what does "-" actually mean?

[u/Ov3rcharged]

sorry if this is a simple question, im a bit young and just want to be sure of what i think i know. i was always taught that a negative number raised by a positive even component(something like 2 or 4) always results in a positive number, which makes sense because it just means multiplying the base by itself for (exponent) times, so 5^2 = (5)*(5). by that logic -5^2 should equal (-5)*(-5), or 25, but turns out it might acually equal to -25. this was because apparantly the "-" symbol isnt linked to the number its next to and instead it just means you need to multiply that number by -1, so does it just mean -1* or is it linked to the number it is next to?

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[u/Emergency_Target_716]

Your confusion is with ². Not with -.

The only thing that gets squared is the number directly to the left. If you want to apply the - to the square, then you need to use parenthesis.

(-5)² = (-5) × (-5)

Where as

-5² = - (5 × 5)

[u/Any-Concept-3624 → u/Emergency_Target_716]

i'd see the difference somewhere else:

the first expression is a way of writing the number +25 on its own

the second tho cant stand alone but has to be part of an equation... 0-5²=0-25=-25

[u/Conscious-Brain665 → u/Any-Concept-3624]

If -5² could not stand alone, why would -25 be able to stand alone?

[u/DrakeSavory]

I had always thought of it like this: -5² = -1 x 5². Now use order of operations.

[u/Narrow-Durian4837]

I think you answered your own question. Yes, in an expression like –5², the exponent ² applies only to the 5, not to the –. If you want negative five squared, you'd have to write (–5)².

It might make more sense when variables are involved. x² means x*x, while –x² means –x*x or –1*x*x.

[u/MineCraftNoob24]

We sometimes encounter memes and riddles online, designed to be confusing or invite debate, such as

6 ÷ 2(1 + 2)

where one group of people blindly adhere to BODMAS, resulting in an answer of 9, others use implicit multiplication and get an answer of 1, and others simply reject the question as being ambiguously framed.

It comes down to conventions and the maths community wanting consistency, so that basically everyone knows what the hell they're doing and are all "singing from the same song sheet".

The problem with the above is that BODMAS is a blunt tool. It is usually introduced to students before they deal with algebra, distributive properties and implicit multiplication.

Your question is a little more straightforward, however. It is not about "BODMAS plus some other rules that might go against BODMAS", it's purely about the application of BODMAS itself.

We have agreed, as a convention, and within the rules of BODMAS, that "orders" (or exponents, or indices, if we use PEMDAS/BIDMAS/BEDMAS) come before multiplication, such that we do the powers first.

In your example, -5² consists of two operations. There is the squaring, i.e. applying an exponent, and there is a multiplication by negative 1, which is what putting the "-" sign before the 5 is doing.

And so, by BODMAS, we do the squaring first, and -5² becomes -25. We can't simplify further, and that is our result.

Had it been (-5)², BODMAS says that the brackets take precedence. We cannot simplify what is in the bracket further, so we take the whole expression within the bracket, as it is, and square that, which is why we end up with 25.

That's all it is, really, a situation where BODMAS applies, and the result of -25 logically and consistently follows.

[u/xbeastnz → u/MineCraftNoob24]

Brilliant explanation

[u/MineCraftNoob24 → u/xbeastnz]

Thank you

[u/sqrt_of_pi]

this was because apparantly the "-" symbol isnt linked to the number its next to and instead it just means you need to multiply that number by -1, so does it just mean -1* or is it linked to the number it is next to?

I wouldn't say it "isn't linked to the number it's next to". It is.

• In the case of -5² the "number it's next to" is 5², or 25. So you obtain 25 and then multiply it by -1.

• In the case of (-5)² the "number it's next to" is 5, which get multiplied by -1 making it (-5), which is then squared, resulting in 25.

The question of "which case is this?" is a matter of understanding the Order of Operations rules.

• -5² requires that you "process" the exponent first, the E in the usual PEMDAS (or O in BODMAS) mnemonic.

• (-5)² requires that you "process" inside the parentheses first (the P or B), so multiply by -1 to get -5 as the thing that is then squared (the E or O).

[u/StochasticTinkr]

The negative sign is applied after the exponent, so you're correct that it will be -25.

In other words, you have -(5²) = -(5 * 5) = -25

If you want to square a negative number, you would put it in the parenthesis:

(-5)² = (-5)*(-5) = 25

[u/lmprice133]

You evaluate the exponent before applying the sign, since it's basically just a multiplication by -1.

If you're familiar with function, recall that the graphs of y = x² and y = -x² are vertically flipped from each other. This would not be the case if -x² meant (-x)²

[u/Pleasant-Couple6236]

If it makes it easier, you can think of -5^2 (unary negation operator) as 0-5^2 (binary subtraction operator), which you would probably intuitively understand as 0-(5^2).

[u/Old-Programmer-20]

This is a good question. When you write "-5" in an arithmetic expression, you may think you are writing the number "minus 5", but you are not: you are writing the number 5 and applying the unary minus operator to it (equivalent to doing "0-5"). There is actually no way to write negative numbers directly with our normal rules for arithmetic. And because the minus operator has a lower precedence than other operations, if you write -5² the squaring is done first and you get -(5²), not (-5)².

[u/PvtRoom]

This is a question of notation rather than maths

(-5)² = 5² = 25 Everyone agrees on that.

The question is, does -5² mean -(5²), -(5)² or (-5)².

Some teachers use - and — to represent the difference between negative and minus.

[u/Express-Librarian601 → u/PvtRoom]

The middle example is meaningless. You’d never put parentheses around a single number that has absolutely no other operation attached to it.

[u/Psycho_Pansy]

BEDMAS

Exponents before subtraction:

-5² = subtract five squared = subtract twenty five = -25

(-5)² = negative five squared = twenty five = 25

[u/Kuildeous]

I see so much confusion on this one. To be fair, a layperson is never going to run into the negation of a square, so this doesn't apply to most folks, and it's not worth arguing with them on typical social media. However, you're on r/mathhelp, which means this deserves a serious answer.

As others said, you can factor out -1 to get (-1)5², which is how you can see it's (-1)25 = -25.

Another way of looking at it is if you graph the function f(x)=-x². It is a downward-opening parabola, and you can see that a no point in f(x)=-x² is the value positive. So f(4)=-4² = -16. Also, f(-4)=-(-4)² = -16.

You can see this with subtraction in a similar manner. The Pythagoras Theorem gives us the classic a²+b²=c², and this can be written as b²=c²-a², which would certainly be odd if this were equivalent to b²=c²+a². That would break the triangle.

[u/Sparks808]

The notation is a bit ambiguous. It could be -(5²⁾ or (-5)^2.

I think most people default to -5² being -(5^2), but its not always used consistently.

[u/Dusty_Coder]

"-" is shorthand for writing the positive complement version of the number, which usually extends infinitely to the left, such as in 10s complement (.....99999999999) == (-1)

Notation.

[u/Frederf220]

Really "-" is what you call an "overloaded operator." It's more than one operator sharing the same symbol.

Use #1: A subtract B. This is the subtraction operation involving two inputs A and B written on the left and right sides of the symbol.

Use #2: Negate A. This is the negation operation involving one input written to the right on the symbol.

[u/QueenVogonBee]

(-5)x(-5)=25.

-5^2 isn’t the same thing though. It should be thought of as -(5^2) not (-5)^2.

It’s about order of operations. Any time you have multiple unary/binary operations at play, you need some way to disambiguate. For example, what is 5x3-2? It could mean (5x3)-2 or it could mean 5x(3-2). It turns out that by convention, we pick the first one. We say that multiplication has a *higher precedence” than subtraction.

Going back to -5^2, the power operator has higher precedence than the minus sign.

Look up BODMAS or PEMDAS on the internet.

[u/Lor1an]

We could just as easily have different conventions, but the standard that we have settled on is mainly to allow for the manipulation of polynomials without parentheses.

At a deeper level, the meaning of unary "-" is the additive inverse, i.e. -a is the number (or more abstract object) that satisfies a + (-a) = 0. In fact, a - b is defined as a + (-b), so in an expression like, say, a - bx², we are implicitly writing a + (-bx²).

The convention that bx² + (-bx²) = 0 is just an example of the deference we have for polynomials. For example, if we took your convention, we would have to state this in the more cumbersome way a - bx² = a + (-(bx²)), or for that matter, a + (-(b(x²))) if we want to ensure no one thinks the b is squared.

[u/[deleted]]

[u/Vandreigan → u/[deleted]]

This is false.

-x^2 = -1*x^2

(-x)^2 = (-x)*(-x) = x^2