How do we define the addition on the set of integers?

Normally it is defined for positive integers (Natural numbers) and then this is extended to all integers. The formal definition of addition (or at least the most common one) is part of Peano axioms - Wikipedia which define how addition and multiplication work. read more

Personally, I define the integers as the ring completion of the semi-ring of natural numbers. Or you may define the integers as the set of equivalence classes of [math]\mathbf{N}^2[/math] where [math](a,b)[/math] is equivalent to [math](c,d)[/math] iff [math]a+d=b+c[/math]. read more

Addition of integers can also be performed on number line. It gives us better understanding. Let us have a look at following problems: 1) 4 + 5 In order to add 4 and 5 on number line, we shall start from zero and move 4 points towards right hand side of zero (since 4 is a positive integer). read more