TITLE: Is the square root of a negative number defined?
QUESTION [24 upvotes]: I have been in a debate over 9gag with this new comic: "The Origins"

And I thought, "haha, that's funny, because I know $i = \sqrt{-1}$".
And then, this comment cast a doubt:

There is no such thing as sqrt(-1). The square root function is only
  defined for positive numbers. Sorry...

This wasn't by ignorance of complex numbers. In the short round of arguing that happened there, the guys insisted that negative numbers don't have square roots, and that $i$ is defined such that $-1 = i^2$, which is not the same as $i = \sqrt{-1}$. In their opinion, too, no respectable math textbook should ever say that $i = \sqrt{-1}$; which is precisely how Wolfram MathWorld defines $i$.
Is there a consensus over if negative numbers have a square root? Are there some gotchas or disputes with the definitions I should be aware of in the future?

REPLY [3 votes]: The square root of a number $k$ may be regarded as the root of a polynomial of the form $z^2 - k$.  The fundamental theorem of algebra tells us that we can write $z^2 - k$ as $(z - m)(z - n)$, where $m$ and $n$ are the roots which this function necessarily has, being of degree two. $z$, $k$, $m$ and $n$ are all complex. It's possible that $m$ and $n$ are the same: that is there is a root with multiplicity two. For $z^2 - k$ this happens when $k = 0$.
So basically, for all $k$ everywhere on the complex plane, $z^2 - k$ has two roots, meaning that $k$ has two square roots, except for $k = 0$, where $k$ has two roots that are both equal to zero.
Square root is defined exactly as much for the real numbers as for the complex numbers. There are two square roots of four, namely ${-2, 2}$ and there are two square roots of $-1$, namely ${-i, i}$.
It is irrationally inconsistent to accept that there is a defined square root over the non-negative real number line, but not elsewhere.
(That kind of thinking seems like a remnant of the suspicion that imaginary numbers themselves are some kind of fraud, a view that is outdated by several hundred years.)