>This is inane: we have a multiplication operation that we are all familiar with and we know exactly what it does, and then somebody tells us that we can use it on some "imaginary" thing (??) such that it times itself equals -1?
How is this more inane than defining 0, or negative numbers? Those didn't always exist either, somebody invented them for various purposes. When a person learns arithmetic, negative numbers are introduced in a similar way. By the time a person learns about complex numbers, they already know how to deal with variables in equations, so i*i=-1 shouldn't be too hard to comprehend.
The fact remains that complex numbers are confusing to an extremely large number of people, and negative numbers and 0 aren't. This is probably because 0 can be given direct meaning in terms of everyday applications: I don't have money = I have 0 dollars. I have 0 dollars and I owe you 5 dollars = I have -5 dollars. But what is i dollars? The switch from real to complex is much more complicated than the switch from the naturals to integers or from the integers to the reals. Algebraically, it is also a completely different kind of extension.
i^2 = -1 should be hard to comprehend, and people should be suspicious when things are presented this way. If your experience with equations make it seem easy, I would argue that you don't fully comprehend the significance of such a definition (for example - lets invent a number j such that 1/j = 0. Does this also seem easy?).
The lack of knowledge people have regarding the reals is different in kind from the confusion surrounding complex numbers, and has nothing to do with what is being discussed.
How is this more inane than defining 0, or negative numbers? Those didn't always exist either, somebody invented them for various purposes. When a person learns arithmetic, negative numbers are introduced in a similar way. By the time a person learns about complex numbers, they already know how to deal with variables in equations, so i*i=-1 shouldn't be too hard to comprehend.