Very minor nitpick: mathematicians and engineers (among others, I suppose) use a symbol i (or j for electrical types) to signify the square root of -1. It is thus possible for a number to be negative, and yet the square of another number.
I have yet to hear an explanation of this that I can fully comprehend, and I would be greatful if anyone can give one.
That’s not really a definition, as it asserts the existence of an object with certain properties.
Not to take this too far afield, but here’s where we get negative numbers from:
Given N (the set of natural numbers–0, 1, 2, 3, …) and addition and multiplication on them, we take ordered pairs of them. We then define (a, b) to be equal to (c, d) iff a + d = b + c. The aim here is to have the integer x represented by the set of pairs of the form (x + a, a). In other words, (a, b) represents a - b.
Define (a, b) + (c, d) to be (a + b, c + d). Under this definition, any pair of the form (n, n) acts as an identity element: (a, b) is equal to (a + n, b + n). From this, you see that (b, a) is the additive inverse of (a, b).
What about (a, b) * (c, d)? Not so simple. That’s defined to be (ac + bd, ad + bc). This is actually pretty easy to remember if you do it as (a - b) * (c - d), bearing in mind that all four variables are positive.
Under our scheme, -1 can be represented as (0, 1). What is (0, 1) * (a, b)? By my calculations, it’s (b, a). Verify that, if you like.
And hence my earlier comment to matt_mcl. Every time you see a negative number, you may replace it by -1 times a positive number, and nothing changes, except the sequence of characters you write/type.
Does any of this make sense?
zev was speaking of real numbers, I believe. [symbol]i[/symbol] doesn’t belong there.
Take ordered pairs of real numbers, (x, y). Define (a, b) + (c, d) as (a + c, b + d). Define (a, b) * (c, d) as (ac - bd, ad + bc). (0, 1) * (0, 1) = (-1, 0). That’s all there is to it–no deep mysteries, just a set-theoretic construction.
Of course, I could say more, but I fear I might lose you. 
Same thing. In the case of -3^2, how are we certain that it don’t mean -9? It’s just convention. The convention says -9.
You think it’s not?
I doubt it.
And infinite regression? What about 1=0+1? Wouldn’t your infinite regression tell you then that 1=0+0+0+0+0…? I guess positive numbers are impossible too?
Arggg…you got me!
-3^2 = 9.
-3^2 is not the same as saying 0-3^2, nor is it the same as saying -13^2. To say that 0-3 = -3 is true and to say that -13 = -3 is true but to replace them in the original equation you must use parentheses to indicate your subsitution correctly, i.e.
-3^2 = (-1*3)^2 = (0-3)^2 = 9
If you really want to go into real and imaginary numbers we can…
A real number is defined as one that can be evaluated using the laws of mathematics, an imaginary number is one that cannot.
If -3^2 = -9 were true then the reverse, i.e. (-9)^(1/2) [square root of -9] would be a real number, but it is not, as the square roots of negative numbers are imaginary : they cannot resolve.
Phlint, if you’re still reading this, then I feel sorry for you, for what has become of your question on eighth grade math. Now, all you need to worry about when dealing with your brother is the order of operations I put in my first post.
However, if you care, then there are people who are not satisfied with the order of operations as they are classically given, and will violate them. There’s nothing all that wrong with this, but don’t do it until you’re comfortable with the rules first. For instance, as David Simmons said, sometimes you will see this:
-1[sup]n[/sup]
This will always mean (-1)[sup]n[/sup], because -(1[sup]n[/sup]) is fairly useless.
I don’t care what anybody says or Dr. Math or that school textbook, minus three times minus three is plus 9 and that’s all there is to it.
When you enter minus three and times it by minus three on any calculator you get plus 9 too, by the way. As my old algebra teacher used to say, “a minus times a minus is a plus, a plus times a minus is a minus, and of course a plus times a plus is a plus.”* Minus three squared is minus three times minus three by definition.
*Mrs. Casey, Algebra I, Taft High School, 1952
Yes. But that’s not what -3[sup]2[/sup] means.
Just to throw something else into the mix, we all know that x[sup]2[/sup] = x * x. Now what happens when x = -3?
My Excel Spreadsheet says the result is -9 and that’s good ebnough for me. I’m not going to argue with Bill Gates.
When x=-3 then x^2=9 but that is not the way -3^2 is parsed or interpreted. By convention the exponentiation is done first so that -3^2 = -(3^2) and if you want it to be otherwise then you need to write it thus: (-3)^2
Why is this so hard? What if x = 3 + 4? Then is x[sup]2[/sup] = 3 + 4 × 3 + 4? Of course not. It’s (3 + 4) × (3 + 4).
Mr Willard
minus three times minus three is plus 9 and that’s all there is to it.
No-one’s disputing this. But the question is how will the specific expression “-3^2” be evaluated, not “minus three times minus three”, which would be unambiguously written as (-3)*(-3) or as (-3)^2. There’s an old acronym that was taught in UK schools when I was a kid, BODMAS, that explains the order of evaluation. BODMAS = Brackets, Order (i.e powers), Divide, Multiply, Add, Subtract. So when you have no brackets to clarify things, it’s the convention that you do the power 3^2 first, then the minus.
The difference is that the `-’ in front of a negative number is it’s own unary operator, which can have a precedence level that is different from addition, multiplication, or exponentiation. I think all here agree that unary negation (i.e. a negative number) is higher precedence than addition or multiplication. And while many people intuitively want it to have higher precedence than exponentiation, any authority we’ve collectively consulted says it does not.
Now, I considered this, and I could not think of a single case where it would make a difference, whether it had a higher precedence or the same precedence. Can you?
Y’know, its funny how many ‘justifications’ for both sides of this argument there are that are totally irrelevant.
The only basis for whether -3^2 is 9 or -9 is entirely on whether or not the order of operations specifies that unary negation has higher or lower precedence than exponentiation (as Achernar says).
As far as I can see (and I Am Not A Mathematician) there can be no appeal to mathematics to solve this conundrum. This is an issue of pure grammatical syntax, and nothing more. It is an issue of convention and convenience.
I (and it seems several other ‘authorities’ in one way or another already mentioned in these posts) seem to agree that the minus sign is not an intrinsic property of the number to its right, but rather is an operator designed to negate the property on its right.
I’d agree with this myself. A unary minus can be applied to any expression in algebra, not just a number, and it is no more or less an intrinsic property of an expression it preceeds than it is a numeric constant. -(x+y)^2, -x^2, and -3^2 all share similar properties: if the negative sign is an intrinsic property of whatever is on its right, then all of the above formulae end up squaring the negativity away. That is, somehow, the - sign is some intrinsic property of x (for example), whatever x is. This to me is juggling definitions for the sake of juggling definitions. It is not necessary, because no matter if you consider the negative sign an intrinsic property or not, it is describable by unary minus being an operator with a precedence.
It is simplest to consider the negative sign as an operator that operates on an expression to its immediate right. If it is an operator, it has precedence along with the rest of the precedence rules.
Furthermore, you can work the mathematics out no matter which way you choose to do the precedence, using parentheses when you want to override the situation, so the choice is down to convention. Period.
So this is entirely about choosing and/or justifying operator precedence, and nothing else.
Justification can come in the form of the ambiguity (or lack thereof) of grammatical syntax, and it can come from conventional use, but it cannot come from pure mathematics. This is all just symbol-pushing, not math. The math comes from evaluating the expression, but with sufficient parentheses, we can force it one way or the other. Justification cannot come from mathematics (square roots, imaginary numbers, whatever). Arguments based on anything other than grammatical syntax convention are meaningless.
For the record, though, as long as opinions are being put out, I’ll put forth my own, based on what I learned in school, and what seems to be on a few math websites.
-3^2 = -9
-x^2 = -(x)^2
-(x+y)^2 = -(x^2 +2xy + y^2) = -x^2 + -2xy + -y^2 = -x^2 - 2xy - y^2
3----1^2 == 4 (that is, 3 - (-(-(-(1^2))), not 3 - (—1)^2)
It is up to convention, entirely. I notice that the FORTRAN programming language places unary minus at the same operator level as subtraction, meaning:
x/-15.0*Y
is grouped as:
x/ -(15.0*Y)
which is counterintuitive, but that’s the order that they placed on it. C/C++ on the other hand puts unary minus above multiplication (there is no exponentiation, so this doesn’t help give us some hints), meaning the above expression would be resolved as:
x/((-15.0) * y)
All of you who ‘think’ you know what the ‘real’ precedence is are pontentially wrong by definition, simply because there is no ‘real’ precedence. Merely human choices.
Having said that, there are, as I’ve said repeatedly, conventions, and fairly strong ones at that. Most of us learned the typical BEDMAS (BODMAS, PEDMAS in some places) order of operations. unary minus doesn’t easily fall into any of these categories by name. M, for multiplication, is about the only one that makes direct sense, as it is justifiable that:
-x
is equivalent to
(-1)x
Not only was this the way I was taught, but it suggests that unary negation has the same precedence as multiplication, is fairly easy to remember, and can be justified by its relative commonness by convention of math textbooks and reasonably easy syntax.
Note:
http://mathforum.org/library/drmath/view/53194.html
which talks about the relationship between unary negation, programming languages, and mathematics.
That website makes a good case for unary negation for being at least as high as addition/subtraction, but no higher than exponentiation. As far as convention goes, it seems to me reasonably clear that in mathematics (programming languages aside) that exponentiation occurs before unary negation, and not after.
Sorry if this felt like a rant, but it seemed like justifications were being stated that really dodged the point entirely.
So for what its worth, I agree with the OP’s textbook, for reasons of pure convention of grammatical mathematics syntax. Surely we can dredge up more ‘ask dr. math’ type websites that say one way or another.
I guess I’m repeating long-windedly what’s already being said. 
Its definition a fascinating discussion. I am enjoying it, rant aside.
The unary negation operator has the exact same effect as multiplying by -1. In fact, there’s no reason not to define it that way.
And like Achernar said, it wouldn’t ever make a difference.
And java or c says it’s -1, so what does that prove :).
(java/c has no exponentiation operator - ^ indicates bitwise exclusive or and is bound lower than unary minus).
What this discussion actually points out is that an expression like that is actually ambigous in much informal use. If you must write such an expression, you might wish to write (-3)^2 or -(3^2) to make your intent clear. Strictly speaking, the exponentiation is supposed to be performed first, but as Dave Simmons pointed out, even many math textbooks may present you with expressions that intend you apply it the other way, especially in power series examples like he was talking about.
It’s a pity we didn’t invent arithmetic using pre- or post-fix notation rather than infix with precedence. If we had grown up doing arithmetic, say, like this:
3-,2^ = or
3,2^- =
the difference would be clear. If we had learned that way, it wouldn’t seem particularly awkward, and we wouldn’t have precedence ambiguities. It might also help if our notation distinguished a “negation flag” to be read as the sign of a literal number (and thus part of the actual numeric token) from the unary minus operation.
The reason -3^2 = -9 is surprising to me is not based on the relative precedence of exponentiation Vs. unary negation. It’s because the minus sign feels like a part of the number, right up there with the decimal point, and not like an operator at all.
BTW, I especially like this bit from the Dr. Math site linked to above: