Definition:Rational Number/Fraction

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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 $1$.



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Vulgar Fraction

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


Proper Fraction

A proper fraction is 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 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 $p > q$.


Mixed Number

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

$p$ and $q$ are integers such that $p < q$
$r = n + \dfrac p q$


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.


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.


Also known as

Some sources use the more unwieldy term fractional number.


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.


Sources


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