Comparing and Rounding Decimals

Date: 08/08/2001 at 00:50:06
From: Justin
Subject: Comparing and Rounding Decimals

The length of each day of the year changes by about 0.000000002 s. 
Write a length of time that could be rounded up to this figure.

Date: 08/08/2001 at 12:23:56
From: Doctor Peterson
Subject: Re: Comparing and Rounding Decimals

Hi, Justin.

Let's try a similar problem, so you can solve your own the same way.

Suppose you wanted to find a number that could be rounded up to 0.04. 
We know it must be less than 0.04, since it is being rounded up. I'm 
also going to assume that you round the number up because that's the 
right thing to do when you round to the NEAREST something, rather than 
just because you were told to round it up rather than down.

Now, what might we have been told to round it TO? Well, 0.04 is a 
multiple of 0.01, so you might be rounding to the nearest hundredth. 
It's possible you are rounding to the nearest thousandth or 
ten-thousandth, instead, since 0.04 is also a multiple of those; but 
we'll take the easiest possibility.

What numbers below 0.04 would round to 0.04 as the nearest hundredth? 
Those that are closer to 0.04 than to 0.03, since otherwise they would 
round to 0.03 (or lower). So any numbers between 0.035 and 0.040 will 
work:

                 |<-->|
    <-------+---------+------>
           0.03      0.04

You can choose any number in this range. It will start with 0.03, and 
the next digit will be 5 or more. So possible answers include 0.035, 
0.03999, and so on.

Do the same sort of thinking for your problem. You'll just have more 
zeros.

If you'd like more information on rounding, check out our Dr. Math 
FAQ:

   Rounding Numbers
   http://mathforum.org/dr.math/faq/faq.rounding.html   

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/