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You're picking the two different branches, since both i and 1 / i when squared yield -1, but themselves differ by a factor of -1, and this is where the problem comes in. So the answer to your question is that you are being inconsistent in how you define sqrt(-1), and if you're consistent in that regard then the problem goes away. Difficulties of this kind are extremely common when working with complex variables, which is part of what makes them confusing.

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Right. To elaborate a bit (for the benefit of the others), you end up with this sort of thing any time you have a multi-valued function. For example, the function x^2 takes the number 1 to the number 1. It also takes the number -1 to the number 1. Thus the inverse of the function x^2 logically could take 1 to 1 or to -1. You must choose a consistent rule for making sense of this (we always choose the positive value without thinking much about it).

With real numbers, this is easy, and remains consistent once the rule is chosen. With complex numbers, however, this "choice" cannot be consistent over the entire complex plane if you are dealing with a multivalued function -- you need to specify where your choice is valid. If you try, for example, to integrate some multivalued function over some line in the complex plane, you must draw a branch cut line over which your line of integration can not cross. Crossing the branch cut would intuitively mean that part of the time you integrate using one rule, and after crossing the branch cut you use another rule. This is bad, since the choice of where you draw the line is somewhat arbitrary -- all that is important is that it must be drawn somewhere to ensure that things are well-defined.