Why sqrt(4) isn't equall to-2?

Question

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Square roots — positive and negative

$\sqrt{4} = -2$. WolframAlpha says "false"!

Now lets take a deeper look to my idea.

Well...we know that,

$$2^2 = 4 \iff \sqrt{4} = 2$$

$(-2)^2 = 4$ so why can't $\sqrt{4}$ be equal to $-2$?

I'm a little bit confused

// Thank you for all your answers, I have answer now. Stepo

Answer 1

By definition, $$\sqrt{x^2} = \vert x \vert$$ for $x \in \mathbb{R}$.

Answer 2

It's just notation most likely. Yes, $(-2)^2 = 4$, but often the $\sqrt{4}$ symbol is reserved for the positive square root, so $\sqrt{4} = 2$. If you want the negative square root, that would be $-\sqrt{4} = -2$. Both $-2$ and $2$ are square roots of $4$, but the notation $\sqrt{4}$ corresponds to only the positive square root.

Answer 3

You are correct that $(-2)^2 = 4$.

The idea is that we want $\sqrt{x}$ to be a continuous, single valued function. But as you noted there are two possibile values of $\sqrt{x}$ for each $x$, so we have to choose a particular branch and that is what we call $\sqrt{x}$.

So the standard choice is to just take positive $\sqrt{x}$ for every $x$.