what makes extention of number system vaild for any model?

Question

Now, I am reading Mathematics: a very short introduction by T.Gowers. In chapter 1 he explains about models. Well, i am quite clear that i understand that part, that a model is not actually whats happening out in the real world but by few assumptions and etc etc we can easily relate to real world in really good terms.But when i come to the chapter 2 Numbers and abstraction i am stuck at few places,

First of all i understand we get equations while working with our models that cant be solved in previous system so something new is needed but what is not entirely clear to me, that why, while extention we follow properties of previous system. i mean i know, that way previous system will be subset of the new one but why would following those previous rules like a+b=b+a,ab=ba etc. will work with our models, as we go further and further with our extentions. i started to get a notion that this process is quite arbitrary that why cant a system with property like (-1)(-1)= -1 be there, what i am pointing out is that the property (-a)(-b)=ab is direct consequence of defining -a ( that when -a is added to a it gives 0) so if i am correct why the number system with properties which derive from natural numbers gives correct results with our models or what motivates us to extend number systems the way we do them not by any other way.

i want you to note that i am ready to just focus on the number properties, which writer of book say u should do for higher mathematics but my only concern is why this form of extention will work for models. please give me a detailed answer i am still in highschool so if u use something as group theory please make it in little easy terms for me.

Answer 1

Actually the way we construct new models like you said, we try to keep the old results follow in our new model as well even though we might generalize it more unless we find some sort of error with the previous model. So it is a question of extension vs fixing. And about construction of numbers, I will give a brief outline, and you can read more about them.

1. God gave us the natural numbers. Revised to peano axiomatic construction based on set theory.
2. We created integers consciously keeping addition the same as with natural numbers. Define numbers in form of e-f as integers such that : a-b and c-d two integers equal if a+d = b+c. And define 0-a =-a and a-0 to be a as shorthand. So minus sign is almost there in the construction itself even though it has no meaning initially.
3. We created rationals keeping addition, substraction and multiplication consistent with the previous systems and define no in form of a/b , b is not zero as rationals similarly defining equality of a/b and c/d if ad=bc. define a/1 = a our familiar integers and so on.
4. We can define real using Cauchy sequence of rationals or cuts .

So you can see a general trend that we are using the previous properties of a number system to define a new one and embedding the previous property inside the new one even if a special case restricted to the old system.