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 a + x = 0. To be able to solve it for one particular a, you would need an element that you do not have. Mathematical point of view is this: We do not have it "really", but we can introduce an abstract object, call it (−a), whose defining property is that a + (−a) = 0. When we do it for all natural numbers a, we get a set of abstract objects whose only property (so far) is the one that defines it. We can form a set consisting of zero, all natural numbers a and all their inverses (−a).

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 (−a) and (−b), we define that the outcome of (−a) + (−b) is the inverse element of the natural number a + b. We have to do similar work for multiplication.

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 −3 in our life. While in this particular case it may look like useless waste of time, it is in fact a general procedure that is very useful when working with objects that are really hard to imagine, for instance when defining complex numbers. Every extension, from integers to rationals, from rationals to reals, from reals to complex etc., is done in this way.

There is another important point to make. Note that we created integers so that we can solve equations a + x = 0 for a a natural number. However, it may actually happen that when we take some negative integer for a, then the equation a + x = 0 cannot be solved. Fortunately for us, this does not happen, but this also has to be proved. In fact, we really do hope that it works, otherwise we would have to introduce still more abstract numbers, thus getting a new larger set, and this could go on forever. Again, this is something that has to be checked with every extension. We enlarge a given set so that we can solve a certain equation in that old set, but we are always so lucky that the equation is also solvable within the new set.