> But the formalism is a barrier for building up a more convenient mental model about what is really going on. Actually, I'd say that complex numbers are a won

> But the formalism is a barrier for building up a more convenient mental model about what is really going on. Actually, I'd say that complex numbers are a wonderful opportunity for illustrating that math is nothing but a mental model, not necessarily with anything "really going on", AND at the same time incredibly useful. IMO the best way to think about complex numbers is to start with the observation that sqrt(-1) cannot exist in the realm of real numbers, quite easily provable. Then you make the bold-assed assumption that it exists anyway and simply call it i. Then play around with it a little, and find out that adding the axiom "there is something called i, where i * i == -1" does not lead to any contradictions; instead you can work with this i just fine, and numbers of the form a+bi are closed under all your everyday operations, just by applying your everyday arithmetic rules. Neat! Then you notice that those numbers can be interpreted as 2D vectors and some operations on them as geometrical transformations thereof. Super Neat! Then someone notices that those numbers and the operations on them can actually be used to model certain aspects of the physical world. Holy Crap! Math is FREAKING AWESOME!