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.
By definition, the notation -x represents the additive inverse of x. That is, -x is the unique number such that x + (-x) = (-x) + x = 0. You can also turn this around and say that x is the additive inverse of -x, since the definition is symmetric in x and -x.
In particular, -1 is the additive inverse of 1, and 1 is the additive inverse of -1, That is,
-(-1) = 1. (*)
That almost looks like what we want, but it isn't, quite.
It's easy to prove that for any x, the additive inverse -x is the same as the product of x and -1. Consider: