Different Techniques for Rounding Off Decimal Numbers
Date: 01/28/2006 at 22:00:57 From: Keith Subject: In rounding decimals why don't you count for the complete # I am a little split on the different ideas of rounding decimals. I understand that in basic math they tell you to take the decimal place just to the right of what you are going to round to (ex: 1.45 to the tenth is 1.5). However what I was taught in advanced mathematics is when you are taking a number that has multiple numbers beyond the decimal place that you should take the complete number in account when rounding. (ex: 1.4457 would round to 1.5 or 2.0). I don't know if it is just the way different teachers teach or maybe it is a different form of math that involves this kind of rounding. Could you help clarify?
Date: 01/28/2006 at 23:02:04 From: Doctor Peterson Subject: Re: In rounding decimals why don't you count for the complete # Hi, Keith. I'm not positive which of several issues you are raising (in part because I don't see that you would ever round 1.4457 to either 1.5 or 2.0), so I'll cover a couple different ideas pertaining to rounding. First, the essential idea in rounding is stated in the words we use when we state a problem fully: Round 1.45 to the nearest tenth. This means Find the multiple of 0.1 that is closest to the number 1.45. If we just do exactly what this says (which does, indeed, take the entire number into account), then we can't help getting the right answer--if there really is one! The two nearest multiples of 0.1 are the tenth above and the tenth below, namely 1.4 and 1.5. If our number had been, say, 1.445, we could find how far it is from 1.4 and from 1.5, and choose 1.4 because that is, indeed, closer. In the case we are considering, however, we find that both numbers are exactly the same distance from 1.45, so there really is NO correct answer. So if we want to have one "correct" answer, we have to arbitrarily choose one of the two. This is where trouble comes in: we have a choice, and we might make different choices depending on our concerns. In teaching children, and in using rounding for simple purposes such as estimation, we want the simplest possible method. A reasonable way to do it is this: Any number between 1.4 and 1.45 will round down; any number between 1.45 and 1.5 will round up. The first group all have the NEXT digit less than 5 (0, 1, 2, 3, or 4); the second group all have their next digit five or more (5, 6, 7, 8, or 9). If we arbitrarily choose to round 1.45 up, then the rule becomes very simple: ANY number with a 5 in the next digit will round up. This rule gives the correct answer whenever there is one correct answer, and gives a valid answer when there are two. In some settings, we care about the statistical properties of our rounding; we don't want to skew averages by always rounding up in this odd situation, so we'd like to round up half the time and round down half the time. One common solution is to always round such "exactly between" numbers so that the last digit is EVEN. Then our 1.45 will round to 1.4 rather than to 1.5. Note that this rule is identical to the other in all cases except exact halfway numbers like 1.45; both methods give the same answer when there is only one valid answer. It makes a different choice in the special case. And it "looks at the entire number" only in the sense that it checks whether there are any other digits following a 5. If there are, then we follow the basic rule; if not, we round to an even number. But there is nothing else beyond the next digit that matters. Even this method can lead to biases (for example, it will lead to too many even answers!); so if that matters, you might need to just randomly decide whether to round halfway numbers up or down. The point is that there are different ways to do it, depending on what is important to you, but the basic idea (and the result in almost all cases) is the same, and is governed by the definition of the problem. For some discussions of various methods, and the reasons for them, see Rounding Numbers http://mathforum.org/library/drmath/sets/select/dm_rounding.html 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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