Date: May 12, 2009 12:37 PM
Author: Bill Dubuque
Subject: Re: -1 x -1 ?
hagman <google@von-eitzen.de> wrote:

> On 12 Mai, 06:59, Kayama <yano...@earth.ocn.ne.jp> wrote:

>>

>> If we can use the expression -1=e^{i\pi}, we can show

>> -1*-1=e^{i\pi}*e^{i\pi}=1 readily:

>>

>> if we rotate once 1 (the vector 01) by \pi rad around the origin anticlockwise on the complex plane, we obtain

>> -1. Further likewise if we rotate -1 by \pi rad once more, we can obtain -1*-1=1.

>>

>> At least I understand -1*-1=1 in this way.

>>

>> In that way we can obtain i (imaginary unit), if we rotate 1 by \pi/2 around the origin anticlockwise.

>>

>> If we cannnot allow to use the expression -1=e^[i\pi}, I don't understand -1*-1=1 vividly.

>

> Well, (-1)*(-1) = 1 holds in all rings, not just in the field C.

> And I doubt you can prove a lot about the exponential function

> without making use of (-1)*(-1) = 1 somewhere ...

More generally, the Law of Signs is simply the special linear case of

the composition of odd functions in near-rings, see my 08 Feb 2008 post

http://google.com/groups?selm=y8z1w7nf41w.fsf%40nestle.csail.mit.edu

--Bill Dubuque