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### Rounding Down to Nothing

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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.
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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.

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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Associated Topics:
Elementary Place Value

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