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

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 Peter Percival Posts: 95 Registered: 12/8/04
Re: -1 x -1 ?
Posted: Sep 16, 1999 5:47 PM

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.

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