Date: Nov 29, 2012 1:31 AM
Author: Peter Duveen
Subject: Re: Some important demonstrations on negative numbers
To demonstrate that 1/-a = -(1/a):

(1/-a) x -a = 1 definition of 1/-a eq.1

- - (1/a) x a = -1 definition of 1/a

- -1 (1/a) x a = -1

(1/a) x -1 x a = -1

(1/a) x -a = -1

- -(1/a) x -a = 1 eq.2

>From eq. 1 and eq. 2:

- -(1/a) = (1/-a).

Perhaps a bit clumsy, but once demonstrated in its generality, the student no longer needs to wonder about this relationship when he sees a negative number in the denominator. As far as even clumsier analogies, that is fine to interpret the result as long as the result has been demonstrated clearly. Seems those who took geometry should easily assimilate such a demonstration (proof) .