Re: Why should -1 multiplied by -1 be plus 1 and not -1

From: George Greene (greeneg_at_greeneg-cs.cs.unc.edu)
Date: 07/30/04 

Date: 30 Jul 2004 17:25:16 -0400

mtx014@linux.services.coventry.ac.uk (Robert Low) writes:
 : Hard to say: of course, the 'real' reason that -1*-1=1
 : is because we want the standard rules of arithmetic
 : for positive integers to extend nicely to the
 : whole set of integers, so any intuitive picture
 : has to side-step that somehow.

That's not my personal intuition.
Mine would be more along the lines of, you have
to ask yourself what "-" "really" means.
-5 arguably MEANS (-1)*5. The "-" in -1
and the "-" in -5 are NOT the same. Multiplying
something by -1 is a special operation, like flipping
a coin. It's in some sense THE ONLY way you CAN get
negative numbers. You could try to get them by
reversing the order on subtraction, but again, that
only reinforces the fact that "multiplying by -1"
is about FLIPPING. If I know what (a - b) is, then what
is (b - a) ? One possible answer is that BY DEFINITION,
(b - a) must be (-1)*(a - b).

 : An argument about
 : 'taking away a debt' being equivalent to 'giving
 : money' might do the trick,

That is more like a - (-b) = a + b .

 But there doesn't appear to be any -1 in that at all,
unless you re-spell (-b) as (-1)*b. If you can get her
to believe that taking away a debt really is like giving
money, then you still have to do a LOT of re-spelling with
this to get to -1 * -1 = +1.

 : then say 'but what's really going on is...'

No, the intuition really is what's really going on.

 : (Of course, my thinking that the axioms of a
 : complete ordered field drive what's really
 : going on,

No, they don't; you can confine yourself to integers
and still do this; you don't need a field. 

-- 
 --- The history of our nation has demonstrated that separate is seldom, if ever, equal.
 --- (Feb.3,2004) Supreme Judicial Court of Massachusetts (4-3), adv.Sen.#2175