01/23/2008, 04:08 PM
Ivars Wrote:If we take a differential of Imaginary unit, dI ,which is imaginary infinitesimal (also mentioned by prof. Bell as i*epsilon):
Ops ... sorry! I didn't get you.
In mathematical analysis, by "imaginary unit" we normally mean a constant, given by the famous sqrt(-1), which actually gives {-i,+i}.
It can also be defined as the solution of equation: -x = 1/x. No real number can satisfy it and therefore, this new mathematical object is around since the 18-th century. It is a constant and, therefore, I presume that we must have d(i) = 0. Take a complex number:
z = x + iy, well then: dz = dx + i dy.
Other strange objects are around, like quaternions, octonions, senonions (I presume so!), with multiple different imaginary units and with complicated inter-relations, properties and symmetries. But, also there, all these different "units" are constant.
The only "imaginary differential" that I can figure is someting like i dx or alike. And, in this case, i dx is always "parallel", so to say, to i. Perhaps I am missing the point.
Gianfranco