# Definition:Fraction

## Definition

A fraction is an expression representing a quotient of one number (or expression) by another number (or expression).

It is usually expressed in the form:

$\dfrac a b$

or:

$a / b$

where $a$ and $b$ are either numbers or expressions.

The term fraction is usually encountered when $a$ and $b$ are integers.

In this case, the fraction $\dfrac a b$ represents a rational number.

### Vulgar Fraction

A vulgar fraction is a fraction representing a rational number whose numerator and denominator are both integers.

### Proper Fraction

A proper fraction is a fraction representing a rational number whose absolute value is less than $1$, expressed in the form $r = \dfrac p q$, where $p$ and $q$ are integers.

### Improper Fraction

An improper fraction is a fraction representing a rational number whose absolute value is greater than $1$.

Specifically, when expressed in the form $r = \dfrac p q$, where $p$ and $q$ are integers such that (the absolute value of) the numerator is greater than (the absolute value of) the denominator: $\size p > \size q$.

### Mixed Fraction

A mixed fraction is a representation of a rational number whose absolute value is greater than $1$, expressed in the form $r = n \frac p q$ where:

$n$ is an integer
$\dfrac p q$ is a proper fraction, that is, $p$ and $q$ are integers such that $p < q$.

### Complex Fraction

A complex fraction is a fraction such that the numerator or denominator or both are themselves fractions.

## Also defined as

Many sources define a fraction solely in the context of numbers.

Specifically, a fraction in such a context will be used to denote a rational number $\dfrac a b$ such that both $a$ and $b$ are integers.

Such sources will typically also demand that the fraction specifically represent a non-integer.

Thus $\dfrac 3 1$ and $\dfrac 4 2$ will not be considered as actual fractions, as they represent the integers $3$ and $2$ respectively.

## Terms of Fraction

The terms of a fraction are referred to as the numerator and the denominator:

### Numerator

The term $a$ is known as the numerator of $\dfrac a b$.

### Denominator

The term $b$ is known as the denominator of $\dfrac a b$.

A helpful mnemonic to remember which goes on top and which goes on the bottom is "Numerator Over Denominator", which deserves a "nod" for being correct.

## Examples

$(1): \quad \dfrac 1 2$ is a proper fraction.
$(2): \quad \dfrac 5 2$ is an improper fraction.

It can be expressed as a mixed fraction as follows:

$\dfrac 5 2 = \dfrac {4 + 1} 2 = \dfrac 4 2 + \dfrac 1 2 = 2 \frac 1 2$

$(3): \quad \dfrac {24} {36}$ is a proper fraction, although not in canonical form.

It is found that when $\dfrac {24} {36}$ is expressed in canonical form:

$\dfrac {24} {36} = \dfrac {12 \times 2} {12 \times 3} = \dfrac 2 3$

its denominator is not $1$.

Hence $\dfrac {24} {36}$ is indeed a vulgar fraction.

## Also known as

Some sources introduce the more unwieldy term fractional number for fraction.

This will be the case only when such a fraction denotes a rational number.

## Also see

• Results about fractions can be found here.

## Historical Note

The consideration of fraction was the next development of the concept of a number after the natural numbers.

They arose as a matter of course from the need to understand the process of measurement.

The convention where a bar is placed between the numerator and denominator was introduced to the West by Fibonacci, following the work of the Arabic mathematicians.

Previous to this, fractions were written by the Hindu mathematicians without the bar.

Thus $\dfrac 3 4$ would have been written $\ds {3 \atop 4}$.

## Linguistic Note

The word fraction derives from the Latin fractus meaning broken.

This is in antithesis to the concept of integer, which derives from the Latin for untouched, in the sense of whole, or unbroken.

Colloquially, informally and rhetorically, the word fraction is typically used to mean a (small) part of a whole, and not in the sense of improper fraction.