Date: Dec 4, 2012 2:25 AM
Author: Paul A. Tanner III
Subject: Re: In "square root of -1", should we say "minus 1" or "negative 1"?

On Tue, Dec 4, 2012 at 12:08 AM, Joe Niederberger <niederberger@comcast.net> wrote:
>>And so to say that that the theorem itself (and not merely such a particular proof of it) is based on such is not right.
>
> I said the sign rule for multiplying negative integers is not based on the fact that integers have a magnitude (together with a sign.) You think it is?
>
> Joe N


You said:

"In fact it emphasizes the fact the rule is based on the signs alone."

This is not correct.

I said :

"One of the standard proofs in abstract algebra textbooks of the theorem (-a)(-b) = ab for all ring elements a,b (including for rings that are not ordered) contains the equality (-a)(-b) = -(-(ab)).

Is this what you meant?

Regardless, there are ways to prove this theorem that does not use the prior theorem of additive groups x = -(-x).

And so to say that that the theorem itself (and not merely such a particular proof of it) is based on such is not right."

That is, the theorem (-a)(-b) = ab is about the elements of the sets and the unary function of taking additive inverses of elements, not about "the signs", which is about how to write "the additive inverse of".