PS What I find a useful way to express quaternions is to distinguish between real and "pure imaginary" portions, where "pure imaginary" means of form bi + cj + dk for real numbers b,c,d. Any non-real quaternion q = a + bi +cj +dk can clearly be expressed as q = a * 1 + s * u where u is the unit pure quaternion u = (bi +cj +dk) / s. and where s is the real number s = √(b^2+c^2+d^2). The subalgebra <1, u> of the quaternions generated over the reals by 1 and u is isomorphic to the complex field C. The map h : C —> <1, u> given by h(1) = 1 and h(i) = u, and extending linearly, is a field isomorphism. Hence anything true in C is correspondingly true of <1, u>. Hence the quaternion q above may be expressed as q = r exp(𝛉u) where r = √(a^2+b^2+c^2+d^2) and tan(𝛉) = s/a (s as above) and exp is the usual function given by exp(t) = 𝝨 t^n / n! (n = 0,1,2, ...). —Dan