Polar coordinates with imaginary radius

Question

Is it possible to construct a mathematical space described by polar coordinates with the real angles and imaginary radius? In the simplest case of just two dimensions, one real angle and the imaginary radius (or vice versa, the real radius and at least one imaginary angle). I don't think so, but if possible, I'd appreciate any reference or ideas.

Answer 1

Short answer, no.

"The radius is a scalar quantity which is fully described by its magnitude. It is defined as the distance between center and any point on the circumference of a circle".

On a complex plane, the value of the distance between two points, whether they have imaginary parts or not can be described with real numbers. For example, 2i is 3 distance units away from 5i, etc. When using Pythagoras's Theorem to find the distance between two points, all units are squared (i.e made positive) and then added before being squared rooted.

Answer 2

I think such a visualization is an intriguing possibility. I have wondered about the possibility of labling each point(r,θ) on the polar plane as either r + iθ or as ir + θ. After all, there is nothing about complex numbers which says they must be represented as a real and imaginary number on the cartesian plane. However, algebraic operations with complex numbers on these two polar planes will be visually different from the conventional (Cartesian) representation of complex numbers.