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### 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
```
Associated Topics:
Middle School Definitions
Middle School Fractions

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