Definition:Rational Number/Fraction

Definition
By definition, a rational number is a number which can be expressed in the form:
 * $\dfrac a b$

where $a$ and $b$ are integers.

A fraction is a rational number such that, when expressed in canonical form $\dfrac a b$ (that is, such that $a$ and $b$ are coprime), the denominator $b$ is not $1$.

Examples

 * $(1): \quad \dfrac 1 2$ is a vulgar fraction.


 * $(2): \quad \dfrac 3 1$ is not a fraction, as $b = 1$, and so $\dfrac 3 1 = 3$ which is an integer.


 * $(3): \quad \dfrac 4 2$ is not a fraction.

Although $b \ne 1$, $\dfrac 4 2$ is not in canonical form as $2$ divides $4$, meaning they have a common factor of $2$.

Furthermore, when $\dfrac 4 2$ expressed in canonical form is $\dfrac 2 1$ which, by example $(2)$, is an integer and so not a fraction.


 * $(4): \quad \dfrac 5 2$ is an improper fraction.

It can be expressed as a mixed number as follows:
 * $\dfrac 5 2 = \dfrac {4 + 1} 2 = \dfrac 4 2 + \dfrac 1 2 = 2 \frac 1 2$


 * $(5): \quad \dfrac {24} {36}$ is a vulgar 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.

Denominator
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.

Historical Note
The convention where a bar is placed between the numerator and denominator was introduced to the West by, 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 $\matrix {3 \\ 4}$.

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.