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A Visual Intro to Complex Numbers

34 points surprisetalk 2 hours ago 11 comments

pratikdeoghare 38 minutes ago | parent

I used complex numbers to solve popular interview question.

It can be used to convince students that they are useful.

https://pratik.deoghare.com/complex-mars-rover.html

jdw64 35 minutes ago | parent

The explanation is easy and fun.

ttoinou 33 minutes ago | parent

I recommend Visual Complex Analysis by Tristan Needham

pratikdeoghare 9 minutes ago | parent

There is proof of this theorem in it that blew my mind some years ago [1]. Someday I want to finish reading the rest of the book.

> Starting with a given convex quadrilateral, construct a square, external to the quadrilateral, on each side. Van Aubel's theorem states that the two line segments between the centers of opposite squares are of equal lengths and are at right angles to one another.

https://en.wikipedia.org/wiki/Van_Aubel%27s_theorem

socializer 30 minutes ago | parent

Really? Of approximately one billion good, visual introductions to complex numbers, we're going to pick an article that appears to be substantially written by an LLM?

arun-mani-j 21 minutes ago | parent

When I first learnt complex numbers in school, I couldn't understand their purpose or why they are called imaginary etc.

I checked where they are used and found examples of current, magnetism etc. But I couldn't understand, electricity is a measurable physical quantity, how can you use imaginary numbers there??

Then I searched internet and found a Stack Exchange answer telling the story and how, instead of, complex (sounds like difficult) or imaginary (sounds like not useful practically), if we had better terms for them, then it would have been easier for beginners to understand them more intuitively.

It's completely true in my case.

After learning usage of vectors, thanks to Godot, for rotation and movement, and then about matrix etc. I now feel like "complex" numbers etc. are the representation of data through multiple numbers where a single value (scalars) is not enough. I may not be right in strict mathematics terms, but at least that's how I was able to understand them.

bananaflag 18 minutes ago | parent

I think it would have been clearer if they weren't called "numbers".

For example, look at matrices [1]: no one asks themselves whether they are "real" or not. We know that we have defined operations on arrays of numbers which are useful to some degree.

If complex numbers were just called "complexes", a lot of confusion would just wither away.

Also, interestingly, quaternions and octonions aren't called "numbers".

[1] Complex numbers may, of course, be seen as special kinds of matrices, but this is beside my point here.

MrBuddyCasino 6 minutes ago | parent

So much of mathematics is just this.

bananaflag 15 minutes ago | parent

The explanation is a bit confusing, since it basically reduces to "some equations don't have solutions, so we have to invent them", which, if you already accept, you don't need complicated equations, just x^2=-1.

One thing which would have worked better (which is briefly mentioned) is Cardano's casus irreducibilis, but that would have needed more work.

reenorap 5 minutes ago | parent

Complex numbers would be a much easier concept to understand if they didn't call it "complex" numbers or use the term "imaginary" numbers. Both of those make it much harder to digest because now you think that it's not real. What they really are are 2 dimensional numbers, meaning they have 2 components to each number.

The same goes for the term "real" numbers. Sometimes math, and its descendants like physics and computer science, can be too cutesy with their names that make it hard for regular people to understand or accept.