Definition:Rational Number/Formal Definition

Definition
The field $\struct {\Q, +, \times}$ of rational numbers is the field of quotients of the integral domain $\struct {\Z, +, \times}$ of integers.

This is shown to exist in Existence of Field of Quotients.

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

Also see

 * Surgery for Rings, which means we may say $\Z \subset \Q$.