Can Negative Fractions Also Be Proper or Improper?

Date: 03/31/2004 at 17:11:53
From: Laurel
Subject: proper and improper fractions

How do you classify negative fractions?  Is a fraction like -3/4 
considered a proper fraction?  Is -8/3 considered an improper 
fraction?  I've generally thought that all proper fractions lie 
between 0 and 1 on a number line.  However, what about the negatives?

Date: 03/31/2004 at 23:27:43
From: Doctor Peterson
Subject: Re: proper and improper fractions

Hi, Laurel.

The usual definitions are aimed at kids who don't know about negative 
numbers; it's rare to see a definition that takes negatives into 
account.  When you consider negative fractions, it is the absolute 
value that has to be less than 1 for a proper fraction; a fraction 
between -1 and 1 therefore is proper.

An alternative way to deal with this might be to restrict the word 
"fraction" to positive numbers, and use "negative improper fraction" 
to refer to "the negative of an improper fraction".  That avoids the 
question, but might sound too tricky to kids.  I can't be sure which 
way people generally think of it when they do talk about negative 
fractions; I just know that -8/3 is improper and -2/3 is proper.

I like to confirm my impression in cases like this, so I searched on 
Google for "negative improper fraction", but I found nothing but 
this, which doesn't clarify the exact meaning:

  Positive and negative numbers
    http://www.mathleague.com/help/posandneg/posandneg.htm 

  What is the reciprocal of -5 1/8? First, we convert to a
  negative improper fraction: -5 1/8 = -41/8, then we switch the
  numerator and denominator, and keep the same sign: -8/41.

Unfortunately, the Math League's definition of improper fractions
ignores negatives:

  Fractions
    http://www.mathleague.com/help/fractions/fractions.htm 

  Improper fractions have numerators that are larger than or equal
  to their denominators. 

  Examples: 

  11/4, 5/5, and 13/2 are improper fractions.

I'm inclined to interpret "larger than" here as "greater in absolute 
value" in order to make the definition work with negatives.  That is, 
p/q is improper if |p| >= |q|.

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

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

Date: 03/31/2004 at 23:48:46
From: Laurel Brenner
Subject: Thank you (proper and improper fractions)

Thank you, Dr. Peterson!  I am a 6th grade math teacher of a top level 
class.  When I asked the students to write where all proper fractions 
lie on a number line, the answer I expected was between 0 and 1.  But 
one student claimed that they were between -1 and 1.  I'd never 
thought of that, and I wanted to be sure before I saw her again that 
it was, in fact, correct.  I appreciate your very prompt reply.