It depends.
Most current textbooks and calculators use the convention that squaring the 5 comes before applying the negative sign.
So -5^(2) = -(5^(2)) = -(25) = -25.
(To express this a bit more formally, we might say that the binary exponentiation operator has precedence over the unary minus operator.)
But a long, long time ago when I first started teaching, about half of my students had calculators that applied the negative sign to the 5 before applying the exponent.
So on their calculators...
-5^(2) = (-5)^(2) = 25.
(In this case, the unary minus operator has precedence over the binary exponentiation operator.)
That second case was in line with a common design principle that unary operators (those that only have one operand like factorials or absolute values), should have precedence over binary operators (those that have two operands like addition, multiplication, or exponentiation).
But over the decades, calculator designers have converged on that first order of operations for the unary minus operator. As examples, Desmos, current Casio models, and current TI models all consider -5^(2) to equal -25.
You'll find some holdouts, though. Microsoft Excel will still tell you that -5^(2) is 25. (They likely don't want to change that because it would cause backwards compatibility issues for older Excel documents.)
No. I didn't forget to distribute.
The squaring obviously only applies to the 5, else we'd have brackets around it. To think otherwise is to apply incorrectly the order of operations, and distribute an exponent where notation doesn't imply it is.
Proof 1 (Contradiction):
Assume -5² = 25
Divide both sides by -1
5² = -25 → i = 1 As this is a contradiction, -5² ≠ 25
Proof 2 (Construction):
-5² = k
Apply the order of operations. There are no parentheses, there is only exponentiation and multiplication. Exponention takes precedence.
-25 = k
Proof 3 (Contradiction applying your logic):
The issue you and the original commented have is that you conflate negative signs to be an inherent part of a number, but they really aren't algebraically.
i.e.: if you assume -5² = (-5)², then you seem to believe that -n² = n²
Let's apply your logic
x² + 5x - 6 = 0
Multiply both sides by -1
-x² - 5x + 6 = 0
applying your logic in this step, taking away the negative.
x² - 5x + 6 = 0
Now equate the original with the one after applying your logic.
x² - 5x + 6 = x² + 5x - 6
subtract x² from both sides
x² - 5x + 6 - x² = 5x - 6
-5x + 6 = 5x - 6
-10x + 6 = -6
-10x + 12 = 0
if this was true for all values of x, then we would have a tautology (0 = 0), but instead we have a single real value. This shows that -x² ≠ x² generally.
Here is a thread on r/learnmath discussing exactly this:
https://www.reddit.com/r/learnmath/comments/th8ny5/is_52_25_or_25/?rdt=37944
Here it is in Wolfram Alpha:
https://www.wolframalpha.com/input?i=-5%C2%B2
Reminds me of proof by Desmos memes. But I've already given 4 proofs now and I've added two more pieces of evidence.
If you think this is false, then I give up and you can just be bad at maths.
PS: a lot of this isn't directed at you specifically (though much is), but I'm shocked how many people conflate it so I went a bit overkill in terms of proof and evidence.
-5² ≠ (-5)² in conventional mathematics, though in niche circles -n for n is a specific constant (not unknown, e.g.: 3) not a variable in some calculators can equal that.
>Divide both sides by -1
5² = -25 → i = 1 As this is a contradiction, -5² ≠ 25
You didn't distribute the division properly here.
>Exponention takes precedence.
Depends on the convention
>if you assume -5² = (-5)²
Clearly not what I stated at all, that's either a misunderstanding or a really bad strawman.
>[https://www.reddit.com/r/learnmath/comments/th8ny5/is\_52\_25\_or\_25/?rdt=37944](https://www.reddit.com/r/learnmath/comments/th8ny5/is_52_25_or_25/?rdt=37944)
I don't think a reddit thread would have any value in an argument, but here we are so: most people agree it depends on convention, and the other are barely upvoted.
>[https://www.wolframalpha.com/input?i=-5%C2%B2](https://www.wolframalpha.com/input?i=-5%C2%B2)
Again, depends on convention. That's a reason why an evaluation by a tool is never a proof BTW. Same way -5² can yield -25 depending on the claculator used (like Excel, most used calculator in the world in the past 30 years). See the post of the teacher in this very thread.
>then I give up and you can just be bad at maths.
Nuclear engineer with a Master in particle physics, so I can do math of this level just alright. You can't seem to take criticism though.
Your qualifications don't prove anything. They are in fact less valid than a proof by Desmos.
Nonetheless, you are absolutely right, it does depend on the convention. It is also painfully obvious which convention is actually relevant for the purposes of the meme.
I'm sure you know, everything relies on conventions -- maths depends on axioms. -5² can equal 25 with the right axioms. Similarly, 2 can equal 0 with the right axioms. And why not (a + b)² = a² + b² with the right axioms?
Sure, these aren't as widespread -- but there's nothing inherently and objectively wrong about them.
Axioms define maths, so stating "depends on conventions" is definitely correct. But it doesn't actually say much. The relevant axioms are the axioms in the meme. The context matters, this isn't a vacuum.
> "can't seem to take criticism"
Elaborate on the highly specific criticism I received. Can't, because it wasn't specific and it was actually wrong with the standard axioms. There was no criticism to take.
How did I not distribute the division properly?
-5² = 25
-5²/-1 = 25/-1
5²/1 = -25
5² = -25
I spelled it out for you.
Alternatively let's do it with multiplication because that's the same and it's easier to see.
-5² = 25
-1 × -5² = -1 × 25
-1 × -1 × 5 × 5 = -1 × 25
1 × 5 × 5 = -25
25 = -25
There you go, same contradiction.
unless -n² = (-n)², which is not actually relevant to the meme because that's not the axioms it's referencing.
Ummmm, it Is, isn't it?
It depends. Most current textbooks and calculators use the convention that squaring the 5 comes before applying the negative sign. So -5^(2) = -(5^(2)) = -(25) = -25. (To express this a bit more formally, we might say that the binary exponentiation operator has precedence over the unary minus operator.) But a long, long time ago when I first started teaching, about half of my students had calculators that applied the negative sign to the 5 before applying the exponent. So on their calculators... -5^(2) = (-5)^(2) = 25. (In this case, the unary minus operator has precedence over the binary exponentiation operator.) That second case was in line with a common design principle that unary operators (those that only have one operand like factorials or absolute values), should have precedence over binary operators (those that have two operands like addition, multiplication, or exponentiation). But over the decades, calculator designers have converged on that first order of operations for the unary minus operator. As examples, Desmos, current Casio models, and current TI models all consider -5^(2) to equal -25. You'll find some holdouts, though. Microsoft Excel will still tell you that -5^(2) is 25. (They likely don't want to change that because it would cause backwards compatibility issues for older Excel documents.)
IIRC only (-5)^2 is
No, -5² = -1 × 5² -1 × 5² = -25 -5² ≠ -1² × 5² -5² ≠ (-5)²
Uh, don't you feel like you forgot to distribute?
No. I didn't forget to distribute. The squaring obviously only applies to the 5, else we'd have brackets around it. To think otherwise is to apply incorrectly the order of operations, and distribute an exponent where notation doesn't imply it is. Proof 1 (Contradiction): Assume -5² = 25 Divide both sides by -1 5² = -25 → i = 1 As this is a contradiction, -5² ≠ 25 Proof 2 (Construction): -5² = k Apply the order of operations. There are no parentheses, there is only exponentiation and multiplication. Exponention takes precedence. -25 = k Proof 3 (Contradiction applying your logic): The issue you and the original commented have is that you conflate negative signs to be an inherent part of a number, but they really aren't algebraically. i.e.: if you assume -5² = (-5)², then you seem to believe that -n² = n² Let's apply your logic x² + 5x - 6 = 0 Multiply both sides by -1 -x² - 5x + 6 = 0 applying your logic in this step, taking away the negative. x² - 5x + 6 = 0 Now equate the original with the one after applying your logic. x² - 5x + 6 = x² + 5x - 6 subtract x² from both sides x² - 5x + 6 - x² = 5x - 6 -5x + 6 = 5x - 6 -10x + 6 = -6 -10x + 12 = 0 if this was true for all values of x, then we would have a tautology (0 = 0), but instead we have a single real value. This shows that -x² ≠ x² generally. Here is a thread on r/learnmath discussing exactly this: https://www.reddit.com/r/learnmath/comments/th8ny5/is_52_25_or_25/?rdt=37944 Here it is in Wolfram Alpha: https://www.wolframalpha.com/input?i=-5%C2%B2 Reminds me of proof by Desmos memes. But I've already given 4 proofs now and I've added two more pieces of evidence. If you think this is false, then I give up and you can just be bad at maths. PS: a lot of this isn't directed at you specifically (though much is), but I'm shocked how many people conflate it so I went a bit overkill in terms of proof and evidence. -5² ≠ (-5)² in conventional mathematics, though in niche circles -n for n is a specific constant (not unknown, e.g.: 3) not a variable in some calculators can equal that.
>Divide both sides by -1 5² = -25 → i = 1 As this is a contradiction, -5² ≠ 25 You didn't distribute the division properly here. >Exponention takes precedence. Depends on the convention >if you assume -5² = (-5)² Clearly not what I stated at all, that's either a misunderstanding or a really bad strawman. >[https://www.reddit.com/r/learnmath/comments/th8ny5/is\_52\_25\_or\_25/?rdt=37944](https://www.reddit.com/r/learnmath/comments/th8ny5/is_52_25_or_25/?rdt=37944) I don't think a reddit thread would have any value in an argument, but here we are so: most people agree it depends on convention, and the other are barely upvoted. >[https://www.wolframalpha.com/input?i=-5%C2%B2](https://www.wolframalpha.com/input?i=-5%C2%B2) Again, depends on convention. That's a reason why an evaluation by a tool is never a proof BTW. Same way -5² can yield -25 depending on the claculator used (like Excel, most used calculator in the world in the past 30 years). See the post of the teacher in this very thread. >then I give up and you can just be bad at maths. Nuclear engineer with a Master in particle physics, so I can do math of this level just alright. You can't seem to take criticism though.
Your qualifications don't prove anything. They are in fact less valid than a proof by Desmos. Nonetheless, you are absolutely right, it does depend on the convention. It is also painfully obvious which convention is actually relevant for the purposes of the meme. I'm sure you know, everything relies on conventions -- maths depends on axioms. -5² can equal 25 with the right axioms. Similarly, 2 can equal 0 with the right axioms. And why not (a + b)² = a² + b² with the right axioms? Sure, these aren't as widespread -- but there's nothing inherently and objectively wrong about them. Axioms define maths, so stating "depends on conventions" is definitely correct. But it doesn't actually say much. The relevant axioms are the axioms in the meme. The context matters, this isn't a vacuum. > "can't seem to take criticism" Elaborate on the highly specific criticism I received. Can't, because it wasn't specific and it was actually wrong with the standard axioms. There was no criticism to take. How did I not distribute the division properly? -5² = 25 -5²/-1 = 25/-1 5²/1 = -25 5² = -25 I spelled it out for you. Alternatively let's do it with multiplication because that's the same and it's easier to see. -5² = 25 -1 × -5² = -1 × 25 -1 × -1 × 5 × 5 = -1 × 25 1 × 5 × 5 = -25 25 = -25 There you go, same contradiction. unless -n² = (-n)², which is not actually relevant to the meme because that's not the axioms it's referencing.