# 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 |

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 $3$ *whole*s 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,

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:

### Fractions

- What Is Fraction?
- Operations on Fractions
- Equivalent Fractions
- Fraction Comparison: An Interactive Illustration
- Compare Fractions: Interactive Practice
- Fraction Comparison Sped up
- Counting and Equivalent Fractions
- Product of Simple Fractions
- What's a number? (Rational number in particular)
- Why 1/3 + 1/4 = 7/12?
- Fractions on a Binary Tree
- Fractions on a Binary Tree II
- Archimedes' Law of the Lever

### References

- S. Schwartzman,
*The Words of Mathematics*, MAA, 1994

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