William, I’ve already conceded to the convention and utility of the accepted order of operations, so you don’t need to convince me of that. I’m just arguing for the sake of consistency, and to try to sort out some of the details.
Actually, ultrafilter, the ordered-pair construction of integers is something that (it seems to me) reinforces my problem. (BTW, your post said you’d define (a,b)+(c,d) = (a+b, c+d). But didn’t you mean (a+c, b+d)?)
See, if you’re going to say that the term “-1” refers to the equivalence class of (0,1), then I should be able to use it this way:
-1^2 = (0,1)^2 = (0,1)*(0,1) = (1,0) = 1
because I just substituted the (0,1) for -1, as they’re just different names for the same number.
Clearly, that’s not what everybody’s claiming the meaning of -1^2 is. I get that point.
Oy, well, Race is evidently starting to employ larger weapons than I have, even if we’ve agreed to some of the issue. (Time to start learning about equivalence classes I guess. )
Just based on your own message, though, Race, you can’t substitute (0,1) for -1 in the formula
-1^2
since there is no -1. The exponentiation comes first. It’s:
-(1^2)
not
(-1)^2
So however equivalence classes work, your substitution doesn’t apply.
It would be, I suppose, using this notation:
-(1,0)^2
Or rather
(0,1) * (1,0)^2
wherein the exponentiation is still done first.
This is, if I seem to be following the rules, (0,1), or -1.
You can’t just substitute the -1 with (0,1) because that implies that the - and the 1 are handled first, which they aren’t.
(Time to start learning about equivalence classes I guess.