In article <firstname.lastname@example.org>, "Guillermo Phillips" <Guillermo.Phillips@marsman.demon.co.uk> wrote: > Hello All, > > Here's something I've always wondered (perhaps in my naivety). Why > should -1 x -1 = 1? > I appreciate that lots of nice things come from this, but what's the > fundamental reason for it? > > Guillermo. > >
I've seen a few proofs on this thread that make unfounded assumtions. One assumes the uniqness of inverses and another assumes 0x=0. Ironically the result (-a)b=-ab is requred to prove 0x=0. I've added proofs of a few propositions below. The one your interested in is Proposition 4. I think Propostions 2 and 3 should be combined to read (-a)b=a(-b)=-ab. I've tried to assume only the axioms of a Ring if anyone sees an unfoundes assumptions please tell me. I also threw in the proof that 0x=0 for the fun of it.
Prop1 additive inverses are unique
Let b,c be additive inverses of a.
a+b=a+c b+a+b=b+a+c 0+b=0+c b=c
Prop 2 a(-b)=-ab
Proof: a(-b)+ab+ab = a (-b + b + b) =a (0+b) =ab
So, a(-b)+ab+ab=ab a(-b)+ab=0 a(-b)=-ab
Prop 3 (-a)b=-ab
Proof: (-a)b+ab+ab=(a+a + -a)(b) =(a+0)b =ab So, (-a)b+ab+ab=ab (-a)b+ab=0 (-a)b=-ab
Prop 4 (-a)(-b)=ab
Proof: let c=-b (-a)(-b)=(-a)c =-ac (Prop 2) =-(a(-b)) =-(-ab) (Prop 3) = ab