Square root of a number - Should we account for the negative

Question

\sqrt{100}

Obviously this is 10

But technically it could also be -10

I've been taking the MasterGMAT course and they claim that for the purpose of the GMAT we only need to worry about the positive case when it comes to square roots.

However on problem 191 of PS in the 13th edition there is a question that clearly uses the negative.

What is the deal here?

Vips0000 · Answer

GMAT scope is very clear on this. Just remember:

1. sqrt(100) = 10
2. solutions for x when x^2 =100, is x=10, -10

I do not have 13th edition handy right now. but if you check the solution, it must be based on the concept stated above.

MacFauz · Answer

Actually, not just the GMAT. Even in traditional mathematics, the square root of a positive number can only be positive and the square root of a negative number is imaginary.

Vips0000 · Answer

Not sure what would you define as traditional maths, but imaginary numbers are also part of maths as much as a negative root of a positive number is. In GMAT for the sake of simplicity and because of nature of arithmatic involved scope has been limited and per-defined.

walker · Answer

It may help: https://en.wikipedia.org/wiki/Square_root

"Every non-negative real number a has a unique non-negative square root, called the principal square root, which is denoted by \(\sqrt{a}\), where √ is called the radical sign or radix. For example, the principal square root of 9 is 3, denoted \(\sqrt{9} \ = \ 3\), because 32 = 3 × 3 = 9 and 3 is non-negative. The term whose root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign, in this example 9.

Every positive number a has two square roots: \(\sqrt{a}\), which is positive, and \(-\sqrt{a}\), which is negative. Together, these two roots are denoted \(\pm\sqrt{a}\) (see ± shorthand). Although the principal square root of a positive number is only one of its two square roots, the designation "the square root" is often used to refer to the principal square root. For positive a, the principal square root can also be written in exponent notation, as a1/2."[/i]

Vips0000 · Answer

Correct.. this article talks about non negative real numbers with postive and negative roots. similarly all negative real numbers have imaginery root that make a complex number of the x+iy form, where i is imaginary part and equal to sqrt(-1).

But for the GMAT -scope is limited to principal root.

MacFauz · Answer

Hmm.. I'm just saying, ever since high school, I was taught

\sqrt{x^2} = |x|

This is followed in the GMAT but I was using this concept even in high school math..

Vips0000 · Answer

I'm sure the high school book must have had the text like this before the formula: "For all real number x / or the prinicpal root of x can be given as:" Anyway.. all said and done, what matters on this forum is what GMAT wants and thats principal root.. :-D

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Square root of a number - Should we account for the negative

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