
Re: 1 x 1 ?
Posted:
Sep 16, 1999 6:19 PM


In article <937516347.13527.0.nnrp14.c2debf68@news.demon.co.uk>, Guillermo Phillips <Guillermo.Phillips@marsman.demon.co.uk> wrote: >Hello All, > >Here's something I've always wondered (perhaps in my naivety). Why >should 1 x 1 = 1? >I appreciate that lots of nice things come from this, but what's the >fundamental reason for it? > >Guillermo.
By definition, the notation x represents the additive inverse of x. That is, x is the unique number such that x + (x) = (x) + x = 0. You can also turn this around and say that x is the additive inverse of x, since the definition is symmetric in x and x.
In particular, 1 is the additive inverse of 1, and 1 is the additive inverse of 1, That is,
(1) = 1. (*) That almost looks like what we want, but it isn't, quite.
It's easy to prove that for any x, the additive inverse x is the same as the product of x and 1. Consider:
0 = x * 0 = x * (1 + (1)) = (x * 1) + (x * (1)) [Distributive Law] = x + (x * (1)) = (x * (1)) + x,
and this means that (x * (1)) fulfulls the definition of the additive inverse of x. That is,
x = x * (1)
for any x. In particular, substitute x = 1 to obtain
(1) = (1) * (1) (**)
or, in words, the additive inverse of the additive inverse of 1 is the same as the product of the additive inverse of 1 with itself.
Combining (*) and (**), we get
1 = (1) * (1).
 Dave Seaman dseaman@purdue.edu Pennsylvania Supreme Court Denies Fair Trial for Mumia AbuJamal

