Definition:Rational Number/Formal Definition

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.