**Definition** We define the [[Equivalence relations]] $\sim$ on $\mathbb{Z}\times \mathbb{N}$ given by $(m,n)\sim(k,l) \iff m \times_{\mathbb{Z}} l = k \times_{\mathbb{Z}} n$where $\times_{\mathbb{Z}}$ represents integer multiplication as defined in [[Set-theoretic construction of the integers|the construction of the set of integers]]. **Proof** We can show $\sim$ is an equivalence relation by showing that it is [[Reflexive Relation|reflexive]], [[Symmetric Relation|symmetric]] & [[Transitive Relation|transitive]]. **Construction** We can define $\mathbb{Q}$ as follows $\mathbb{Q}=(\mathbb{Z}\times \mathbb{N}) /\sim$ (i.e the set of [[Equivalence relations|equivalence classes]] of $\sim$). We can define arithmetic operations on $\mathbb{Q}$ as follows (rewriting $(m,n)$ as $m/n$): - $[m/n]+[k/l]:=[(m\cdot l)+(k \cdot l)/\,n\cdot l]$ - $[m/n]\cdot[k/l]:=[m\cdot k/n\cdot l]$ - $[m / n]< [k / l] \iff m \cdot l <_{\mathbb{Z}} k \cdot n$. The additive inverse of $[m /n]$ is $[-m /n]$ where $-m$ is the additive inverse of the integer $m.$