Date: Sep 16, 1999 5:47 PM
Author: Peter Percival
Subject: Re: -1 x -1 ?



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.