Date: Sep 16, 1999 6:19 PM
Author: Dave Seaman
Subject: Re: -1 x -1 ?
In article <firstname.lastname@example.org>,
Guillermo Phillips <Guillermo.Phillips@marsman.demon.co.uk> wrote:
>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?
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:
0 = x * 0
= x * (1 + (-1))
= (x * 1) + (x * (-1)) [Distributive Law]
= x + (x * (-1))
= (x * (-1)) + x,
and this means that (x * (-1)) fulfulls the definition of the additive
inverse of x. That is,
-x = x * (-1)
for any x. In particular, substitute x = -1 to obtain
-(-1) = (-1) * (-1) (**)
or, in words, the additive inverse of the additive inverse of 1 is the
same as the product of the additive inverse of 1 with itself.
Combining (*) and (**), we get
1 = (-1) * (-1).
Dave Seaman email@example.com
Pennsylvania Supreme Court Denies Fair Trial for Mumia Abu-Jamal