Chess and mathematics

#16

The idea that negative numbers don't exist (but ordinary positive numbers do) is all garbage philosophy and bullshit.

If you don't believe in negative numbers, you still can define relative numbers without any circular argument in the following way.

Let N the set of natural numbers; +,x denotes the usual operations. 
A "relative number " is a subset X of N^2 (the set of pairs of numbers) having the following property:
there exist a pair (a,b) in X such that for every pair (c,d), (c,d) belongs to X if and only if a+d=b+c.

if X,Y are relative numbers in the above sense we define X+' Y as the set of all pairs (p+q,r+s) where (p,r) is in X and (q,s) is in Y. We also define X *' Y as the set of all pairs (p*q+r*q, p*r+q*s).
If n is an element of N we define f(n) as the set of pairs (k+n,k) where k is any natural integer.
Finally we say that the relative integer X is strictly smaller than the relative integer Y and we write X <' Y if for some (a,b) in X and (c,d) in Y, a+d < b+c (it turns out that when it is the case then for every(a',b') in X and (c',d') in Y,we also have a'+d' < b'+c'). 

If you're motivated you can check that

1°) if X and Y are relative integers, so are X +' Y and X *' Y.

2°) The set of relative integers with the operations +',x' defined as above is a ring with f(0) being neutral for the sum+' and f(1) neutral for the product x' (i.e usual algebraic properties hold)

3°) for every p q natural integers, f(p+q)= f(p)+' f(q) and f(p x q)= f(p) x' f(q)

4°) for every p, q natural integers, if f(p)= f(q) then in fact p = q ( 3° and 4°  allow to view N as a subset of the set of relative integers with the same operations)

5°) <' is a total strict order and for every natural integers k l, k < l if and only if f(k) < f(l)

6°) for every X relative number, if by  -'X we denote the set of all pairs (q,p) where (p,q) is in X, then -'X is also a relative integer and X + (-' X) = (- X')+ X = f(0). -'X is the only relative number to have this property.

7°) for every relative number Y, f(0) <' Y or Y= f(0) or Y <' f(0). Y <' f(0) if and only  f(0) <' -'Y;  and  -'Y <' f(0) if and only if f(0) <' Y.

8°) if A,B are relative numbers A <' f(0) and B<' f(0), then f(0) <' A x' B (NB: so in that construction, the infamous fact that "the product of two negative numbers is a positive number" can be deduced).

The set of relative integers is usually abbreviated  by the letter Z.

how would you prove it. don't say stuff like its obviously true. you have to prove it without resorting to a calculator, just using algebra.

     Those that are good in a field and especially those who have PhD never improve(or almost never).

     First, they never put serious effort because they think they can do it the easy way. They think they will read something once and they will understand it , they don't need to put any effort or they will play some blitz and they will go from nothing to 2200 in a semester.

    Second, they always think that they can find correlation of their field with chess which never happens. Their accomplishments are useless in chess.

   Third, they always think they can doubt established methods of training although they are ignorants.They even think they can make their own. People that have never succesfully trained themselves think they are qualified to have an opinion on training others(and themselves). 

   Fourth , they never follow instructions. It's impossible for them to get into the mentality of kindergarten kid and just shut up and follow instructions.

     Till now I have seen no one showing a meaningful improvement while a regular housewife started from scratch and 6 years after  is close to 1900 FIDE. When someone comes in the chess club and the first he says is "I am a doctor" or "I have PhD in paleoanthropology" or "I was first in my class in Law school" we know he is a lost case.

     Those that are good in a field and especially those who have PhD never improve(or almost never).

     First, they never put serious effort because they think they can do it the easy way. They think they will read something once and they will understand it , they don't need to put any effort or they will play some blitz and they will go from nothing to 2200 in a semester.

    Second, they always think that they can find correlation of their field with chess which never happens. Their accomplishments are useless in chess.

   Third, they always think they can doubt established methods of training although they are ignorants.They even think they can make their own. People that have never succesfully trained themselves think they are qualified to have an opinion on training others(and themselves). 

   Fourth , they never follow instructions. It's impossible for them to get into the mentality of kindergarten kid and just shut up and follow instructions.

     Till now I have seen no one showing a meaningful improvement while a regular housewife started from scratch and 6 years after  is close to 1900 FIDE. When someone comes in the chess club and the first he says is "I am a doctor" or "I have PhD in paleoanthropology" or "I was first in my class in Law school" we know he is a lost case.