What Is Fraction?

That boy who holds three quarters of his heart and all the best of him. While I hold one third and all the worse.

Daphne de Maurier
My Cousin Rachel
Black Dog & Leventhal Publishers, 2006, p. 117

A fraction (sometimes, a common fraction) is a way of expressing a number that is a ratio of two integers:

$\displaystyle\frac{p}{q}$ or $p/q$ or even as $p\div q.$

The top (or the first) number is called the numerator, the bottom (or the second) number is called the denominator. The numerator answers the question, How many? The denominator specifies the answer to that question, How many of what? Accordingly, $p/q$ is pronounced "$p$ $q$-th," meaning $p$ parts each equal to the $q$-th part of a whole. (This we may meaningfully refer to as the u-property or u-aspect of a fraction, that is interpreted as the number of fractional units.) When the numeric aspect of the fraction needs to be emphasized, "$p/q$" is pronounced simply "$p$ over $q$" or "$p$ divided by $q$."

What is the whole? It depends. What is $1,$ or what is $5?$ These are abstractions that are used to indicate quantity: $1$ apple, $1$ acre, $1$ dollar, $5$ apples, $5$ acres, $5$ dollars. In the same manner $2/3$ is an abstraction that indicates a quantity of apples, areas, or the balance in a bank account. $2/3$ of an apple is two parts of an apple that was divided into three parts. To determine $2/3$ of $5$ apples one needs to divide $5$ apples into three equal parts and take any two of them. How does one divide $5$ apples into three equal parts? Here's one possibility: divide each of the apples into three parts and combine five of the pieces at a time. (This of course assumes that the apples are equal and one is sufficiently skillful to divide each into three equal parts.) Incidentally, it shows that $2/3$ of $5$ apples equals $10/3$ of an apple. When Mr. Smith's backyard is said to be $2/3$ of Mr. Johnes' the intention is to their areas: the latter is bigger than the former and, more accurately, if Mr. Johnes' backyard measures, say, $3$ acres, then that of Mr. Smith's is $2$ acres. $2/3$ of a bank account balance is $6$ if the whole is $9,$ and is $60$ if the whole is $90.$

According to S. Schwartzman

fraction (noun), fractinal (adjective): from Latin fractus, past participle of frangere "to break," which is the native English cognate. The Indo-European root is bhreg-, of the same meaning. Related borrowings from Latin include fragile (= breakable), diffraction (breaking up into colors) and fragment. A fraction is literally a piece broken off something. In fact, in 16th century English mathematics books, fractions were sometimes referred to as "broken numbers."

(The term Algebra derives from the Arabic gebr which means "reunion of broken parts.")

The number expressed by a fraction has infinitely many fractional representations. Three fifths of a whole is the same as six tenths or nine fifteenth (of the same whole), etc. So that

$3/5 = 6/10 = 9/15 = ... = 81/135 = ...$

Going backwards, we conclude that, unless the numerator and the denominator of a fraction, are mutually prime, i.e. if they have no common factors except for $1,$ the fraction can be simplified. In other words, for a fraction $p/q$ in which $p$ and $q$ have common (non-trivial) factors, there is always another fraction, say, $r/s$ equal to $p/q,$ with $r$ and $s$ smaller (in absolute value) than $p$ and $q,$ respectively. For example, $81/135 = 9/15$ because $81$ and $135$ have $9$ as a common factor. The fractions that can't be simplified are said to be in lowest terms. For such fractions, the numerator and denominator have no common factors, in other words, are mutually prime or coprime. Every fraction can be reduced to lowest terms. For example, $81/135 = 3/5.$

Fractions that represent the same number are said to be equivalent, or simply equal (as in, say, $3/5 = 81/135.)$ Numbers represented by fractions are called rational. A rational number whose denominator in lowest terms is $1$ is integer: $p/1 = p.$

A fraction with $0 \lt p \lt q$ is called proper or simple. The denominator is usually positive. If $|p| \ge |q|,$ the fraction is improper. $\frac{2}{3}$ is a proper fraction; $\frac{10}{3}$ is improper. Improper fractions are often written in a mixed form, for example $\frac{10}{3} = 3\frac{1}{3},$ meaning $3$ wholes and $1$ additional third. Indeed, $10$ apples may be put into three groups of three apples with an apple left over. Similarly, $10$ thirds equal three times three thirds with one third left over. But three thirds is $1,$ a whole. Thus indeed $\frac{10}{3} = 3\frac{1}{3}.$ Omitted here is the symbol of addition.

Two fractions $p/q$ and $r/s$ can be made to have common denominator, for example,

$\displaystyle\frac{p}{q}=\frac{ps}{qs}$ and $\displaystyle\frac{r}{s}=\frac{rq}{sq}$

This an extremely useful operation that helps define arithmetic operations on fractions. Observe that as there are infinitely many ways to represent a rational number as a fraction, i.e., there are infinitely many equivalent fractions, there are also infinitely many ways to write two fractions with a common denominator. Taking their product $qs$ is only one possibility. Sometimes, but not always, one needs to find the representation in which the common denominator is the least possible. For example,

$\displaystyle\frac{5}{6} + \frac{3}{8} = \frac{40}{48} + \frac{18}{48} = \frac{58}{48}.$

But it is also true that

$\displaystyle\frac{5}{6} + \frac{3}{8} = \frac{20}{24} + \frac{9}{24} = \frac{29}{24}.$

Clearly the two results are equal and either way is good enough to add the fractions. However, (finding and) using the least common denominator results in a fraction in lowest terms which may make it more amenable to further handling.

Fractions with $100$ in the denominator, like $3/100,$ $71/100,$ and so on, play an important role in practical matters, business, finance, statistics, etc. They have a more convenient typographical notations: $3/100 = 3%,$ $71/100 = 71%.$ In this form, the fractions are called percents: $3$ percent, $71$ percent.

Addition

$3$ apples plus $5$ apples make $8$ apples. $3$ cars plus $5$ cars make $8$ cars. $3$ tenths of an apple plus $5$ tenths of an apple make $8$ tenths of an apple. In general, $3/10 + 5/10 = 8/10.$ And even more generally,

$\displaystyle\frac{p}{q}+\frac{r}{q}=\frac{p+r}{q}.$

And what if the fraction do not have the same denominator? Well, make them to have one and, subsequently, add the two:

$\displaystyle\frac{p}{q}+\frac{r}{s}=\frac{ps}{qs}+\frac{rq}{sq}=\frac{ps+rq}{qs}.$

Subtraction

The same holds for subtraction. If the fractions have the same denominator, subtract their numerators. Otherwise, first arrange for the fractions to have a common denominator:

$\displaystyle\frac{p}{q}-\frac{r}{s}=\frac{ps}{qs}-\frac{rq}{sq}=\frac{ps-rq}{qs}.$

Division

Being able to produce fractions with a common denominator is especially useful in defining division of two fractions. If a group of $10$ apples is to be divided into groups of $2$ apples each, then there are going to be $5$ groups (of $2$ apples). This is because $(10$ apples) divided by $(2$ apples) gives $5.$ Similarly $(10$ sevenths) divided by $(2$ sevenths) gives $5.$ And of course $(10$ thirteenths) divided by $(2$ thirteenths) also gives $5$ as does $(10$ hundredths) divided by $(2$ hundredths), and so on.

So to divide $p/q$ by $r/s,$ first make them have a common denominator: $p/q = ps / qs$ and $r/s = rq / sq.$ On the second step, simply divided their numerators:

$\displaystyle\frac{p}{q}\div\frac{r}{s}=\frac{ps}{qs}\div\frac{rq}{sq}=\frac{ps}{rq}.$

Multiplication

Multiplication of the fractions is in a sense the simplest of the four arithmetic operations:

$\displaystyle\frac{p}{q}\times\frac{r}{s}=\frac{pr}{qs}.$

The reasoning behind this definition is this. $1$ third of $1$ fifth is $1$ fifteenth, meaning that $\frac{1}{3} \times \frac{1}{5} = \frac{1}{15}.$ When we multiply two "units" we create a new unit. Some unit products do not make much sense, but some do. For example, distance × force = work. Luckily, fractional units can always be multiplied: $\frac{1}{q} \times \frac{1}{s} = \frac{1}{qs}.$ If there are $p$ of $q\mbox{ths}$ and $r$ of $s\mbox{ths,}$ then there are $pr$ of $(qs)\mbox{ths,}$ which just reiterates the definition. (An interactive tool to practice multiplication of fractions is available on a separate page.)

Observe the relation of the definitions of division and multiplication:

$\displaystyle\frac{p}{q}\div\frac{r}{s}=\frac{ps}{rq}=\frac{p}{q}\times\frac{s}{r}.$

This explains the notoriously infamous rule for the division of fractions: write the second fraction upside down and multiply.

Comparison of Fractions

The matter of comparing fractions is simple: if their difference is positive the first fraction (minuend) is greater than the second (subtrahend). The way to find the difference is to see to it that the fraction are written with a common denominator. With a caveat, this can be stated a little differently; there is a separate page with an interactive gizmo devoted to the comparison of fractions.

Here are additional pages related to the definitions, properties of and operations on, fractions:

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References

  1. S. Schwartzman, The Words of Mathematics, MAA, 1994

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