How would I find ?

What is the square root of 9?

Well, it's 3 of course.

Remember when you learned that? Perhaps you discovered it yourself? If you did, it wasn't because you followed some magical recipe for finding square roots. You simply knew that 3\times 3=9, and then you learned that multiplying something by itself is called “squaring”, so 3^2=9, and then you were taught that “square root” means “that which, when squared, gives the number”, and you recognized 3 as the square root of 9.

Later, you noticed that (-1)(-1)=1, and as a result (-3)^2=9 as well. So 9 really has two square roots, and so does 16, and so on. In fact, whenever anything has a square root a, it also has -a for a square root, since (-a)^2=a^2.

What about the square root of 18?

Well, 18=2\times 9, so whatever the square root of 2 is, we can multiply it by 3 and get a square root of 18. Simply,

(\sqrt{2}\times 3)^2=2\times 9=18.

In the same way, \sqrt{90}=3\sqrt{10}, and so on.

Ok. Now, you were introduced to those new complex numbers, made from a square root of -1, called i. Before we attempt to answer hard questions like square roots of i, we should do what we did when we were kids: square random stuff and see what we get.

What's i^2? Well it's -1, of course.

What's (2i)^2? Clearly -4.

Now, how about something more complicated like 1+i? The most important thing here is to get used to such manipulations. The usual laws of algebra still apply. All you need is to remember that i^2=-1.

So:

(1+i)^2=(1+i)(1+i)=1+i+i+i^2=1+2i-1=2i

Cool. We didn't quite find a square root of i, but we did find a square root of 2i. Can we fix it?

Sure, just like we handled 18. If 1+i squared is 2i, we just need to divide it by the square root of 2 and we’re home free.

\displaystyle \left(\frac{1+i}{\sqrt{2}}\right)^2=\frac{(1+i)^2}{2}=\frac{2i}{2}=i

And there you go. You've found the square root of i just by fooling around with squaring simple numbers, the same way you've found the square root of 9.

As usual, the negative of this number is also a square root of i. So the two square roots of i are now known to us.

When you get further into it, you learn better ways of finding square roots (and higher roots) of complex numbers, and you learn the nice geometric interpretation they have, and the connection to the exponential function.

But to get started, all you need to do is what you did when you first learned arithmetic: just play.