A complex beginning

· by · in Mathematics, Physics. ·

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:

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.

18 responses to “A complex beginning”

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  2. Having been educated in the U.S. system, I can tell you that complex numbers are unfortunately not a significant part of mathematics curricula. We are taught how they arise, but what potential applications comes out of complex numbers is not mentioned (at least not in general level mathematics courses.) Do you know what is the use of complex numbers? And how are they related to or different from imaginary numbers?

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    • Hi Mehak,

      I guess it just seemed like the grass was greener on the other side then! Thank you for your comment. When I left school I was none the wise but having been through university once I am a little better informed!

      In answer to your questions – the main difference is that the imaginary numbers are a set of numbers contained within the complex numbers. In fact the complex number set is about the biggest one we define. If you take any complex number z=a + bi where a and b are non-zero real numbers, then we say Re(z) = a and Im(z) = bi. That is, the real part of z is a and the imaginary part of z is bi. When you draw an Argand diagram to represent complex numbers (are you familiar?) you take the y axis to be the imaginary number line and the x to be the real number line. You can think of the numbers on you y axis as you imaginary numbers [we often call these pure imaginary numbers]. So many would say 4 is a real number, 4i is an imaginary number and 4 + 4i is a complex number. Note: Mathematicians often use the terms complex and imaginary interchangeably anyhow!

      What are they used for? This is the exciting bit. When you study complex numbers you think do these things even exist… they are called imaginary! Well they do and the reason we know they do is they are used to solve real life problems, like in electrical engineering. The best thing about complex numbers is that they make it REALLY easy to take roots . If you go to about 30 mins in in the lecture I linked there is a neat example of using complex numbers in order to solve a real integration problem in a much easier way than with calculus. And finally, quantum mechanics – there are very many fields in quantum mechanics which you can’t even solve without complex numbers!

      I could go on for hours but my day job just started – let me know if you have any further question!

      Keep on rationalising.

      Joseph

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  5. Born and raised in the US (except for a couple of years in SA), I was age 50 when I finally went to college and my 14 year old son had to help me with the prerequisite maths; I never took calculus or anything other than the algebras in high school, either! Funny, I never enjoyed mathematics much until I got to be an old fossil!

    Brilliant site! Mama always said that you learn something new every day; I’m going to have to put your site on my daily Start Menu, just out of validation!!

    Thanks so much for visiting my blog and following…all the best to you 🙂

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    • Glad to hear you have enjoyed your visit – and no such thing as too old to learn. I am convinced that constant learning nurtures the brain and keeps us happy and healthy. It is very rare I come across anything I consider bad learning. Hats off yo you for getting to college! All the best 🙂

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  6. Hi!

    Thanks for following my blog. Your blog heading attracts me. If only we could rationalize the biggest mathematical wonder called the Universe.Have you ever thought it that way?

    Mathematics always gave me the jitters. But I think it’s not so formidable as it seems.

    All the best to you and your explorations.

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    • Thank you very much! The universe is truly mind bending and I am in a personal mission to understand as much of it as I can while there is blood in my body. It is such a wonderful opportunity to be in this thing we call the universe.

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  9. One primary use of complex numbers is in signal processing, control systems engineering, and electrical engineering.

    They can be used to determine unstable or meta-stable roots in equilibrium calculations in Chemical Engineering – for example supersaturated solutions or for nucleation phenomena (see spinodal decomposition).

    They are not often used, but when they are, they can be powerful. For example, if you look into the Mandelbrot set the fractal patterns are formed by stable solutions – the black background are regions where there is no converged solution.

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  10. I was a math and science teacher. Our students struggle with the basics in both so to “catch up” with Europe we force feed them higher level maths like complex number so they can cheat their way throug this too…lol.

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