Significant Figures rev 2

# Significant Figures rev 2 - A brief tutorial on significant...

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A brief tutorial on significant figures Suppose you want to measure the hypotenuse of a right triangle, and further suppose that you cannot measure it directly. Perhaps it is a few cm longer than your ruler, or it is too delicate to put a ruler next to or some other contrived reason why you cannot measure it directly. You can calculate the length of that hypotenuse if you measure the two other legs of the triangle, and we will suppose that these can be measured. So, intrepid explorer that you are, you measure the legs of this triangle (using a ruler incremented in millimeters) and arrive at the following two measurements: The horizontal leg is 10.25 cm and the vertical leg 7.35 cm. (How to deal with the unstated uncertainties associated with these measurements will be discussed later.) We can now use the theorem of Pythagoras to calculate the length of our hypotenuse. Figure 1 222 CA B =+ , or 22 B . See figure 1 above. This results in (10.25 ) (7.35 ) cc m c m . Using our scientific calculator, it returns an answer for c of 12.6128902 cm. Now, if we tell someone that the hypotenuse of that triangle is 12.6128902 cm, they would be justified in thinking that we spent great efforts to measure that hypotenuse to within a few wavelengths of visible light, because we are telling them that we know something about the 10 -9 meter decimal place with the answer we’ve given. Clearly, a plastic meter stick (see figure 2) from the local office supply company is not capable of making measurements with that sort of precision. What we need, then, is a set of standards to guide us in how to round our calculated answers so that we are not alleging to know things that we cannot possibly know. This is where proper significant figure techniques come into play. Ruler Fig.2

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What defines significant figures? Ultimately, the number digits we report in our final answer (in other words, how many digits we round to) is determined by how many digits are in the inputs to our calculation, and how large our uncertainty is. In order to determine the number of significant figures in a number, we need to understand what a significant figure is, and we need to understand how to count them. What is a significant figure? Informally, it is any digit in a reported number that we have some degree of certainty about. For example, if I tell you a length is 7.35 cm, I am telling you that I have some degree of confidence that I know each digit of that length. I know that the length exceeds 7 cm, but not 8 cm. I know that it exceeds 7 cm by about 0.3. I also know that it is half way between 7.3 cm and 7.4 cm. But how do we decide how many significant figures there are in a number? We have some simple rules for determining this.
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Significant Figures rev 2 - A brief tutorial on significant...

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