You should know the basics of the subject before stating that everyone is wrong and that the current standard is "ridiculous". You apparently confused "complex number" and "imaginary part". The latter made sense in the historical context, which was finding real solutions of quadratic equations (with real coefficients, of course).
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.
> keep in mind that complex numbers are not rotations
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).
Thanks for this. It always bugged me that quaternions didn't fit well with the linear algebra of computer graphics. You absolutely need quaternions, but every time you do a rotation you have to convert to quaternions, do the rotation, then convert back.
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.
Matrix algebra and geometric algebra are two different languages for many of the same topics/structures. If you like you can embed either one in the other (though embedding geometric algebra into matrix algebra is extremely cumbersome, sort of like rewriting your Python programs in assembly language).
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.
I suspect that if people in the 1800s had been developing 3D computer graphics rather than electromagnetic theory, Hamilton's camp would have won the battle. But most people would probably say that vectors and matrices are a better fit for EM work.
Clifford’s geometric algebra/calculus is a much better fit than the Gibbs/Heaviside concept of "vector" analysis for doing electrodynamics problems. Especially if you try to handle quantum mechanics and special relativity.
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.
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.