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### What Exactly is a Fraction?

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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.
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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

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.

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/
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Associated Topics:
High School Number Theory
Middle School Fractions

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