Date: Sep 16, 1999 6:19 PM
Author: Dave Seaman
Subject: Re: -1 x -1 ?



In article <937516347.13527.0.nnrp-14.c2debf68@news.demon.co.uk>,
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:

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 dseaman@purdue.edu
Pennsylvania Supreme Court Denies Fair Trial for Mumia Abu-Jamal