Reexport Base.imag?

[jiahao]

Currently, Base.imag is not imported and re-exported, which causes shadowing errors when intermixing Complex and Quaternion numbers in arithmetic computations. Since im^2 == -1 in both algebras, the algebras themselves are not in conflict, and the analogous mixed-algebra computations should be similar to Real * Complex in Base. At least on paper, I don't foresee any problems in allowing Quaternions.imag to extend Base.imag.

[JeffBezanson]

The imag function here returns a vector of the 3 imaginary components. Does that really implement the imag function? Doesn't seem right to me.

[chakravala]

If you'd like to gain a better geometric or algebraic understanding of imaginary units of quaternions and complex values, I recommend you play around with it on Grassmann.jl a bit.

julia> using Grassmann

julia> i,j,k = hyperplanes(ℝ^3)
3-element Array{SValue{⟨+++⟩,2,B,Int64} where B,1}:
 -1v₂₃
 1v₁₃ 
 -1v₁₂

julia> (ℝ^3)(I) # this is the 3D pseudoscalar
1v₁₂₃

julia> (ℝ^3)(I)^2 # squares to -1 also
-1v

In the quaternion algebra, there is an i,j,k which are all planes which square to -1 but there is another capital I which is usually called the volume element which is a single imaginary unit.

Quaternions are generated by a 3D vector space, but complex numbers are generated by a 2D vector space combining a scalar value with a pseudoscalar value, i.e the volume element V(I).

julia> (ℝ^2)(I) # this is the 2D pseudoscalar
1v₁₂

julia> (ℝ^2)(I)^2 # squares to -1 as imaginary unit
-1v

As you can see, the pseudoscalar V(I) in 2D is v12 while in 3D it is v123.

It so happens that the pseudoscalar v12 from 2D is equal to k from the i,j,k of quaternions also.

Thus, you could either:

• equate im with v12, the bivector plane element, which would require a sign change for 3D!
• equate im with V(I), the pseudoscalar element, which is more natural for n-dimensions

If you associate it with the v12 element alone, then the Base.imag function for quaternions should only return the k component of the quaternions instead of all 3 values. Otherwise, if you choose to equate im with the value of the pseudoscalar volume element V(I), then there is simply no comparison between the im value and the i,j,k components of the quaternions, they are fundamentally different quantities.

I hope that this explanation provides some insight about the relationship between im and I and i.

Basically, im is a 2D pseudoscalar, while i,j,k is both a 3D pseudovector and also a paravector.

[goretkin]

The definitions at

Quaternions.jl/src/Quaternion.jl

Line 15 in 01d45b4

Quaternion(z::Complex) = Quaternion(z.re, z.im, zero(z.re), zero(z.re), abs(z) == one(z.re))

and

Quaternions.jl/src/Quaternion.jl

Line 21 in 01d45b4

convert(::Type{Quaternion{T}}, z::Complex) where {T} =

would be consistent with defining

Base.imag(q::Quaternion) = q.v1

However, while I can personally attest to finding it useful to have conversion/promotion from Real to Complex or Real to Quaternion, I do not have the same experience with regards to mixing Complex and Quaternion numbers in operations. Does anyone have an example where having the preceding definitions were beneficial?

[hyrodium]

As discussed in #61, we will have imag_part function which returns Tuple instead of Vector. (#77)
The original Quaternions.imag will be deprecated with the PR, so I'll close this issue.