Rounding Down to Nothing
Date: 10/23/2000 at 22:02:46 From: Susan Swarts (Teacher) Subject: Rounding In my third grade class, we are working on rounding numbers. A question came up about rounding the numbers 0, 1, 2, 3, and 4 to the nearest 10. The nearest 10 would be zero, but that seems to say that you would be "rounding" down to nothing. That seems inaccurate, since as you do have "some." Would you round down to 1? Would you consider negative numbers here? We can't seem to find information on this subject.
Date: 10/23/2000 at 22:57:15 From: Doctor Peterson Subject: Re: Rounding Hi, Susan. The problem with rounding small numbers down to zero is not that the rounding itself is wrong, but that one would not ordinarily want to do it. As you say, rounding loses a lot of accuracy - in fact, it loses all the information you have. For precisely that reason, we would rarely round a number that way. For example, suppose you measured the height of everyone in your class, and got numbers like, say, 1.234 meters. If I asked you to round them to the nearest ten meters, you'd probably question my choice, since they would all round to zero. Even if we round to the nearest meter, we'll lose all our information, since all the numbers will be 1. Instead, we would probably choose to round to the nearest centimeter, in order to avoid losing data. But if you were measuring the heights of mountains, with some numbers in the kilometers and others (say, in Delaware) only a few meters, then rounding to the nearest ten meters would make sense, even if some "mountains" (sand dunes?) rounded to 0. You would still have useful information; the zero would tell you a lot about the height compared to real mountains. Someday your students will learn about significant digits, which are very relevant here. Typically we would choose to round not to some arbitrary amount, like the nearest meter, but to a certain number of significant digits. The number of significant digits indicates the amount of information you have; the numbers 1.234 m, 123.4 cm, and 0.001234 km (with four significant digits) represent the same amount of information, though the numbers are very different. We might round them to three digits, making them 1.23 (the nearest hundredth), 123 (the nearest one), and 0.00123 (the nearest hundred-thousandth). In other words, we choose to round in a way that retains a reasonable amount of information, while still simplifying the data. So rounding to the nearest ten always takes you to the nearest multiple of ten, even when it's zero (or negative, if, say, you are rounding a temperature). The answer can't be 1, since that isn't a multiple of ten. If the result is zero, it's not the answer that's wrong, but the question. You should probably be rounding to the nearest one, or tenth; or using a smaller unit, which amounts to the same thing. You can find information about both rounding and significant digits in our FAQ on Rounding, at http://mathforum.org/dr.math/faq/faq.rounding.html - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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