> To me, it makes sense to define abs(x) = sqrt(x^2) i.e. ignoring the negative solution enforces the positive result.
Why does this make sense to you? You have some notion of what an absolute value should be, on an intuitive or conceptual level, and the mathematical definition you give is consistent with that (in the one dimensional case).
Now taking this valid definition for the 1-d case and generalizing that to higher dimensions is where you run into problems.
Instead, you can go back to the conceptual idea of the absolute value and generate a definition for higher dimensional cases from there.
Interpreting absolute value as the distance from the origin yields the same concrete definition of abs(x) = sqrt(x^2) for the 1-d case, but generalizes better to higher dimensions: abs( (x,y) ) = sqrt(x^2 + y^2) for the 2-d case equivalent to complex numbers.
Why does this make sense to you? You have some notion of what an absolute value should be, on an intuitive or conceptual level, and the mathematical definition you give is consistent with that (in the one dimensional case).
Now taking this valid definition for the 1-d case and generalizing that to higher dimensions is where you run into problems.
Instead, you can go back to the conceptual idea of the absolute value and generate a definition for higher dimensional cases from there.
Interpreting absolute value as the distance from the origin yields the same concrete definition of abs(x) = sqrt(x^2) for the 1-d case, but generalizes better to higher dimensions: abs( (x,y) ) = sqrt(x^2 + y^2) for the 2-d case equivalent to complex numbers.