S.O.S. Mathematics CyberBoard

Here's a little bit of premise:

I'm working on a research project trying to design a distance function that generates both complex-valued and real "distances". I would like to see if for all real values my function produces if the euclidean triangle inequality holds. I was wondering if someone might help with what such an experiment would look like. Given three values with real distances from the origin, would I examine d(0,a), d(0,b), and d(0,c)?

Any insight appreciated

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let:
a=b
ab=b^2
ab-a^2=b^2-a^2
a(b-a)=(b+a)(b-a)
a=b+a
a=2a
1=2

What do you mean "generates complex-valued and real 'distances' "? You cannot have a complex-valued distance, the complex numbers are not an ordered field. What do you mean by "generates"?

_________________
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Well I suppose the formula in question produces distances in name only right now. I'm playing around with functions f:CxC-->C. As far as the ordering, I'm aware of that problem and it's why for now I'm only trying to verify the triangle ineq for the real-values the function produces.

_________________
let:
a=b
ab=b^2
ab-a^2=b^2-a^2
a(b-a)=(b+a)(b-a)
a=b+a
a=2a
1=2

You realize that there is already years and years worth of mathematical research on generalized notions of distance, called "metrics" and that they are very well axiomatized, and all are nonnegative real-valued functions. If you want functions which happen to satisfy the triangle inequality on the real numbers that's fine, but I would refrain from going near the word "distance" when describing them, as anyone who knows any basic analysis will either ignore you or throw a hissy-fit over how what you're talking about is not and cannot be a distance unless it is real-valued.

_________________
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