Date: Dec 3, 2012 7:43 PM
Author: Paul A. Tanner III
Subject: Re: In "square root of -1", should we say "minus 1" or "negative 1"?
On Mon, Dec 3, 2012 at 6:22 PM, Joe Niederberger <firstname.lastname@example.org> wrote:
> "Minus times minus makes a plus" makes perfect sense.
> In fact it emphasizes the fact the rule is based on the signs alone. The fact that negative numbers also have a magnitude is of no account whatsoever in regards to this sign rule. Think of the implied subjects and objects here being the signs, not the numbers. Or, don't, I don't care.
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.