Kerry wrote: << [I wrote:] << There is absolutely no mathematical way to distinguish between [the two complex square roots of -1]. Perhaps they should only be referred to as a pair, and never one at a time?
Ok, I'll show my ignorance here. How can we distinguish between 1 and -1 and not be able to distinguish between i and -i? Isn't the simple fact that they're distinct numbers (or distinct points in the complex plane) enough? What am I missing?
Well, 1 and -1 have different algebraic properties. E.g., 1 is the unique complex number whose product with any other number z is z. -1 is the unique complex number that's unequal to 1 but whose square is 1. On the other hand, there is no distinguishing property that holds for one of i,-i but not the other. This follows from the fact that there exists a field isomorphism f:C -> C interchanging i and -i. (The simplest example is complex conjugation: f(x+iy) = x-iy.) --Dan