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 proper 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 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.

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.

Also known as
Some sources use the more unwieldy term fractional number.