5 ms·

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 th

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".

deleted 2y ago
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HKH2 2y ago
Thanks. I had no idea. Seems counterintuitive at first, but it makes sense, since ’-3’ is '0 - 3'.
- setopt 2y ago
  This is a good way to think of it. It would be very confusing if “0 - 3^2” and “0 + (-3^2)” should mean different things.
  - zmgsabst 2y ago
    But what about (0-3)^2? You said that “-3” = “0-3”. 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.
    - RandomThoughts3 2y ago
      > (0-3)^2? It’s entirely unambiguous due to the parenthesis. 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.
      - zmgsabst 2y ago
        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).
        RandomThoughts3 2y ago
        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.
    - setopt 2y ago
      > You said that “-3” = “0-3”. No, I didn’t.
DiscourseFan 2y ago
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.
- RandomThoughts3 2y ago
  You are mistaking the notation for the object. 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).
  - DiscourseFan 2y ago
    I mean, all equations are true if all terms are set to equal zero, so the term itself must have some sort of ontological distinction.
spockz 2y ago
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.
- naniwaduni 2y ago
  Precedence rules are mostly a formalization of the underlying basic convention that the operations whose terms are written closer together are done first.
  - zmgsabst 2y ago
    Mostly to make polynomials easy. 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.
- bikenaga 2y ago
  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. :-)
- Izkata 2y ago
  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.
tsimionescu 2y ago
-3^2 is not a standard mathematical convention. -3² is, but that is completely different in many other ways.

deleted 2y ago

[deleted]

HKH2 2y ago

Thanks. I had no idea. Seems counterintuitive at first, but it makes sense, since ’-3’ is '0 - 3'.

setopt 2y ago
This is a good way to think of it. It would be very confusing if “0 - 3^2” and “0 + (-3^2)” should mean different things.
- zmgsabst 2y ago
  But what about (0-3)^2? You said that “-3” = “0-3”. 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.
  - RandomThoughts3 2y ago
    > (0-3)^2? It’s entirely unambiguous due to the parenthesis. 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.
    - zmgsabst 2y ago
      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).
      - RandomThoughts3 2y ago
        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.
  - setopt 2y ago
    > You said that “-3” = “0-3”. No, I didn’t.

DiscourseFan 2y ago

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.

RandomThoughts3 2y ago
You are mistaking the notation for the object. 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).
- DiscourseFan 2y ago
  I mean, all equations are true if all terms are set to equal zero, so the term itself must have some sort of ontological distinction.

spockz 2y ago

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.

naniwaduni 2y ago
Precedence rules are mostly a formalization of the underlying basic convention that the operations whose terms are written closer together are done first.
- zmgsabst 2y ago
  Mostly to make polynomials easy. 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.
bikenaga 2y ago
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. :-)
Izkata 2y ago
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.

tsimionescu 2y ago

-3^2 is not a standard mathematical convention. -3² is, but that is completely different in many other ways.