Definition:Rational Number

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Informal Definition

A number in the form $\dfrac p q$, where both $p$ and $q$ are integers ($q$ non-zero), is called a rational number.

The set of all rational numbers is usually denoted $\Q$.


Thus:

$\Q = \set {\dfrac p q: p \in \Z, q \in \Z_{\ne 0} }$


Formal Definition

The field $\left({\Q, +, \times}\right)$ of rational numbers is the quotient field of the integral domain $\left({\Z, +, \times}\right)$ of integers.

This is shown to exist in Existence of Quotient Field.


In view of Quotient Field is Unique, we construct the quotient field of $\Z$, give it a label $\Q$ and call its elements rational numbers.


Canonical Form of Rational Number

Let $r \in \Q$ be a rational number.

The canonical form of $r$ is the expression $\dfrac p q$, where:

$r = \dfrac p q: p \in \Z, q \in \Z_{>0}, p \perp q$

where $p \perp q$ denotes that $p$ and $q$ have no common divisor except $1$.


Fraction

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 $1$.


Geometrical Definition

In the words of Euclid:

With these hypotheses, it is proved that there exist straight lines infinite in multitude which are commensurable and incommensurable respectively, some in length only, and others in square also, with an assigned straight line. Let then the assigned straight line be called rational, and those straight lines which are commensurable with it, whether in length and in square or square only, rational, but those which are incommensurable with it irrational.

(The Elements: Book $\text{X}$: Definition $3$)

And let the square on the assigned straight line be called rational and those areas which are commensurable with it rational, but those which are incommensurable with it irrational, and the straight lines which produce them irrational, that is, in case the areas are squares, the sides themselves, but in case they are any other rectilineal figures, the straight lines on which are described squares equal to them.

(The Elements: Book $\text{X}$: Definition $4$)


Linguistic Note

The name rational number has two significances:

$(1): \quad$ The construct $\dfrac p q$ can be defined as the ratio between $p$ and $q$.
$(2): \quad$ In contrast with the concept irrational number, which can not be so defined.
The ancient Greeks had such a term for an irrational number: alogon, which had a feeling of undesirably chaotic and unstructured, or, perhaps more literally: illogical.
The proof that there exist such numbers was a shock to their collective national psyche.


The symbol $\Q$ arises from the construction of the rational numbers as the $\Q$uotient field of the integers $\Z$.


Also denoted as

Variants on $\Q$ are often seen, for example $\mathbf Q$ and $\mathcal Q$, or even just $Q$.


Also see

  • Results about rational numbers can be found here.


Sources