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/