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Origin of the Term "Improper" Fraction

Date: 07/28/2006 at 13:15:24
From: AnneMarie
Subject: origin of the term "improper" fraction

I love your website and use it often.  Do you know why fractions with
larger numerators than denominators are called "improper"?  Where did
Just curious! 

Thank you.  AnneMarie

Date: 07/28/2006 at 14:28:05
From: Doctor Ian
Subject: Re: origin of the term 

Hi AnneMarie,

It might help to consider some other uses of "proper" in mathematics.  

For example, what are the divisors of 12?  They are

  1, 2, 3, 4, 6, 12

But a number is _always_ a divisor of itself, so we use the term
"proper divisor" to refer to the _other_ divisors.

Similarly, a set is always a subset of itself, so we use "proper
subset" to refer to subsets missing at least one element.  

You can find other examples by searching for the keyword "proper" at


So by extension, we might consider "proper" to be used, when talking
about fractions, to exclude the analogous case, i.e., where the
numerator isn't at least one less than the denominator.  

Does that make sense? 

- Doctor Ian, The Math Forum

Date: 07/28/2006 at 15:31:35
From: Doctor Peterson
Subject: Re: origin of the term 

Hi, AnneMarie.

To add a bit to what Dr. Ian said, I have two slightly different ways 
to approach the question.

First, historically, if I want to strictly answer your question, I 
should tell you WHERE the usage first occurred.  For that, we look 

  Earliest uses of mathematical words

That site says

  PROPER FRACTION appears in English in 1674 in Samuel Jeake
  Arithmetic (1701): "Proper Fractions always have the Numerator
  less than the Denominator, for then the parts signified are less
  than a Unit or Integer" (OED2). 

Second, to find WHERE a word comes from, and WHY, we can look in a 
dictionary.  I looked up "proper" and found that it originally meant 
"belonging to one" or "peculiar to something"; then came to mean 
"appropriate or suitable" and "strictly correct".  That last meaning, 
I think, is where the common mathematical use originated.  A "proper 
fraction" is one that really is a fraction (a broken part, as it is 
etymologically) in a strict sense, rather than following the 
extended meaning mathematicians have given it (anything expressed as 
a/b).  In same way, a "proper subset" is REALLY a subset, strictly 
speaking (a part, not the whole), and a "proper divisor" is STRICTLY 
a divisor, not just the same number, and so on. 

Another related usage is "the city proper", meaning "the city itself, 
taken in a narrow sense rather than a broad sense".  In m-w.com, where 
I looked, this is defined as "strictly limited to a specified thing, 
place, or idea".  Of all the definitions they give, I think this is 
the closest to the general mathematical usage.

Finally, taking Dr. Ian's advice, I found this page:


  In general, the opposite of trivial. 

  SEE ALSO: Proper Divisor, Proper Ideal, Proper Subset, Proper
  Class, Proper Submodule, Proper Fraction, Proper Subfield, Proper
  Extension, Proper Superset, Trivial. 

Taking that general sense of the word, one might wonder whether it 
would also exclude "trivial" cases at the other end of the spectrum: 
are the fraction 0/1, the subset {}, and the divisor 1 "improper"? 
Usually, the answer is no; but sometimes one might want to take it 
that way:

  Proper Divisor

  To make matters even more confusing, the proper divisor is often
  defined so that -1 and 1 are also excluded.  Using this
  alternative definition, the proper divisors of 6 would then be
  -3, -2, 2, and 3, and the improper divisors would be -6, -1, 1,
  and 6.

We've more than answered your question, but I think it's interesting!


- Doctor Peterson, The Math Forum
Associated Topics:
Elementary Definitions
Elementary Fractions
Middle School Definitions
Middle School Fractions

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