OK, my main point restated: A number is whatever we want it to be, whatever it is useful to treat as a number. That's it. That's the whole gag. Saying somethi

OK, my main point restated: A number is whatever we want it to be, whatever it is useful to treat as a number. That's it. That's the whole gag. Saying something isn't a number is pointless: It is if it can be, and it isn't if it's more interesting to treat it as something else. This leads to something more fundamental: Math is all about modeling. Math is a language for making models that are logically consistent. Confusing a model with the thing being modeled is wrong. Saying complex numbers aren't numbers because they're rotations is going at it backwards: Complex numbers can be used to model rotations. That doesn't tie them to that one model. They can be used in other ways, too.