Definition:Rational Number

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

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

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

Thus, $$\Q = \left\{{\frac p q: p \in \Z, q \in \Z^*}\right\}$$.

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.

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 $$\frac p q$$ 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 meant "chaotic", "unstructured" etc., with a negative connotation. The proof that there exist such numbers was a shock to their collective national psyche.