Complex numbers and functions, Quaternions and Riemann Surfaces

Below post is a back-up of my posts in my Quora blog. I take a copy of them to here in case if something goes wrong in there.

I was planning to write a long post and add some visuals on the subject for some time. But after watching the available videos in internet, I have decided that they don’t require any additional explanation and do a much better job than I could do. Therefore I will only give a brief and give links to those amazing videos.

(These videos are comparatively long for today’s video consuming standards so they can be pretty quickly boring for you. Watch them only if you are interested with this subject and want to learn what the concepts in the title really mean. Try to take your time and enjoy while watching. If you will put a timer in your brain to finish, you will never do.)

First let’s start with this awesome explanation about what the imaginary and complex numbers actually are and what they are useful for:

Once you understand the concept of imaginary and complex numbers, inevitably you will ask what about more dimensions than 2? How can I deal with rotations in higher dimensions? Apparently, this question gave trouble to great mathematician William Rowan Hamilton and after 10 years of looking for the solution for rotation in 3 dimensions, he finally has figured out that the answer actually lies in 4 dimensions instead of 3 and he came up with the idea of Quaternions. Here is another amazing video from the great Youtube channel of Numberphile:

If you will get deep enough understanding of what the complex numbers and quaternions are and continue to go deeper, you may figure out that each quadratic equation in even in their basic forms like y=x²+1 must have complex roots on the imaginary numbers axis. This can show you a way to imagine how can a function with a 2nd order have 3 dimensional visualization.

Even better, there are videos which garnish their explanation with very cool 3D visualizations like the one below. Ok, this is not a single video but a series of 13 great videos. What they show with that colored 3d quadratic equation in first video that you can see below is very cool, indeed. But what really has blown my mind is what they have did with that Phyton code while they were explaining the complex functions starting from video 10. If you will be patient enough and give yourself to what is being explained from the beginning, I promise that you will be rewarded. It will be as enlightening as the moment when Morpheus was explaining Neo what the matrix is. Here they are; Welch Labs presents, Imaginary Numbers are Real:

I forgot to say that starting from video 12, they explain what a Riemann surface is and how they are useful for our understanding. Especially, if you are into physics and Relativity, this is a must see for you. Those videos have finally gave me some clues about how to visualize the Riemann Surfaces.

That’s all for now. If I come to know more cool videos in future, I will keep updating this post with them.