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Re: In "square root of -1", should we say "minus 1" or "negative 1"?
Posted:
Dec 4, 2012 2:25 AM
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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".
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