Absolutely agree.

What's your preferred terminology? I like "Forward" and "Reverse" for positive and negative number and "Lateral" for the imaginary unit, since it is just perpendicular to the Forward and Reverse numbers.

That's pretty close to what Gauss said they should be called:

"If ... +1, -1, and √-1 had been called direct, inverse and lateral units, instead of positive, negative, and imaginary (or impossible) units, such an obscurity would have been out of the question"

(https://shitohichiumaya.blogspot.com/2016/10/gausss-quote-fo...)

I just think of them as different axes on a 2D Cartesian plane. That’s basically how they work.

EDIT: similarly, quaternions as a number in 4D, with that extra dimension being handy for avoiding the instability problems when you get near “gimbal lock” in 3D rotations.

But the quaternions sort of make the real component special dont they?

That it more an effect of how we use them in computer geometry. Unit quaternions with ijk plus real map to an xyz tilt plus a rotor. But I believe you could pick any such mapping.

On the other hand, dual quaternions do care which is the dual component, since multiplying by complex numbers looks like true rotation, while multiplying by a dual looks like rotation around a point at infinity, aka translation.

So in that case maybe there is something special about the real component.

Good and evil.

Spooky and strange?

Human and Werewolf.

Spooky Werewolf Algebras on the Monster group. Coming to arxiv this October.

I also like the name "lateral unit", maybe call the field "planar numbers"?

Mathematicians have largely abandoned the term "imaginary number" and use "complex number" instead. I never used the term "imaginary number" when I was teaching (except to point out that it wasn't very good terminology).

But the old term still survives in some areas. For instance, the real and imaginary parts of a complex number:

  re (3 + 4 i) = 3  and  im (3 + 4 i) = 4.

> They are terrible choices of name, coined by people who didn't really understand them, and didn't want you to understand them either.

Sorry to be blunt, but your comment is horrible. There is far more going on here than you realize. Real numbers have an isomorphism with our reality. We thought we didn't need imaginary numbers to describe reality. They were inteded to be used more as a mathematical tool.

Another good example is euclidean vs non-euclidean geometry. They are both incredibly useful, but the former one was though to be "real" and the latter "imaginary". But it turns, physics is okay with us living in a non-euclidean universe (https://en.wikipedia.org/wiki/Shape_of_the_universe).

> Real numbers have an isomorphism with our reality.

You can probably get a nobel prize by proving this. :)

In quantum physics complex numbers are used to represent circular symmetry, so they are more like modulus and phase.

Do you have a suggestion for an alternative to "real number"?

Real number are called 实数 in Chinese which is literally `solid number`.

I responded to a similar comment above with a link to what Guass called them.

Continuous numbers?

Imaginary numbers are also continuous, are they not?

So they are, so are many other spaces.

I'd avoid calling complex numbers "numbers" personally. In the context of plain english (and not currently in the context of mathematics) I think of the word "number" as saying that this is somehow a measure of a quantity (We have "a number" of brands of cereal in the store). Complex numbers are not a measure of a quantity.

I'd probably replace the "numbers" part with "field". I don't have a good replacement for the "complex" part, but I think that should change also (because telling people things are complex is almost as bad as telling them that they are imaginary). If no one proposed something better and I was king of the universe I guess I'd just name them after whoever discovered them.

Would you consider -3 a number? It's certainly not a measure of quantity. A better way of thinking about it is that it's 2 things: a magnitude and a direction. Complex numbers just expand the direction into two dimensions. As long as you accept negative numbers, then you already accept the fact that there can be more than one number of a given size. The complex numbers just contain infinitely many numbers of a given size, instead of only 2.

In plain english, no I don't think I would particularly consider -3 a number. If I say someone gave me a number of horses I don't mean they stole my horse. I think it is the right call to include them in the mathematical definition though. Unlike complex numbers they are almost always useful (they are needed to turn the object into a field, which is needed if you want everyday operations like subtraction to always remain within the set), have a much more intuitive definition of multiplication, are a smaller generalization, etc.