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can we consider SQRT(n) be negative?

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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by gabriel » Mon Dec 24, 2007 10:14 am
sqrt of a number is always positive... the common notion that sqrt of a number like 4 is both 2 and -2 is wrong .. 2 and -2 are the roots of the equation x^2-4=0, whereas the sqrt of 4 is only 2 ..

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by StarDust845 » Mon Dec 24, 2007 8:04 pm
Are you talking about mathematics in general? Or are you talking about what I should consider for GMAT? because

https://en.wikipedia.org/wiki/Square_root

In the above link it says sqrt(9) could be -3 or 3.
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by gabriel » Mon Dec 24, 2007 8:36 pm
I guess I should have been a little more specific, the nonnegative sqrt of a number is called the principal sqrt of that number and unless specified, when we talk about a sqrt of a number we are talking about the principal sqrt, this is true for both general mathematics and GMAT.

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by srawat_itpro » Fri Dec 28, 2007 2:56 am
gabriel wrote:I guess I should have been a little more specific, the nonnegative sqrt of a number is called the principal sqrt of that number and unless specified, when we talk about a sqrt of a number we are talking about the principal sqrt, this is true for both general mathematics and GMAT.

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Not necessarily so!! I have seen a few questions where it was not explicitly specified to consider -ve root also but still the answer was dependent on it. I guess unless explicitly mentioned such as X^2 > 0 or X^2 is positive , one should consider both +ve and -ve roots. Generally in questions which require one to determine factors of X^2 , we should consider only +ve factors.

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by gabriel » Fri Dec 28, 2007 5:13 am
srawat_itpro wrote:
gabriel wrote:I guess I should have been a little more specific, the nonnegative sqrt of a number is called the principal sqrt of that number and unless specified, when we talk about a sqrt of a number we are talking about the principal sqrt, this is true for both general mathematics and GMAT.

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Not necessarily so!! I have seen a few questions where it was not explicitly specified to consider -ve root also but still the answer was dependent on it. I guess unless explicitly mentioned such as X^2 > 0 or X^2 is positive , one should consider both +ve and -ve roots. Generally in questions which require one to determine factors of X^2 , we should consider only +ve factors.

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I think the question you are talking about are questions based on inequality, in which case both the roots have to be considered. If the question is based on a simple equation and we are asked the root of a positive number then we do consider the positive root unless specified, there are reasons for it (Hint : Pythagoras theorem) which is beyond the scope of this question.

I am absolutely sure that this is also mentioned in OG 11, but I dont have mine any more, if any one knows about the question in the explanation of which this property is mentioned then please do post the question number over here.

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by simplyjat » Tue Jan 08, 2008 10:27 pm
Square root of a number is always positive and negative. i.e. sqrt(9) = +3 & -3. And GMAT specifically checks that you consider both the alternatives. The equation "X^2 -9 = 0", can be simplified to X^2 = 9 and X = sqrt(9); giving the answer X = +3 or -3.

Now there are situations (exceptions), such as length, where the negative number does not make any sense. And in these cases the negative value can be discarded. i.e. if the area of a square is 9, then after calculations, we can omit -3 as length is never negative.
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by StarDust845 » Wed Jan 09, 2008 11:36 am
simplyjat wrote:Square root of a number is always positive and negative. i.e. sqrt(9) = +3 & -3. And GMAT specifically checks that you consider both the alternatives. The equation "X^2 -9 = 0", can be simplified to X^2 = 9 and X = sqrt(9); giving the answer X = +3 or -3.

Now there are situations (exceptions), such as length, where the negative number does not make any sense. And in these cases the negative value can be discarded. i.e. if the area of a square is 9, then after calculations, we can omit -3 as length is never negative.
You are right, but what gabriel was trying to get at is this:

If n is a positive integer 4, what is the sqrt(n)? We consider only +2.

But for your question for something like x^2 - 4 = 0. What is X? It can be +2 or -2. There is a subtle difference.

(As gabriel wrote earlier, in math review in OG11, they gave an example).

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by solaris » Wed Jan 09, 2008 11:39 am
If a question on the GMAT refers specifically to a number which is sqrt(x), it is ALWAYS referring to the positive root of 'x'.

Obviously, if it talks about x^2 then you have to assume both +ve and -ve values of x.
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by Matt@VeritasPrep » Fri Aug 18, 2017 1:25 pm
AFAIK this is a convention dating back to the ancient Greeks, who didn't use/believe in/consider negative numbers, but did make great use of square roots. √16 = 4, always and forever, because we used the √ symbol.

On the other hand, x * x = 16 has two roots, x = 4 and x = -4.
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