Placement of the Negative Sign in a Negative Mixed Number

Date: 10/25/2004 at 16:35:09
From: Julia and Sam
Subject: Negative Fractions

When you have a negative improper fraction, does the negative sign 
automatically distribute throughout your fraction? For instance, is 
-1 3/5 the same as 1 -3/5 and 1 3/-5?  We have an intuitive sense that
the fractions are equal, but we are having trouble proving it. 

If this is true, why is it the case?  Your help is much appreciated!

Julia and Sam

Date: 10/25/2004 at 22:51:24
From: Doctor Peterson
Subject: Re: Negative Fractions

Hi, Julia.

Mixed numbers don't play well with negative signs, or with algebraic
notation in general.  When we write 1 3/5, we really mean (1 + 3/5),
and the negative -1 3/5 means -(1 + 3/5), which distributes as

  -1 + -3/5

So

  -1 3/5 = -(1 + 3/5) = -8/5

  1 -3/5 = 1 - (3/5) = 2/5

  1 3/-5 = (1)(3/-5) = -3/5

Or at least that's how I would interpret each expression if I came
across it out of context!  Only the first contains cues that strongly
suggest a mixed number is intended, since we never write mixed numbers
with negative fractional parts.

Generally, it's better to use improper fractions instead of mixed
numbers in algebra, to avoid not only that source of confusion, but
also something like 1 3/5 x, where the first space means addition, the
second means multiplication, and parentheses have to be supplied.

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


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