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

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 Bill Dubuque Posts: 1,739 Registered: 12/6/04
Re: -1 x -1 ?
Posted: May 12, 2009 12:37 PM

> 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
>>
>> 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

--Bill Dubuque

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
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1/21/09 Tay
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1/21/09 lwalke3@lausd.net
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1/22/09 Dave L. Renfro
1/22/09 mensanator
5/12/09 Kayama
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5/12/09 Bill Dubuque
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5/13/09 Cobra
5/13/09 Cobra