Explain the difference between these square roots

[greenneub]

Hey guys, I was just wondering what the difference between these two statements are:

V¯(x) = ± 4

V¯(x) = - 4 ---> does not exist.

This is the quote from my text, "...we remind you of a very important agreement in mathematics. The square root sign V¯ always means take the positive square root of whatever is under it. For instance, V¯(4) = 2, it is not equal to -2, only 2. Keep this in mind in this section, and always. "

Maybe I've been staring at the pages too long, but how is (4) different from (-2)²? And why can we right ± 2, but not -2? I know this is basic but I'm embarrassingly confused about this.

 

[qntty]

$\sqrt{x}$ means the positive square root of x (this way you can refer to $\sqrt{x}$ and only be talking about a single number, not two numbers). If the author says $\sqrt{16}=\pm 4$, he is just making the point that both $4^2$ and $(-4)^2$ equal 16, however the correct notation is$\pm \sqrt{16}=\pm 4$.

 

[HallsofIvy]

As qntty said, the first, [tex]\sqrt{4}= \pm 2[/itex] is simply <b>wrong</b>. $$\sqrt{4}= 2$$ because $$\sqrt{x}$$ is <b>defined</b> as the <b>positive</b> number y such that $$y^2= x$$. That is why we must write the solution to $$x^2= a$$ as $\pm\sqrt{a}$- because $\sqrt{a}$ does not include "$$\pm$$".[/tex]

 

[AlbertEinstein]

And also because that we wish that square-root should be a "function", and for being a function it has to be defined like that only. By definition, a function takes a value from a set A and maps it into B, and no two numbers in A can map to the same number in B.