What Exactly is a Fraction?

Date: 10/15/2001 at 07:17:48
From: Sridhar Rajagopalan
Subject: What (exactly) are fractions? Is 3 a fraction?

What is a fraction? Is 3/1 a fraction? Is 5/sqrt(2) a fraction?

I tried to check some internet glossaries, and there does not seem to 
be unanimity. The TUSD Math Glossary says:
"Fraction: a numeral representing some part of a whole; a numeral of 
the form a/b (meaning a divided by b) where b is not zero."
  http://instech.tusd.k12.az.us/Core/glossary/mathglossary.htm   

The _Math to Build On_ glossary says:
"Fraction - A number that expressed a portion of a whole. The 
denominator of a fraction represents the number of the portions the 
whole has been divided into, and the numerator expresses the number 
of the portions measured. The fraction 1/4 could be stated as 1 our 
of 4 parts of the whole."
   http://mathforum.org/~sarah/hamilton/ham.glossary.html   

The MathPro Press On-line Mathematics Dictionary says:
"Fraction: An expression of the form a/b."
   http://pax.st.usm.edu/cmi/inform_html/glossary.html#F   

(This reminds me of the definition of a trapezium. Some say that a 
trapezium is a quadrilateral with one pair of sides parallel. Some 
say that plus, the other pair of opposite sides should not be 
parallel.)

Personally, I am comfortable with having identical definitions of 
fractions and rational numbers. But I have seen many sources defining 
fractions in such a way that -3/5 is not a fraction, only a rational 
number...

Regards.

Date: 10/15/2001 at 08:37:20
From: Doctor Rick
Subject: Re: What (exactly) are fractions? Is 3 a fraction?

Hello, Sridhar.

You've asked a good question, but since there is no official 
international governing body for mathematical definitions (as far as I 
know), you'll only get another viewpoint from me, not a definitive 
answer.

I am not particularly comfortable with having identical definitions of 
fraction and rational number, for the simple reason that then we would 
have no reason to keep both words in our lexicon.

I am not entirely pleased with any of the definitions you list, 
because they all omit an essential requirement. I would go with this 
version:

  A fraction is a representation for a rational number, in the form
  a/b, where a is an integer, and b is a *positive* integer. The 
  number so represented is the result of dividing a by b.

Do you see what was missing in your definitions? This is NOT a 
fraction:

  1.2
  ---
  4.8

Your first definition comes closest to mine. I like the word "numeral" 
in that definition, as distinct from "number." The same *number* can 
be represented by the *numerals* 3/2 or 1.5; only the first is a 
fraction. Likewise, 3 can be represented by the numeral (fraction) 
3/1, but the number itself is not therefore a fraction.

I have no problem with calling -2/5 or 3/2 a fraction; both are 
improper fractions, representing numbers outside the range 0 to 1. 
This makes me somewhat uncomfortable with the phrase "representing 
some part of a whole," which strictly limits the definition to proper 
fractions. This description is helpful in introducing the concept of 
fractions, but once improper fractions are introduced, it no longer 
belongs in the definition.

I see no need to have the definition so broad that 2/-5 is a fraction. 
As it stands, every rational number can be represented as a fraction. 
(In fact, *only* rational numbers can be represented as a fraction, so 
this makes for a simple definition of rational numbers.)

I see no need, on the other hand, to restrict the definition so that 
every rational number has a *unique* representation as a fraction. 
Thus, 4/6 and 2/3 represent the same rational number. There is, 
however, a unique representation as a fraction *in lowest terms*.

These are my opinions. I will leave the question for other Math 
Doctors to respond to if they have different or additional viewpoints.

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

Date: 10/15/2001 at 09:29:37
From: Doctor Peterson
Subject: Re: What (exactly) are fractions? Is 3 a fraction?

Hi, Sridhar.

I would like to take a slightly different perspective from Dr. Rick's 
on this. I fully agree with him on the most precise definition of 
"fraction": it is a particular representation of a rational number. 
But it is worth noting that, like many other words, it actually has 
several different meanings depending on the context.

The root meaning of "fraction" is "broken piece," which is the source 
of the idea that it must be a part of a whole. This is common in 
informal use; if I say "only a fraction of the people here understand 
what the word means," I mean "less than the whole," and probably 
much less. I am saying nothing about whether those people are 
"rational." ;-) In mathematical terms, this concept arises in the 
phrase "proper fraction" - that is, a mathematical fraction that fits 
the informal sense that it should be less than one.

Once we get into the mathematical realm, a fraction always refers to a 
way of writing a number, using numerator and denominator. Most 
narrowly, these must be whole numbers (or integers, once children are 
introduced to negative numbers). This kind of fraction is more fully 
called a "common" or "vulgar" fraction, and this is what Dr. Rick 
defined for you. When we use the word "fraction" alone, we usually 
mean this.

From here we find a broader definition, given in my American Heritage 
dictionary as "an indicated quotient of two quantities." This retains 
the concept that a fraction is a way of writing something, as distinct 
from its actual value, but ignores questions as to what sort of 
quantities are being divided. This allows for "fractions" like 1.2/3.4 
or (x+1)/(y-1). The term is in fact used in these senses; the latter 
might be called an algebraic fraction. In this realm we can 
distinguish between "simple fractions" and "complex or compound 
fractions" like (1/2)/(3/4); common fractions are always simple.

Finally, we see the phrase "decimal fraction" used for non-integral 
decimal numbers; this drops the "indicated quotient" concept and 
retains only the "broken" (non-integral) aspect of the most basic 
definition. I don't think this meaning is ever intended when we use 
the word "fraction" without qualification; in fact, we more often drop 
the word "fraction" and just call it a "decimal," which can be a 
dangerous practice!

Clearly the term "fraction" has a somewhat different meaning in each 
case. 

To answer your specific questions, 3/1 is a fraction (specifically a 
simple, common, but improper fraction), while 5/sqrt(2) would only be 
called a fraction in an algebraic context. On the other hand, 3 is not 
a fraction, even though it is a rational number.

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