exact) I used dimensional analysis, assuming that the "one troy ounce" I was starting with was inexact (because it is a measurement), so I assumed the answer should have only ONE significant digit, but the answer key said the answer should have 3 significant digits because of the 0.0648. I am really lost now, and I don't know how to distinguish between a measurement that is inexact (which I thought a measurement always is) and one that is exact (which I thought they never were, until this problem). The only explanation I can think of is that the "one troy ounce" wasn't actually measured, they just wanted us to use it for the sake of converting. PLEASE HELP!!!!!!!!!!!!!!! Thanks.
Date: 07/25/2005 at 22:37:23 From: Doctor Peterson Subject: Re: Exact Numbers and Conversion Statements Hi, Jason. The general rule is that a number is exact if you are told it is exact. There is nothing more general that can be said. Any formula you are given should make it clear whether the constants in it are exact. Generally, within a system (such as the 20 and 24 in your example), all numbers are exact integers; between systems, such as your 0.0648, the numbers are usually approximate, and that should be indicated by using an "approximately equal" sign, or saying "to three significant digits", or something like that. But since it is so unusual for a conversion factor between systems to be exact, you can take it the other way and, in that setting, assume a number is inexact unless it is explicitly stated, as it will be for 2.54. You asked about "infinite, infinite repeating, fractions, pi". You didn't really mean "infinite numbers", but rather "infinite decimals". If a decimal is given to you indicating how it repeats, as with a bar over it, then that is exact because you are being told every digit in the number. If you are just given some decimal places and are not told how the rest of it behaves, then obviously it can't be considered exact. That is true of pi, which as an irrational number can't be expressed exactly. As for fractions, they are exact unless you are told that the fraction is approximate. For example, the 5/9 or 9/5 in conversion between Fahrenheit and Celsius is an exact fraction. This means that in such cases you don't need to consider the number of significant digits. Other fractions, such as 22/7 for pi, are approximations; to use this with significant digits, you would have to determine HOW accurate it is, by writing it as a decimal and comparing to a good decimal approximation to pi. A decimal approximation makes a lot more sense in such a context. Now, in your question about "the mass of a troy ounce in grams", there is no number given! There is no measurement here! It is "the mass of an exact troy ounce", not "a mass measured as 1.0 troy ounces". That means you can't take "1" as having a specific number of significant digits; so you have to take it as exact. So the only number in your calculation that has a specific precision is the last conversion factor, and only that limits the precision of your answer. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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