It does indeed. I have a love/hate relationship with math. Love because it's fascinating and intergal to an understanding of our universe, hate because I had some horrible math teachers over the years...plus I do admit to being slightly lazy.

There is no why. It just is. This is the worst thing about being human :)

http://www.smbc-comics.com/index.php?db=comics&id=1914

Nice trick. It works in any even base b:

    xy = b((x - b/2) + (y - b/2)) + (b - x)(b - y)

For b = 2 in particular, x - b/2 and x - b/2 are the negative 1's complements of x and y, and b - x and b - y are the 2's complements. More generally, they are the same kinds of complements generalized to base b.

Because of that symmetric relationship with complementation, I suspect there is a conceptual proof waiting to be found. By conceptual I mean something that isn't just blind formula crunching. That would get us much closer to "why".