Argument of a Dual Number

[safiire]

The following method always seems to return 0 + 0ɛ for positive duals.

angle{T<:Real}(z::DualNumbers.Dual{T})
angle{T<:Real}(z::Dual{T}) = z ≥ 0 ? zero(z) : one(z)*π

This is a lot different than the definition of argument that I'm familiar with. In the generalized complex numbers, parameterized by p and q:
z = x + iy (x, y ∈ R) where i^2 =iq + p (q, p ∈ R)

(so i, ɛ, λ, whatever we call the imaginary unit)

Ignoring q and setting it to zero, and setting p = 0 we get the Dual numbers, p = 1 gives Double numbers, and p = -1 giving the regular Complex numbers. There is a general definition for angle or argument, magnitude or norm, and a very weird concept of unit circle, that is only actually what you would call a circle for p = -1.

So the unit "circle" is everywhere ||z|| = 1, which for duals is just at 1 and -1.

So for duals the argument is y / x. I will link to a paper, and just paste an image of the relevant part here.

Here is a link to the paper this is from:
https://people.rit.edu/harkin/research/articles/generalized_complex_numbers.pdf

The paper also shows how to implement the sinp, cosp, and tanp trig functions for any value of p.

[dlfivefifty]

angle refers to the argument in the complex plane, and so the current definition is correct as the argument does not change if we perturb a real number by a small real ε perturbation. Two other examples:

angle(dual(im,im)) # returns π/2+0ε as the argument stays fixed
angle(dual(1,im)) # returns 0+ε as the argument changes

[dlfivefifty]

Your proposed angle could be implemented as a separate function DualNumbers.dualangle. Overriding Base.angle would be confusing in the context of complex numbers. This confusion is the reason that Base.real overrides were replaced with DualNumbers.realpart.