Date: May 12, 2009 4:17 AM
Author: hagman
Subject: Re: -1 x -1 ?
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 ...