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Topic: -1 x -1 ?
Replies: 29   Last Post: May 13, 2009 9:01 AM

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 Dave Seaman Posts: 2,446 Registered: 12/6/04
Re: -1 x -1 ?
Posted: Sep 16, 1999 6:19 PM

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

Date Subject Author
9/16/99 Guillermo Phillips
9/16/99 Ian A. Mason
9/16/99 Peter Percival
9/16/99 Dave Seaman
9/17/99 Steve Leibel
9/17/99 Dale Henderson
9/18/99 Peter Percival
9/18/99 Bill Taylor
9/18/99 John Savard
9/19/99 carel
9/20/99 Jon Haugsand
9/20/99 HH
9/20/99 Jon Haugsand
9/20/99 Jonathan Hoyle
9/20/99 Jim Hunter
1/21/09 Tay
1/21/09 mensanator
1/21/09 lwalke3@lausd.net
1/22/09 Henry
1/22/09 David R Tribble
1/22/09 Dave L. Renfro
1/22/09 mensanator
5/12/09 Kayama
5/12/09 mensanator
5/12/09 hagman
5/12/09 Bill Dubuque
5/12/09 Bill Taylor
5/13/09 Cobra
5/13/09 Cobra