No - in a subtle way, you're assuming that the unary minus has precedence, when the point is that it doesn't.
You're right that "thing^2" means "thing times thing", but in "-3^2", what is it that is being squared? To write it, as you did, as "(-3) x (-3)", assumes that in "-3^2" the thing being squared is "-3". But that in turn assumes that the unary minus is done before the square. By the standard mathematical convention, in "-3^2" the thing being squared is "3". So you do "3 * 3", then you negate the result and get "-9".
Interesting. In our high school math class when -3^2 was hand written it was ok to interpret it as (-3)^2. But maybe that could have been because spacing would be used as some informal parentheses.
Leaving "-3²" to be "interpreted" according to context is not a good idea. Mathematical notation should be unambiguous and context-free as much as possible, not "do what I mean". A computation involving "-3²" will be right or wrong depending on what it means; that shouldn't depend on who is reading it or under what circumstances.
Not to unduly slight your teacher, but it could be they weren't sure about what "-3²" means. Everyone who teaches has gaps in their knowledge -- I sure did. :-)
Precedence rules are mostly a formalization of the underlying basic convention that the operations whose terms are written closer together are done first.
But if we’re doing math mostly on computers, we should adopt rules that make writing on computers easy — not pedantically insist typing code follow the rules of handwriting polynomials.
It was the same for me. The question seems to be whether "-3" here represents the operation "negate" or the number "negative 3". If it's the latter, then -3^2 is (-3)^2.
So we have “-3^2” is “(0-3)^2” is 9. Agreeing with -3^2 = 9.
You’re performing a sleight of hand when you define “-3” to be “0-3”, but move the parenthesis to get your second equation. You have to insert your definition as a single term inside parenthesis — you can’t simply remove them to change association (as you have done). That’s against the rules.
So if you think “-3” is “0-3”, then you should agree the answer is 9.
I don’t think the rest of your argument actually makes sense.
There is no sleight of hand required. The original argument is entirely related to having unary minus and binary minus which are different operators conceptually have similar precedence as being less surprising.
My point is that you can’t define “-3” as “0-3” to make it work: you’re assuming exactly what’s being debated (via sleight of hand) when you insert the terms with brackets in the way you do rather than the way I do.
When you try to swap in the unary operator without that to make it “less surprising”, you get 9.
Precisely what you said was wrong about the unwary operator (in Excel).
But no one is defining -3 as (0-3). You are entirely missing the point. I am going to quote myself again:
> The original argument is entirely related to having unary minus and binary minus which are different operators conceptually have similar precedence as being less surprising.
And no, you don’t get 9 when you swap the unary operator. That’s the whole point and why it’s surprising that Excel did reverse the precedence for implementation easiness.
Isn't this treating 3 as something that can be analyzed from -3? Like -3 is the union of 3 and the (-), and not something onto itself. Ought it to be the case that negative is more than just a sign? Like, I think 3 should be ontologically distinct from -3, even if the addition of 3 and -3 = 0. Idk, I don't have a reason for this, it feels right for some reason.
There is indeed two ontologically different elements 3 and (-3) in Z. The question is however purely about what is the meaning of the ambiguous without precedence rules representation -3^2.
Note that it gets more complicated quickly if you want to keep thinking about it in that mathematicians often consider ontologically different but equivalent operations as the same when it’s irrelevant to what they are doing or the results trivially extend to both case. See for example 3-3 and 3+(-3).
You're right that "thing^2" means "thing times thing", but in "-3^2", what is it that is being squared? To write it, as you did, as "(-3) x (-3)", assumes that in "-3^2" the thing being squared is "-3". But that in turn assumes that the unary minus is done before the square. By the standard mathematical convention, in "-3^2" the thing being squared is "3". So you do "3 * 3", then you negate the result and get "-9".