Rounding Negative and-a-Half Up or Down

Date: 11/10/2010 at 05:37:31
From: Geoff
Subject: rounding negative decimal numbers

To whom it may concern,

We are grade 7 students from Riffa Views International School in Bahrain.
We are currently studying a unit on fractions, decimals and rounding in
math -- and we have a small dilemma.

We are trying to figure out if you have a negative decimal like -3.5,
should you round it to -4 or -3.

We have already learned that for positive numbers, five tenths or more
should be rounded up, while four or less should be rounded down. We think
that 5 goes with 6, but we have searched the Internet and found a number
of different answers for -3.5. Some say you should round up and some say
you round down.

Also, we wonder if negative being opposite means that the rules for a
negative number should be opposite as well. If so, -3.5 should be rounded
to -4. But if .5 should be rounded up, then it would be -3 because -3 is
more than -4.

We appreciate your answers and look forward for your reply.

Best Regards,
Grade 7 students at RVIS
Riffa Views International School
Bahrain
  http://www.rvis.edu.bh/ 

Date: 11/10/2010 at 13:07:12
From: Doctor Peterson
Subject: Re: rounding negative decimal numbers

Hi, Geoff and class.

That's a good question, and you've done some great thinking.

The quick answer is that it doesn't matter much, if at all. Here's why:

Think first about the meaning of "rounding to the nearest integer," which
is the full name of what you are doing. You are looking for the closest
integer to the given number. For example, you round 3.42 to 3 because it
is closer to 3 than to 4. You round -4.78 to -5 because it is closer to 5
than to 4.

Now, what about 3.5? It's the same distance from 3 and 4, so there really
IS no nearest integer. Either 3 or 4 would be equally good as an answer.
The same is true of -3.5: it's exactly between -3 and -4.

Whatever answer you give, it is an arbitrary choice -- and there are
several ways to make that choice. It's common to choose to round 3.5 up to
4 (which appears to be the rule you have been taught) simply because it's
the easiest rule to learn and use. Any other number with a 5 in the tenths
place, such as 3.51, will round up because it is closer to 4 than to 3; so
we lump 3.5 together with all those and say we'll round up whenever the
tenths place is 5 or more. Following that rule (keeping the goal of
simplicity), we would round -3.5 to -4 because -3.51 rounds to -4. So the
rule is to round "away from zero" when the tenths digit is 5 or more.

But there are other choices. Here are some pages on which we have talked
about this (aimed at people at various levels, so some will be of more
interest to your teacher or parents than to you):

  Rounding Decimals: Even/Odd Issues
    http://mathforum.org/library/drmath/view/58972.html 

  Are There Times When Following the Rounding Rules Isn't Best?
    http://mathforum.org/library/drmath/view/70605.html 

  Rounding Negative Numbers
    http://mathforum.org/library/drmath/view/71202.html 

  Teaching Rounding Rules
    http://mathforum.org/library/drmath/view/63989.html 

Ultimately, there is no one standard way to round negative numbers that are exactly 
halfway; you can choose your preferred rule and use that. Probably you'll find that 
the one I suggested is easiest.

If you have any further questions, feel free to write back.

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