What is a Fraction, Really?

(An archive question of the week)

One of the things I have learned as a Math Doctor is that it can be dangerous looking up a definition online. Sources vary — not because they are wrong, but because definitions depend on context, so you can easily find what appear to be contradictions because they refer to different usages. Today’s question not only reflects a variety of definitions, but evoked different answers from two of us!

Looking for the “correct” definition

The question, from 2001, referred to three sources, none of which still exists, but the ideas are typical:

What Exactly is 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 expresses a portion of a whole. The denominator of a fraction represents the number of 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 out 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...

Sridhar’s examples include 3/1 (an improper fraction that simplifies to an integer), 5/√2 (a fraction including an irrational quantity), and -3/5 (a negative proper fraction); the quoted definitions don’t mention such issues. On reading these definitions now, my own first concern is the emphasis on “parts of a whole”, which has some unintended implications, restricting one’s thinking to a particular model of a fraction (which is appropriate for early grades, but needs to be left behind eventually). But the main issue is that, like many definitions, they are incomplete. That may be intentional.

A better definition

Doctor Rick answered first, focusing on refining the definitions to include some essentials that they all omit:

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

This definition removes the model of “parts of a whole”, replacing it with an abstract model; and adds a restriction that the numerator and denominator are integers. At lower levels, the latter is probably just assumed.

He continues:

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.)

The definitions are probably intended to be used before learning about negative numbers (or in contexts where nothing is negative), so it makes some sense not to mention that possibility; but they really should not exclude improper fractions like 3/2.

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.

The root meaning: proper fractions

I had been pondering the question myself, and an hour later submitted my answer. I focused on variation in usage:

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.

So, one meaning in everyday use is “less than a whole”; and this root is reflected in our calling a fraction that is greater than one “improper” – it doesn’t fit the root idea of what a fraction is, but is an extension.

The usual meaning: common fractions

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.

So the everyday mathematical use is just what Doctor Rick discussed: an integer over an integer. Again, there is a special term reflecting this particular usage; beyond “proper” fractions, we have the larger category of “common” fractions.

As for “vulgar fraction”, that derives from an early use of the word “vulgar”, referring as it did in Latin to the “ordinary” or “common” people, as opposed to the nobility. In common use, the word now means “unrefined” or worse; but occasionally it still has this less judgmental meaning.

Generalizing: algebraic fractions

Now we move beyond arithmetic into algebra, where “fraction” takes on a larger meaning (which is hinted at especially in the third definition above):

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.

Here we move beyond parts and wholes, and recognize fractions as representations of division, as I have discussed elsewhere. We can now divide any sort of quantities, not just integers. The term “indicated division” reflects what I have said elsewhere, “a fraction is simply a division we haven’t bothered to perform yet, a division problem frozen in time.”

What about “decimal fractions”?

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!

It turns out that I lied earlier, when I said, “a fraction always refers to a way of writing a number, using numerator and denominator”. This usage is generally restricted to mathematicians who want to be precise – they can’t call “.123” a “decimal”, as we usually do, because that literally means “base ten”, and says nothing about being less than one! (Properly speaking, a “decimal fraction” should be one whose denominator is a power of ten, but still written as a fraction: “123/1000”; but we have no better term.) This conundrum becomes most confusing when deciding what to call a “decimal” form in a non-decimal base; is it, say, a “base-two fraction”? It certainly can’t be called a “binary decimal”!

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.

Again, “fraction” refers to the way a number is written, not to the number itself and its properties. In fact, this is why we need the term “rational number”, so we can talk about any number that can be written as a fraction, without caring how it happens to be written at the moment.

Extras

For more about the term “proper fraction”, see

Origin of the Term "Improper" Fraction

For the difference between fractions and rational numbers, see

Introduction to Rational Numbers for Middle School Students

Why Do We Need to Study Rational Numbers?

For the difference between numbers and numerals, see

Numbers and Numerals

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