How do I compute step by step this formal definition of the multiplication operation $\cdot_{\mathbb Z}$ in $\mathbb Z$
$$(m,n)\cdot_{\mathbb Z}(p,q)=(m\cdot p+n\cdot q,m\cdot q+n\cdot p) ?$$
What is in fact $(m,n)$ in $\mathbb{Z}$ i.e. when $(m,n)\sim(m',n')$?
The definition of multiplication in $\bf Z$, seen as $\mathbf N\times\mathbf N$ is the following: $$(m,n)(p,q)=(mp+nq,mq+np)$$ Just remember that $(m,n)$ is implicitly $m-n$ – except that in $\mathbf N$, this difference is defined only if $m\ge n$.