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

Also see

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