Some years ago, when I first learned about imaginary numbers, I asked myself the inevitable question – “What’s the square root of i?”

Okay, so i is the square root of -1. I had come to grips with that concept. But after that, when you start thinking about it… surely the square root of i must be an even more imaginary number, j? But it isn’t! In fact, you can express the square root of any complex number as another complex number!

You might be thinking right now: “Well that’s trivial! You just take the Argand plane, and then use the something-or-other theorem and rotate some dots around the origin and voila! You get the answer!” But I didn’t know that back then.

But I had it on good authority that I didn’t need to invent another super-imaginary number. Complex numbers would do. So I decided to find the square root of i!

And today, I’d like to share that journey with you.

How I found the square root of i

Since the square root of i was a complex number, it could be written as “a + bi” with a and b being real numbers.

SQRT(i) = a + bi

I squared both sides

i = a^2 + 2abi – b^2

And here came a nice logical leap (at least I thought it was nice at the time). When you have two complex numbers that equal each other, the real parts must equal each other, and the complex parts must equal each other. So:

i = 2abi
a^2 – b^2 = 0

And from there it was just a bit of trivial algebra, finally yielding 1 + i over the square root of 2. And by squaring that, I verified that it was indeed equal to the square root of i! Eureka! I found it!

1. yoursurprise-bellatio-3 Says:
   November 29th, 2011 at 11:55 pm

   Hi there! I know this is somewhat off topic but I was wondering which blog platform are you using for this site? I’m getting tired of WordPress because I’ve had issues with hackers and I’m looking at alternatives for another platform. I would be great if you could point me in the direction of a good platform.

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