Is the Fraction N/N Considered Proper or Improper?

Date: 10/31/2003 at 09:22:47
From: Richard Menor
Subject: Fraction terminology

I am developing a technical math review program for adults and have a
question regarding the correct term to use when attempting to identify
the fraction n/n when n is greater than 0.  It does not seem to fit
into any of the common fraction categories, (ie, proper, improper,
mixed).  I understand that it can be identified as "1", but was
wondering if n/n is just a unique situation that does not fit into any
of the fraction groups.  It would seem that it could be identified as
a mixed number, (n/n = 1 0/n but that seems too obvious).  Any
comments would be appreciated.

Date: 10/31/2003 at 10:30:56
From: Doctor Peterson
Subject: Re: Fraction terminology

Hi, Richard.

It certainly is not "mixed", since it has only a fractional part, and 
no whole part.  The numerator of a "proper" fraction is less than the 
denominator, so I would call this an improper fraction.  You can find 
definitions of "improper" that say the numerator has to be greater,
not just equal; but I think that is wrong.  See this page, which is a
little ambivalent:

  Improper Fraction
    http://mathworld.wolfram.com/ImproperFraction.html 

  A fraction p/q > 1. A fraction with p/q < 1 is called a proper
  fraction. Therefore, the special cases 1/1, 2/2, 3/3, etc. are
  generally considered to be improper.

According to Merriam-Webster (m-w.com), an improper fraction is

  a fraction whose numerator is equal to, larger than, or of equal
  or higher degree than the denominator.

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

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

Date: 10/31/2003 at 13:02:17
From: Richard Menor
Subject: Thank you (Fraction terminology)

Thanks for quick response.  The books that I referenced indicated that
improper fractions had numerators larger than denominators.

Thanks again.

Richard