Square root is a fundamentally geometric idea: you need to have a concept of area before you can find the ratio of an area to the side of a square. Much better than talking about what is or isn’t absurd in terms of pure symbols (like √-1 or whatever) is to give some geometric motivation of the definition of area in terms of orthogonality, which brings in the link to rotation, and finally results in complex numbers. By embedding complex numbers in so-called “geometric algebra”, you put them in their proper coherent and comprehensible context. http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
Can you recommend an introduction to geometric algebra & calculus? (preferably online) I've quickly read some but they were vague and in general quite unlike other mathematical texts. For example the document you linked to doesn't actually define the operations it uses. You can sort of conjure up a definition by piecing various statements together (by using the laws plus the operations' behavior on basis elements), but surely there must be a clearer introduction available? Other introductions were just plain wrong, for example one claimed that the curl of the curl is always 0. All that I've seen were strongly trying to persuade the reader of the awesomeness of GA rather than teaching it.
