> 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).

In my mind abs(x) = x*sign(x) which is why the above formulation seems correct. This formulation is useful, for example, in formulating reflections.

> Instead, you can go back to the conceptual idea of the absolute value and generate a definition for higher dimensional cases from there.

This is an interesting idea...how would you define sign(x) in a higher dimension? Wouldnt sign in a higher dimension be a component-wise function? E.g. the reflection would happen on one axis but not the other.

> Interpreting absolute value as the distance from the origin

This seems to make sense in that it is a different interpretation of abs which seems simpler than reflection in higher dimensions, but seems like a different definition.

I know that there are applications of complex numbers in real systems. In such systems, the complex definition seems to not be as valuable. E.g. if I'm solving a laplace transform, the real number definition seems more applicable than the complex number definition, right?

I've asked wolfram alpha to solve the equation and it lists both answers: one using the formulation of sqrt(x^2) and the other using sqrt(re(x)^2 + im(x)^2) so it seems like there is merit to both...

I suppose in the laplace example, we are actually operating in one dimension and the imaginary component is approximating something non-real, but doesnt actually exist. I.e. any real/observable effect only happens when the imaginary component disappears meaning that this is still technically one dimension. So, since we're still in one dimension, the one dimensional formula still applies. Is that correct?

Your explanation has been the most helpful though, thanks.