Hacker News Reader

Oversimplifications like this are not helpful. There are a ton of ways that complex numbers are interesting, and they have important generalizations as e.g. an even subalgebra of G^2 (leading to quaternions as the even subalgebra of G^3, and so on). They’re not just “a circle”.

> EDIT: I would also say the lie that we live in a 3+1D world (X, Y, Z + Time), helps confuse the mater. We probably live in a 6D + 1 world: X, Y, Z, Pitch, Yaw, Roll, and Time.

If your model doesn’t treat time the same way as space up to the metric tensor, it’s probably not very well generalized. There should be an even number of parameters.

Rotation including the time dimension is just acceleration. The math is simpler if you’re in a ++++ metric - rotations are a bit weird in +++- because rotations into the time dimension are hyperbolic. I suggest reading Dichronauts, a book covering a ++-- tensor.

What does the ++++ and +++- notation mean? Looking at info about tensors I don't see much. I did order the book out of interest.

I really like this explanation. The negative square roots explanation never sat right with me in high school and I think this touches on why.

The way I think about it:

- To express direction in Cartesian space, you need signed numbers

- To express 2D rotation, you need imaginary numbers

- To express 3D rotation, you "need" quaternions (in the sense that they're the smallest way to represent an arbitrary rotation)

> quirky work around for negative square roots. I would say that is more of a side effect.

I agree it's not a perfect motivation, for teaching purposes. When you introduce an answer to the square root of a negative number, questions of soundness might arise. Obviously it's sound, but is that a distraction at the level where complex numbers are introduced?

Besides, many people like geometric interpretations: they might feel more 'real' or relevant than an algebraic motivation.

/Nonetheless/, I can't agree well defined square roots is just a side effect. The complex numbers are (up to isomorphism) the one and only algebraic closure of the reals. That's a very natural and and useful way look at them for, say, university level maths.

> introduced imaginary numbers as rotations initially

How would you do this without developing the exponential first?

Well any complex number a + bi can be described as a transformation matrix

    |a -b|
    |b  a|

So you could start by just asking what it means to transform the plane

Ok but in a college algebra class it’s very easy to ask “what is the inverse of the squaring operation?” Not so clear a path to the action of a matrix on the plane.

I have successfully discussed this in an advanced calculus class.