We want sums to follow those of natural numbers.1
Consider [(a,b)],[(c,d)]∈Z. We define the integer sum of [(a,b)] with [(c,d)] as [(a+c,b+d)].2
We denote the sum of [(a,b)] and [(c,d)] by [(a,b)]+[(b,c)] So if x,y∈Z then the sum of x and y is x+y.