> ... since a positive number, no matter how small, is greater than any negative number ...
Strictly speaking, not "greater", which is an absolute measure, but more positive. -1 is absolutely greater than +1/2, but +1/2 is more positive than -1. Magnitudes don't have signs.
Consider a polar vector, which possesses a magnitude and one or more angles. For such a vector, the magnitude is always positive, on the basis that it is defined this way when shaped from Cartesian elements:
m = sqrt(x^2 + y^2 + z^2 ...)
In this convention, for any given value of m, the signs of the individual components are lost in the conversion, which means when discussing magnitude, only the absolute value of m matters.
But, because Mathematica has a "Greater[]" function, and because that function simultaneously uses the word I chose and pays attention to scalar signs, I must be wrong about this.
> That seems an odd definition of "greater" ...
I respect the people behind Mathematica, so I have to agree -- it's not what I thought. I should have qualified what I said more carefully.
Strictly speaking, not "greater", which is an absolute measure, but more positive. -1 is absolutely greater than +1/2, but +1/2 is more positive than -1. Magnitudes don't have signs.