RE: Mathematics - All about Complex numbers
Hi @Drifter1,
Thank you for this great lesson about complex numbers. Very well structured. I do not have time right now to read it fully till the end but I will!
Why am I so interested in your article? Because of the enthusiasm I felt when reading the first lines. Hey, maybe a life long block in my scientific life could be lifted here!
It hasn't.
Let me explain: Since high school I resisted i^2 = -1 So I just took it as a tool, but more I advanced in my studies up to the postdoc (Physics), realizing how complex numbers work so well in many domains, more I got confused.
I tried to understand it geometrically, like for instance two perpendicular lines will have the product of their slopes = to -1. As a complex number can be seen as a vector when drawn in a reference of two perpendicular axis, I tried to correlate this with that. Resulted in just further confusion.
For example, I though using the imaginary number i was just a practical way to write down coordinates, separating one coordinate from the other. So why don't we have 3D complex number hanging around, of the form z = a +bi +cj ? with j having special properties too...
So my question to you: can you convince me of the reality of i^2=-1. If I got that, that would really help me in quantum mechanics. A topic I love, but for which I can't deep dive in sometimes when equations involving complex numbers appear. Complex numbers prevent me getting a picture of what is really going on when reading the equations... Now the problem has become psychological... lol!
Off topic: I loved what I read up to now, would you be interested in joining me in my effort to promote education for all by making a video for the @openschool ? Maybe about complex numbers ;-)
Complex numbers are indeed a very interesting topic of mathematics.
They are used in many different science branches and are very useful.
The imaginary unit "i" is meant to be unreal and can't be real cause i^2 = -1 <=> i = root(-1) which can't be defined in the real number space, because the subroot is negative...
I guess that "3d complex numbers" could be interesting, but I can't think of an application right now.
Complex numbers are like vectors. They have a real and a imaginary part. It's not so complicated if someone thinks about it that way.
Lastly, I'm busy with my university and I already thought of making some videos, but there is so much uploaded to YouTube already that it isn't necessary I think..
Thank you for your kind answer.
Yes, that is also how I see them, but what confuses me is why they are limited to 2D... not seeing complex numbers generalized to more dimensions (3,4 ,5 etc...) implies to me that it is just a lucky convenience, not a valid representation...
And I still do not know why i^2 = -1... (of course, I understand it mathematically as well as how to use it, but not conceptually). It is at the heart of the whole complex number tool box, but it sounds so 'invented'... For me, this very thought contradicts why complex numbers are so effective in Physics. If they are so effective as tools to describe the physical world, there must be an element of physical or geometrical truth in i^2=-1. Any clue?
You didn't understand my request. This would not be any video, like there are thousands (with many good ones) on Youtube. This would be to directly teach kids (via Skype or Dlive) by getting invested into the @OpenSchool initiative. But of course, your studies should take precedent.
Be well