# Squaring Negative Numbers

1. Jan 9, 2014

### AlfieD

Hi,

In BIDMAS (brackets, indices, division, multiplication, addition, subtraction) it shows clearly that indices should happen before subtraction, so why does -52=25? Because the real working (according to BIDMAS) should put the square first and then put the negative sign on, which would mean the answer is -25. And then for $\sqrt{negative numbers}$, you have to use i; imaginary numbers. And i is equal to the square root of -1. But surely the $\sqrt{-25}$ is -5, not 5.

I was told that BIDMAS was correct in every situation and should be applied to ALL mathematics. Is this wrong? If so, is there anything else it shouldn't be applied to?

Last edited: Jan 9, 2014
2. Jan 9, 2014

### Staff: Mentor

I would read -52 as -(52)=-25. Why do you think it would be 25?
This interpretation is even more obvious with formulas like b2=c2-a2.

$\sqrt{-25}=\pm 5i$, it is neither -5 nor 5.

3. Jan 9, 2014

### Staff: Mentor

-52 is -25, no problem with that.

(-5)2 on the other hand is 25.

Edit: beaten by mfb.

4. Jan 9, 2014

### AlfieD

Haha, I think that because I was taught that if you square a negative number it becomes a positive because I was told that -52 is the same as -5*-5, and if you multiply two negatives you get a positive.

5. Jan 9, 2014

### AlfieD

You two contradict each other here I'm pretty sure; mfb said that $\sqrt{-25}=\pm 5i$ but then Borek, if you say that -52=25, your logic would suggest that the reverse of this would mean that $\sqrt{-25}=-5$. What's odd is that you both agree that -52=-25, you just don't agree on the reverse. Am I incorrect somewhere here, because my brain is fixated on the fact that if 52=-25 that $\sqrt{-25}=-5$?

6. Jan 9, 2014

### Office_Shredder

Staff Emeritus
AlfieD, -52 is -(5*5), not (-5)*(-5) (this is what everyone above is saying). In particular when you calculate -52 you are NOT squaring something and getting -25, you are squaring something and then doing an additional operation (taking negatives) to get to -25.

7. Jan 9, 2014

### AlephZero

$-5^2 = -25$, and $-\sqrt{25} = -5$.

$-\sqrt{25}$ and $\sqrt{-25}$ are not the same.

8. Jan 9, 2014

### AlfieD

Ah, OK, thanks for the clarification. I will make note to tell my teacher he was W.R.O.N.G. wrong the next time I see him! :D

9. Jan 9, 2014

### AlfieD

Thanks, never even considered the two different ones (stupid brain). Does $\sqrt{}$ fall under a category in BIDMAS though (indices for example)? Or is it completely separate and undefined to BIDMAS?

10. Jan 9, 2014

### Staff: Mentor

There is no ambiguity involved in the calculation order for square roots, they are similar to brackets. Calculate everything under the square root, then take the root out of this value.

11. Jan 9, 2014

### AlfieD

Thanks for all the clarification! Big help!

12. Jan 9, 2014

### Staff: Mentor

To add to the comment of mfb, note that the square root √ is better (in the sense of "more general") expressed as 1/2. Thus, changing $\sqrt{a+b}$ into $(a+b)^{1/2}$, you can easily apply BIDMAS.

13. Jan 9, 2014

### DrewD

As a teacher, I would advise that you be careful. Just as you were wrong about mfb and Borek's responses because you misunderstood them, you may also have misunderstood your teacher. I can't tell you how many times a student has argued about how I was wrong and what they wrote was exactly why I had told them, only to go back and look at my notes to see that they had just misunderstood it.

That being said, I have also been both wrong and unclear in my presentation, so you should bring it up to your teacher, but be respectful. Just like your future bosses, teachers are fallible and (almost) omnipotent when it comes to things that are important to you (your grades and your paycheck). That is very dangerous for you even if your teacher was completely wrong. This is a common mistake that I see coworkers make. I have even seen it in books, so your math teacher, who may not have a degree in math if the UK is anything like the US, may be teaching you exactly what he learned was correct.

14. Jan 10, 2014

### AlfieD

I understand your concerns but I definitely know this is what our said because he makes a point of it every time negative numbers come up (this is a lot obviously). I won't just shout out that he's wrong, I'll just subtlety slip into conversation. The only explanation I can come up with as to why he said this to us (I highly doubt it's because he didn't know) is because he doesn't think that's what the questions are asking, I think he wants us to imagine a (-5)^2 situation rather than -(5^2) (I know the second brackets weren't necessary I just put them in in order to show clearly what I meant. I'll ask.

15. Jan 10, 2014

### DrewD

I'm sure he just learned that $-5__^2$ should be interpreted as $(-5)^2$. When my coworkers get this wrong I ask whether $1-x^2$ is always positive. That usually convinces them that the parentheses are necessary. Good luck.

16. Jan 10, 2014

### 1MileCrash

The order of operations is always consistent. The square of a negative number is positive. You were not understanding notation.

17. Jan 10, 2014

### AlfieD

No, we established earlier that -52=-25. My teacher said that it was 25. I don't think you understood the notation properly.

18. Jan 10, 2014

### hddd123456789

AlfieD, -5^2=-25 would be true if interpreted as "the negative of 5 squared is negative 25". But even if there is a misunderstanding of notation, the square of a negative number is definitely positive as in (-5)^2=25.

19. Jan 10, 2014

### Staff: Mentor

-52 is not the square of a negative number. It is the negative of a square.

The square of negative numbers is positive.

20. Jan 10, 2014

### 1MileCrash

Your teacher said that the square of -5 is 25, and that is correct.

-5² is -25. -5² is not the square of negative 5, it is the negative square of positive 5, this is why I said you are not understanding the notation properly.

21. Jan 10, 2014

### AlfieD

Ok I'm getting lost with all of these messages and quotes so I'm just gonna list the facts below:

1) My teacher said that -52=25, not the square of -5, not anything like that; he said that -52=25.
2) He is wrong because we've established that -52=25 is -25.

I spoke to him today and he said that textbooks at our year of maths don't appreciate that -52=25 and they mean (-5)2; the square of -5.

I hope this de-clutters it a bit for you, it does for me anyway.

22. Jan 10, 2014

### Staff: Mentor

No we didn't establish that. -52 = -25. It's not equal to 25.

23. Jan 10, 2014

### 1MileCrash

Ok

As long as you see that the square of a negative is positive, and that your book is using -5² to be (-5)² (which is not standard but likely won't cause many problems for you right now) I think you are fine.

Many of these textbooks introduce notation that is different from the norm for the sake of simplicity, but I think it has the opposite effect. Either that or they don't explain the notation. I knew a physics major who only two years ago thought that the inverse of a function is actually the function raised to the -1 power.

24. Jan 10, 2014

### 1MileCrash

I think he just made a typo there.

25. Jan 10, 2014

### Staff: Mentor

Moral of the story: use parentheses everywhere. Even if they look excessive. They remove ambiguity.

Doesn't mean -(5)2 won't start another round.