Rational Number
Definition1
Let the set of integers $\mathbb{Z}$ be given. Consider the following Cartesian product.
$$ \begin{align*} S &= \mathbb{Z} \times (\mathbb{Z}\setminus \left\{ 0 \right\}) \\ &= \left\{ (a, b) : a \in \mathbb{Z} \text{ and } b \in \mathbb{Z} \setminus \left\{ 0 \right\} \right\} \end{align*} $$
And define the equivalence relation $\sim$ as follows.
$$ (a, b) \sim (c, d) \iff ad = bc $$
The quotient set of $S$ by $\sim$ is defined to be the set of rational numbers and is denoted as follows.
$$ \mathbb{Q} := S/\sim $$
An element of $\mathbb{Q}$ is called a rational number.
Explanation
By definition, a rational number is the equivalence class $(a, b)/\sim$ of the ordered pair $(a, b)$ consisting of two integers $a$ and $b \ne 0$. It may be represented simply by the ordered pair $(a, b)$, but more commonly the slash (/) or dash (−) notation is used; such an expression is called a fraction.
$$ \frac{a}{b} = a/b = (a, b)/\sim $$
Relationship with integers
Each integer $n \in \mathbb{N}$ can be expressed as the rational number $n/1$. That is, it can be regarded as $\mathbb{Z} = \left\{ n/1 : n \in \mathbb{Z} \right\}$, and the set of integers is a subset of the set of rational numbers.
$$ \mathbb{Z} \subset \mathbb{Q} $$