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Rounding Negative Numbers

Date: 05/23/2007 at 17:32:59
From: Paul
Subject: Rounding negative numbers

I was wondering what -0.5 is rounded to the nearest integer?

My first impression was -1, but when we round 0.5 we round up, which 
would suggest -0.5 should be rounded to 0.  Can you even round negatives?

I asked a few mathematics graduates and did not get a convincing 
response, with both 0 and -1 being given.  I thought that possibly you
round the magnitude of the number and then deal with the negative,
giving -1.  However, we round 2.5, 1.5, 0.5 upwards, so to follow the
pattern we would have to round -0.5 to 0.

Date: 05/24/2007 at 10:36:00
From: Doctor Vogler
Subject: Re: Rounding negative numbers

Hi Paul,

Thanks for writing to Dr. Math.  The answer to your question is that 
there are many different ways of rounding.  Some include

  (1) floor: round down to the next integer
        x-1 < floor(x) <= x

  (2) ceiling: round up to the next integer
        x <= ceiling(x) < x+1

  (3) truncate: round toward zero
        trunc(x) = floor(x) when x >= 0
        trunc(x) = ceiling(x) when x <= 0

  (4) antitruncate: round away from zero (this is more rare)
        trunc(x) = ceiling(x) when x >= 0
        trunc(x) = floor(x) when x <= 0

Then there are also the more familiar methods of rounding to  the 
"nearest integer."  But the trouble is that when you have half of an 
odd number, then there are two integers that are equally "near," so 
the above definition is ambiguous.  I think that the most common 
method for resolving this confusion is to round halves up to the next 

  (5) round(x) = floor(x + 1/2)

But you could also argue that this rule was placed so that you only 
have to look one decimal digit past the decimal point, and so you 
want 1.5 to round to the same number as 1.59.  By this logic, you 
would also want -1.5 to round just like -1.59, in which case you 
might switch conventions when you have a negative number:

  (6) round(x) = floor(x + 1/2) when x >= 0
      round(x) = ceiling(x - 1/2) when x <= 0

Of course, there is no reason why you can't do the opposite of either 
of those methods:

  (7) round(x) = ceiling(x - 1/2)

  (8) round(x) = ceiling(x - 1/2) when x <= 0
      round(x) = floor(x + 1/2) when x >= 0

And in more special cases, you might prefer to do something else, like 
round down if the fraction is less than 1/3 and round up if the 
fraction is at least 1/3, as in

  (9) round(x) = floor(x + 2/3)

The bottom line is that "round" is not as well-defined a term  
(mathematically) as the floor (sometimes called "greatest integer") 
and ceiling functions are, and you can choose to use whichever 
definition works best for you.

But my personal opinion, when no more precise definition is given, 
would be to use definition (5), despite my argument for definition 

If you have any questions about this or need more help, please write 
back, and I will try to offer further suggestions.

- Doctor Vogler, The Math Forum 

Date: 05/24/2007 at 15:06:55
From: Paul
Subject: Thank you (Rounding negative numbers)

Thanks for the help.
Associated Topics:
High School Negative Numbers
High School Number Theory

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