>>11577371

It's not bullshit because it allows you to expand the set of numbers whilst not breaking the previous rules. Addition and multiplication stay the same in positive integers whether or not you see them as a subset of all integers.

And by the way, the rigorous way of introducing negative integers isn't even by assuming that for every integer n there exists m such that n+m=0: in number theory, you take the set NxN of the couples (a,b) of positive integers and create an equivalence relation ~ with the following rule: given two couples (a,b) and (c,d)

(a,b)~(c,d) iff b+c = a+d

Then you take the set (NxN)/~ of all equivalence classes and you give it an operation * so that (a,b)*(c,d) = (a+b,c+d)

and you notice that if you do

(a,b)*(b,a) you get (a+b,b+a)=(a+b,a+b)

but by the rule defined by ~ a+b+0=a+b+0

and therefore (a+b,a+b) is equivalent to (0,0)

We obtained the desired rule of for every element (a,b) of this set, we can find another one (b,a) such that they add to (0,0). So we built a set that is structurally identical to integers by using couples of positive integers and without the need to introduce the - symbol.

Analogous processes are used to build thr set Q of ratios (from positive integers) and the set C of compled numbers (from real numbers). A more difficult problem is how the set R of real numbers is built from Q.