I really wish we'd just call them "rotation numbers" or something. "Imaginary" was always a ridiculous name.
So what can we contribute, do you know any authors that have already found some new and better terminology and promoted it?
"Imaginary numbers" is something from a highschool curriculum.
And please keep in mind that complex numbers are not rotations, and they do not map well to them. For instance, which rotations would be represented by the complex numbers "3", "4", "1-2i" and "8i"? You can map {plane rotations} to { r, |r| = 1 } using z ↦ r*z, but that's a circle, not a 2D space.
To summarize, complex numbers can be thought of plane vectors, with an obvious geometric way to add them, and a non-obvious way to multiply them. The set of "rotations" is too vague and loosely related to complex numbers, e.g. 3D rotations are often represented by quaternions, which are more "complex" than complex numbers.
Complex numbers are (isomorphic to) "amplitwists": similarity transformations between plane vectors. If you want a pure rotation, you need a unit-magnitude complex number.
The complex number 3 represents scaling by 3. The complex number 4 represents scaling by 4. The complex number 1 – 2i represents a scaling by √5 combined with a rotation clockwise by arctan(2). The complex number 8i represents a scaling by 8 combined with a quarter-turn anticlockwise rotation.
> complex numbers can be thought of plane vectors
No, (physics-style displacement) vectors and complex numbers are distinct structures and should not be conflated.
Complex numbers are best thought of as ratios of planar vectors. A complex number z = v/u is a quantity which turns the vector u into the vector v, or written out, zu = (v / u)u = v(u \ u) = v. (Concatenation here represents the geometric product, a.k.a. Clifford product.)
Mixing up vectors with ratios of vectors is a recipe for confusion.
> non-obvious way to multiply them
Multiplication of complex numbers is perfectly “obvious” once you understand that complex numbers scale and rotate planar vectors and i is a unit-magnitude bivector.
> 3D rotations are often represented by quaternions, which are more "complex" than complex numbers.
Analogous to complex numbers, quaternions are the even sub-algebra of the geometric algebra of 3-dimensional Euclidean space. Used to represent rotations, they are objects R which you can sandwich-multiply by a Euclidean vector v = RuR* to get another Euclidean vector, where * here means the geometric algebra “reverse” operation. Those of unit magnitude are elements of the spin group Spin(3).
For more, see http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
Very cool. I'd never thought of them that way but it makes perfect sense, given that rotation and scaling will, indeed, turn one vector into another.
With geometric algebra (Clifford algebra), it's all just one system with the quaternions being a special case within that system. All of it makes more sense and computationally it's only slightly more work than linear algebra.
Why don't we use Clifford algebra instead of linear algebra today? Because back in the 1800s there was something of a war between the people who liked Hamilton's formulation (quaternions, tensors, Clifford algebra) and those who liked vectors and matrices. Except in a few fields like relativity, the vector/matrix crowd won the war. And they probably shouldn't have.
You absolutely do not. Quaternions are (usually) the best solution, but they're not the only one.
> And they probably shouldn't have
Geometric algebra may or may not be better in certain low-dimensional applications like CG, but there's a whole lot more to linear algebra than that.
Geometric algebra has a much richer geometrical structure, but in some contexts the more basic structure of linear transformations is all you need, and matrices work just fine.
In geometric algebra, you can extend a linear transformation of vectors by "outermorphism" to act on arbitrary multivectors.
Cross products, pseudovectors, etc. are tremendously confusing for students. Then throwing it all out the following school term in favor of cumbersome matrix representations makes things even worse.
I suspect if Clifford had lived past ~35 years old we would have avoided a lot of the confusion of the 20th century.