Complex numbers are wonderful, and frankly not popular enough. My understanding is that in America they form an integral part of the high-school mathematics education (correct me if I am wrong); in the UK you would need to take the Further Mathematics course, and in reality this is simply flirting with a beautiful subject.
For me, I must be honest this was a refresher – so the time devoted above may be misleading; I had some very solid foundations. In addition; as with all of the topics which are discussed on this blog, until further notice, generally I am looking to get to first class undergraduate standard.
The front end of Professor Binney’s notes provided a great starting point – although I must say there were areas where I found I needed a little more simplicity first. I think in reality this is the result of trying to follow lecture notes without being in the lectures; lecture notes are certainly not a transcript of what was said in the lecture notes. Something that really helped bring it all back was this lecture by Prof Arthur Mattuk. Aside from these aids I also used the following resources:
- General internet searches for word definitions (never be ashamed!)
- This lecture series and problems
- And this from the Art of Problem Solving
Aside: The Art of Problem Solving; amazing place and wonderful name.
I think my main lessons were 1) as I had always known, problem solving is the key to understanding and 2) the internet is better than I thought.
Taking 1), mathematics without problem solving is like trying to learn a language without speaking it… learning to paint without painting… learning to cycle without cycling.. you get the idea. You need to feel these things out for yourself. Problem solving exposes weakness. It also transforms what looks like an over-complicated expression or formula and turns it into a logical and natural part of the mathematical world.
Taking 2), it is a dawning realisation that every day people can, with nothing more than internet access take materials from some of the world’s greatest academic institutions (see Oxford notes above). I knew I would be able to find the information I needed – but honestly I expected it to be harder, and I expected the resources to be of lower quality.
I think for anyone picking this up for the first time, or even refreshing don’t hope to understand without being as comfortable as a human can be with expressing z = a + bi as z= mod(r)(cos(x) + sin (x)) = mod(r)e^(ix). I will endeavor to get proper equations on here soon. For me this is the roots and fruits. Understanding this underpins everything else and it is worth spending time truly convincing yourself of that fact.
There isn’t a lot more for me to say on the subject – other than if you are a beginner give this number system time and respect – they are very useful and very very exciting. For me I will now naturally progress to revisit differential equations.