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The convention is that $-3^2 = -9$. But I think this implies that "-3" does not refer to the integer that is a solution to $3+x=$0. Otherwise $-3^2$ would mean "$-3$ times $-3$", the same as $(-3)^2$.

Secondly, I used the following logic to prove that $-3^2$ is in fact the same as $(-3)^2$:

$(-3) = -3$ and $-(3) = -3 \implies -(3) = (-3)$ which implies that $(-3)^2 = -(3)^2$. But this is incorrect.

What am I missing? And more importantly, what does "$-3$" mean and why does it seem like brackets change what "$-3$" means?

Edit: Seems like I have been bad at explaining what I want to know. My problem here is that there is apparently no way to write a negative number as one unit (for a lack of a better term). "$-n$" does not mean a number directly, but rather the number that is the result of a unary operation and a number $n$. So why is this the way it is? Why doesn't our notation allow us to directly write a negative number as one unit?

I literally started the post by saying that $-3^2 =-9$ so the answers pointing to a supposed duplicate is not helping.

Тyma Gaidash
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wafredi
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    $-x$ is the additive inverse of $x$. When $x=3^2$, $-x$ is written just as $-3^2$, instead of $-(3^2)$. That’s all. – azif00 May 15 '23 at 22:40
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    One simple argument is that, in the order of operations, exponentiation takes precedence over multiplication and hence $-3^2 = -1 \cdot 3^2$. After all, you wouldn't suggest that $3-x^2$ really means $3+(-x)^2 = 3+x^2$, would you? – PrincessEev May 15 '23 at 22:45
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    Surely something like this has been asked here before. – Shaun May 15 '23 at 23:22
  • @SarveshRavichandranIyer it does not. I'm aware -3² = -9 – wafredi May 16 '23 at 11:07
  • @wafredi Ok, then I will leave it as a related question and retract the vote. However, the reason why I was so confident is because I felt it might cover some part of what you're asking. For example, I don't get the logic in your first paragraph : you say that $-3^2=-9$ is correct (ok), but then there isn't any "$-3$" at all in there. In fact, when you claim that $-3^2 = -9$, the squaring $\cdot^2$ takes precedence over the negation $-*$, so that (in terms of brackets) you're effectively saying $-(3^2) = -9$ (minus of (square of $3$) = minus of 9). – Sarvesh Ravichandran Iyer May 16 '23 at 11:23
  • So the left hand side doesn't have a "$-3$" at all, because the squaring separates them. If you think of "minus" and "square" as two separate operations, then the problem with what you write in your third paragraph is that even if $-(3) = (-3)$, it is not true that $-(3^2) = (-3)^2$ because in the LHS, you first squared and then came the minus, while on the RHS, you first did the minus and then came the square. You can't interchange the square and the minus because they don't commute. Once again, the RHS has a clear "(-3)" in it, but on the LHS , there is no $-3$ but "minus of square of $3$" – Sarvesh Ravichandran Iyer May 16 '23 at 11:26
  • @KurtG. No, not according to the OP here. – Sarvesh Ravichandran Iyer May 16 '23 at 11:29
  • Coming to the new question, the present notation is flexible. Readers should eventually be comfortable reading $-3$ as both "the negative number $-3$", and "the operator minus/additive inverse applied to $3$". If one isn't comfortable, one resorts to context. For example, in a chapter about "negative numbers" in a textbook, $-3$ is the negative number $-3$ (because no one knows what the "minus" operator is, then). On the other hand, when an equation like $-3^2 = -9$ is present somewhere, a charitable interpretation is that the $-$ refers to the "minus sign applied to $(3)^2$". – Sarvesh Ravichandran Iyer May 16 '23 at 11:35
  • @SarveshRavichandranIyer no, I said that "-3" does not refer to the integer that solves 3+x=0. I know -3² = -133, I thought that would be obvious from the very first sentence. I can also assign a := -3 and then a² = 9, no problem there. So why doesn't our notation allow us to write a negative number without using an operator whose precendence needs to be accounted for? In other words, why do parentheses change what "-3" means? – wafredi May 16 '23 at 11:38
  • @SarveshRavichandranIyer Thank you for the latest answer. Now you are getting to what I want to know. If "-3" indeed is how you write the negative number, isn't it problematic that its meaning is contextual? – wafredi May 16 '23 at 11:42
  • @wafredi Ultimately, at the highest level of mathematical generality, a negative number is the result of "some operation" (in a group, that is the "additive inverse", in the integers that's "applying the minus sign", and in the nonzero rational numbers that's "taking the reciprocal") applied on the original number. You cannot talk about what "negativity" means without talking about the operator that "produces" that negativity i.e. which needs to be applied to a quantity (in our case, $3$) to make it negative. Now, this operation may not behave well with respect to squaring, for example. – Sarvesh Ravichandran Iyer May 16 '23 at 11:43
  • Which is why you need parentheses : to decide whether to square first, or whether to apply said "operation" first. I agree that the meaning of $-3$ being contextual is problematic (and is the origin of a lot of comic capers on social media!). The answer to that debate is that, to the best of my knowledge, I find very rarely the usage of the $-$ sign without signifying its precedence. – Sarvesh Ravichandran Iyer May 16 '23 at 11:45
  • @SarveshRavichandranIyer Thank you! This is what I was looking for. – wafredi May 16 '23 at 11:50
  • @wafredi Thanks. I'm really for suggesting the wrong dupe candidate and misunderstanding your question. I'm glad that I could clarify the actual question here. – Sarvesh Ravichandran Iyer May 16 '23 at 11:53

2 Answers2

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The expression $-3^2$ involves two operations:

  • Negation, aka “additive inverse” or “unary minus”. (Although it uses the same $-$ symbol, this is a distinct operation from subtraction, which takes two operands.)
  • Exponentiation. (When the exponent is 2, this is given the special name “squaring”.)

The question is, which one takes precedence?

  • If negation has higher precedence, then $-3^2 = (-3)^2 = 9$.
  • If exponentiation has higher precedence, then $-3^2 = -(3^2) = -9$.

The standard convention is that exponentiation takes higher precedence than negation. A nice feature of this convention is that it simplifies writing polynomials with negative leading coefficients, e.g., $- x^2 + 6x - 9$. If negation had precedence, then we'd have to write $-(x^2) + 6x - 9$ instead.

Thus, $-3^2 = -9$.

Dan
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    Then this means that there's no way for me to actually write a negative number. I can only write the negation of a positive number. – wafredi May 16 '23 at 11:12
  • @wafredi The negation of a positive number is not a negative number? Please elaborate. – Kurt G. May 16 '23 at 11:42
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    The negation of a number is a number accompanied by an operator, not simply a number. The precendence of that operator must be taken into account when doing anything with the number. This just goes back to the problem with -3². This is not the number "-3" squared, it's the number 3 squared, multiplied by -1 because exponentiation takes precendence over unary minus. – wafredi May 16 '23 at 11:47
  • @wafredi: Indeed, that's the way the grammar is defined in may programming languages: 3 is in an int literal, but -3 is an expression. – Dan May 16 '23 at 13:51
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The mistake in your logic is where you say that “$-(3)=(-3)$ implies that $(-3)^2 = -(3)^2$.” If two things are equal, their squares are equal, but the square of $-(3)$ is not $-(3)^2$. The square of $-(3)$ is $(-(3))\cdot(-(3))$, which is different from $-3^2$, because exponentiation has higher precedence than the unary minus operator.

Steve Kass
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