Definition:Rational Number

Informal Definition
A number in the form $\displaystyle \frac p q$, 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:
 * $\displaystyle \Q = \left\{{\frac p q: p \in \Z, q \in \Z^*}\right\}$

A rational number such that $q \ne 1$ is colloquially and popularly referred to as a fraction. The similarity between that word and the word fracture is no accident.

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

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 simply pick a quotient field of $\Z$, give it a label $\Q$ and call its elements rational numbers.

We note that $\left({\Z, +, \times, \le}\right)$ has a total ordering $\le$ on it.

From Total Ordering on Quotient Field is Unique, it follows that $\left({\Q, +, \times}\right)$ has a unique total ordering on it that is compatible with $\le$ on $\Z$.

Thus $\left({\Q, +, \times, \le}\right)$ is a totally ordered field.

Comment
The name rational has two significances:
 * 1) The construct $\displaystyle \frac p q$ can be defined as the ratio between $p$ and $q$.
 * 2) In contrast to 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. The proof that there exist such numbers was a shock to their collective national psyche.