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Hi The GRE prep test is asking for the square root of a number.. for example $\sqrt{16}$. It says the answer is $4$. Couldn't the solution be both $4$ and $-4$?


marked as duplicate by user147263, Joel Reyes Noche, drhab, Math1000, hardmath Aug 17 '15 at 13:59

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  • $\begingroup$ The solutions of $x^2 = a$ are $x = \sqrt{a}$ and $x = -\sqrt{a}$. The square root itself is defined on nonnegative numbers. $\endgroup$ – Slowpoke Aug 16 '15 at 22:55
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    $\begingroup$ Ah, the GRE... fond memories. $\endgroup$ – wltrup Aug 16 '15 at 22:59

To amplify what @hcl14 said:

The square-root function is defined on the set of nonnegative real numbers by saying that $\sqrt{x}$ is the unique non-negative real number whose square is $x$.

There's also a notion of the "square root of a complex number," but in making that definition, you must either (a) decide that you want to admit the possibility that the square root produces either a singleton set (when the argument is zero) or a two-element set (when it is not), or (b) pick one of the two possible square roots for each nonzero number in a way that's "as consistent and continuous as possible", and recognize that in doing so, you'll end up with a function that's discontinuous on some arc containing $0$; it's typical to make the definition continuous everywhere except on the non-positive portion of the real line; in this case, the extended definition matches the usual definition on $\mathbb R$.

If the problem had been "What numbers, when squared, give 16?", then the correct answer would be $\{4, -4\}$, but that's not how the square root is defined, no matter what people say they remember from highschool, etc. :)

  • $\begingroup$ Right.It's a matter of convention. Conventions are not theorems, just conveniences with wide-spread usage. Like the definition of prime number,which excludes 1. $\endgroup$ – DanielWainfleet Aug 16 '15 at 23:24
  • $\begingroup$ Well said .. I should have made that point myself Thanks! $\endgroup$ – John Hughes Aug 16 '15 at 23:26

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