Using juxtaposition for multiplication, by association: [ab + a(-b)] + (-a)(-b) = ab + [a(-b) + (-a)(-b)] Then by distribution, the definition of "-", a0=0a=0, a+0=0: above = ab + [a + (-a)](-b) = ab + 0(-b) = ab. Likewise: [ab + a(-b)] + (-a)(-b) = a[b + (-b)] + (-a)(-b) = a0 + (-a)(-b) = (-a)(-b). A fortiori the result follows.
So, it's not "the fundamental reason" but lots of fundamental reasons. Can a simpler proof be given? Perhaps the special case (-1)(-1) = 1 can be proved more simply. A supplementary: is there an interesting algebraic system in which it's false?
Guillermo Phillips 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.