OOT: Negative number multiplication

[Florian Diesch]

Groundhog <linuxmurah at gmail.com> wrote:

> few weeks ago I had a question, acctually really simple question, but
> until today nobody can give me answer. okay here is my question:
>
> "as we know since Elementary School:
> (+) x (+) = (+)
> (+) x (-) = (-)
> (-) x (+) = (-)
> (-) x (-) = (+)
>
> now when I change with numbers:
> 5 x 3 = 5 + 5 + 5 = 15 (everybody know this very well)
> 5 x (-3) = ....................... = -15 (everybody getting confused)
> (-5) x (3) = (-5) + (-5) + (-5) = -15 (no problem with it)
> (-5) x (-3) = ....................... = 15 (then, everybody get frustated)
>
> could someone give answer about this 'elementary' question??

First let's define what negative numbers are:

  a.  (-1) = 0-1
  b.  For every number n:  (-n) = (-1)*n

So what's -3? 
It's (-1)*3 = (0-1)*3 = 0*3-1*3 = 0-3 = 0+((-1)*3) = 0+(-3) = -3
Looks good.
 
Let's look at 5*(-3):
 5*(-3) = 5*((-1)*3)
        = 5*((0-1)*3)
        = 5*((0*3)-(1*3))
        = 5*(0-3)
        = 5*0-5*3
        = 0-15
        = (0-1)*15
        = (-1)*15
        = -15

And now (-5)*(-3):
(-5)*(-3) = ((-1)*5)*((-1)*3)
          = (-1)*(-1)*5*3
          = (0-1)*(0-1)*15
          = (0*0-0*1-1*0-1*(-1))*15
          = (0-0-0-(-1))*15
          = (-(-1))*15
          = (-(0-1))*15
          = ((-0)+1)*15
          = 1*15
          = 15




   Florian
-- 
<http://www.florian-diesch.de/>