# Math complex numbers.

I have heard that $i=\sqrt{-1}$ and I have also read about it here http://www.mathsisfun.com/numbers/imaginary-numbers.html.

Now I want to ask why in example $\sqrt{-4} = 2i$ as $i=\sqrt{-1}$.

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possible duplicate –  mezhang Feb 8 '13 at 8:45
Where is the duplicated question please? –  Enve Feb 8 '13 at 8:45
I do not recall the exact one, but this is similar question that has all the answer you might seek. math.stackexchange.com/questions/49169/… –  mezhang Feb 8 '13 at 8:46
$\sqrt{-4}$ are the complex numbers $x$ that satisfy $x^2=-4$. Thus $x=2i$ or $x=-2i$. –  Stefan Hansen Feb 8 '13 at 8:47
@StefanHansen also x = -2i –  mezhang Feb 8 '13 at 8:48

It is because $(2i)^2 = 2^2 i^2 = 4 \cdot -1 = -4$. Thus $2i$ is a possible answer to $\sqrt{-4}$ (though perhaps not the only one. What about $-2i$?)

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Ok. Thank @mixedmath I understood now. –  Enve Feb 8 '13 at 8:55

If you remember the basic definition

$$x=\sqrt a\Longleftrightarrow x^2=a$$

then

$$(2i)^2=2^2i^2=-4\Longleftrightarrow\sqrt{-4}=2i$$

Of course, also $\,(-2i)\,$ makes the job.

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Thanks. This answer is correct. But I accepted @mixedmath answer. He was first. –  Enve Feb 8 '13 at 9:01
You don't need to explain your decision as one chooses whatever answer seems better, but I thank you for doing it. –  DonAntonio Feb 8 '13 at 9:10
You're being a bit careless with the equivalence arrows here, since also $x=-\sqrt{a}$ satisfies $x^2=a$... –  Hans Lundmark Feb 8 '13 at 10:15
To talk about $\sqrt{-1}$, which is not good, please refer to this page:
$i^2$ why is it $-1$ when you can show it is $1$?
I'd say that choosing the positive root of a positive real number is an agreement more than a definition, and it is done for very good reason (e.g., to make $\,\sqrt x\,$ an actual function), yet it can possibly be defined, and I think it usually is, as I and mixedmath wrote in our answers and everything's fine with the complex square root in this case. Other thing is to talk about arguments and branches, but this can wait until later.\ –  DonAntonio Feb 8 '13 at 9:05