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- Sep 3rd 2012, 04:49 PM #1

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## Can you tell me if I'm right? :)

My sister and I are currently arguing about something regarding square roots. We'd like someone here to tell use who's right and why?

Basically, the question is: Is is true that Square root of 16 = ±4?

My reasoning is pretty simple:

Since Square root of 16 is 16^1/2, using the opposite operation, we could prove that both 4 and -4 are equal to 16^1/2 since (4)² = 16 and (-4)² = 16.

My sister's reasoning (well it's not hers since she's too stubborn to write it down...) is pretty much this:

First of all, she brought that the square root of -16 doesn't exist. I don't see the point in that, but she did... What she's basically saying is that you are finding the square root of (±4)² which then means that in can only be 4 for some magic... Honestly, I doubt she even understands her own logic...

When answering, please take into account that I want an answer that is useful for maths a 16 years old would do. Please don't include Ph.D. level reasoning!

Thank you!

- Sep 3rd 2012, 05:02 PM #2

- Sep 3rd 2012, 05:05 PM #3

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## Re: Can you tell me if I'm right? :)

- Sep 3rd 2012, 05:15 PM #4

- Sep 3rd 2012, 05:31 PM #5

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## Re: Can you tell me if I'm right? :)

The definition of "square root" as a function is commonly defined to be specifically the

*POSITIVE*number that when squared gives the original number. Yes it is an extremely asymmetrical and imbalanced definition, but it is agreed upon so that "square root" can be a*function*. Functions gives only*one*value to each number.

So the square root of 16 is 4.

But if you define it as a multiple-valued function, it can be -4 as well. So it can depend on the context. Honestly the argument is simply a question of notation or definition, much like arguing whether the first positive number is 1 or 0. Its not really a mathematical question. On a different planet, it may very well be that the square root of 16 as a function is -4.

I agree with the convention thing but say you had a test that asked you whether square root of 16 = ±4 is true or false, what would you answer? We're not talking about a variable by the way.

But the square root if -1 doesn't equal 1, the absolute value of*i*.

- Sep 3rd 2012, 05:53 PM #6

- Sep 3rd 2012, 05:57 PM #7

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## Re: Can you tell me if I'm right? :)

Ah thanks for the correction. I don't know too much about the precise definitions either. :P In that case it is true that the square root of 16 is both 4 and -4, but 4 is the principal square root.

- Sep 3rd 2012, 06:22 PM #8
## Re: Can you tell me if I'm right? :)

- Sep 3rd 2012, 06:53 PM #9

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- Sep 4th 2012, 06:01 AM #10

- Sep 4th 2012, 06:18 AM #11

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- Sep 4th 2012, 10:56 AM #12

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