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Originally Posted by dk5sm5luigi
This is exactly why the proof that I posted with e^(i*pi) doesn't work.
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No. This is quite different. Quaternions are an extension of the real numbers and you don't need to even mention them to talk about e^(i*pi). With quaternions, ab != ba in some circumstances. There are even octonions with a(bc) != (ab)c.
The reason why your e^(i*pi) thing doesn't work is that you are taking the log of a negative/complex value incorrectly. Why does log(exp(3pi*i) = i*pi? Because exp(3pi*i) = exp(pi*i) = -1 and in the natural extension of the log to the complex plane log(-1) = i*pi. So one must be very careful when using natural logs on complex arguments. There are several good books on the subject, in fact MIT has a complex analysis class available entirely free. This is necessary before one "proves" anything involving complex arguments using functions defined on the reals.