We have lived with integers for so long that we usually do not feel that there is a big difference between positive integers, zero and negative integers. Natural numbers are in some sense "real" objects, they represent how many parts you have in a whole, and zero can be also interpreted in this way. Admittedly it is an abstraction, but a rather simple one. However, what object can you point to in order to show what "−3" is? There is a way to explain, but it is not so simple any more. So there definitely is a difference and indeed, many thousands of years stand between the "discovery" of natural numbers and negative numbers.
Now imagine that you only have natural numbers and wonder about solving an
equation
Now we have a set, but we do not know how to work with it, for instance we
do not know how to add in this set when we involve one of those new abstract
guys. Thus a mathematician needs to do another step, namely specify how to
add two such abstract elements and how to add good old natural numbers with
these new guys. In fact we can define it any way we feel like, but then most
likely the operation would not have any nice properties and thus it would be
useless. So naturally we want to define addition in such a way that it fits
well with what we already have. For instance, having two abstract inverses
After making these definitions, it is necessary to check that they make sense (that one cannot get to conflicting results in this way) and also prove that "new addition" and "new multiplication" have all the properties we hope for (commutativity etc.). Since we used as inspiration our experience with negative integers, it is to be expected that things will work out, but note that proofs must be made without refering to "the usual negative integers", only using definitions. After we are through with it, we work on "comparison" by defining and investigating inequality.
The main point is that while we do proper definition of integers, we have to
pretend that we never heard of numbers like
There is another important point to make. Note that we created integers so
that we can solve equations