Square root of complex numbers

I’m studying complex analysis.

I was reading a section about roots of complex numbers, and I found that $\sqrt{i}$ has two values. ($i$ : imaginary unit) However, for a non-zero real number, $\sqrt{a}$ is always one value. Moreover, $\sqrt{a}$ is a real number.

However, $\sqrt{i}=\pm(\sqrt{0.5}+\sqrt{0.5}i)$.
This is not a number !! I think a number should have one value. If $\sqrt{i}$ is the number having positive sign, then it shouldn’t have the negative sign unless it is a zero.

I understand those are the numbers satisfying $x^2=i$.
But.. what does $\sqrt{i}$ represent for?

Is it different from the sqaure root of real numbers?

Answer

Every complex and real number except $0$ have two square roots. If $r$ is one of the square roots $-r $ is the other.

So it is not true that "$\sqrt i $ has two values". It's that, "$i$ has two values as square roots. And it's not true that positive real numbers have one square root. They have two.

So what does the symbol $\sqrt {} $ and what does "the square root" mean? Well, nothing really. It's convenience to have a single value "square root function" so we arbitrarily chose that the positive value of square roots of positive real would be "the" square root. And the negative square root was the "other".

We could do the same for complex square roots. We could arbitrarily decide the one with a no negative real component was "the" square root and the one with a negative component would be the "other". But what would be the point?

The main reason we do this for the reals is because the real numbers is convenience really. And none of that convenience is useful in the complex numbers. The complex numbers don't have an greater/less than ordering. You are going to learn that exponentiation is cylic and there are multiple logarithms of each number (don't worry about that; you'll learn it later).

So basically we say $\sqrt z$ to mean the set of the two complex numbers, $w $ and $-w $, so that $w^2=(-w )^2=z$. Or we write $\sqrt z =\pm w$ to mean that the number to be consider a square root of $z$ could be either of $w$ or $-w $.