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/