To understand well exp ( i.pi )
To understand well exp ( i.pi ) :
The definition and domain of exponentiation has been changed several times. The original operation x^y was only defined when y was a positive integer. The domain of the operation of exponentation has been extended, not so much because the original definition made sense in the extended domain, but because there were (almost) unique ways to extend exponentation which preserved many of what seemed to be the « important » properties of the original operation. So in part, these definitions are only convention, motivated by reasons of aesthetics and utility.
The original definition of exponentiation is, of course, that x^y = 1 * x *x * … * x, where 1 is multiplied by x, y times. This is only a reasonable definition for y = 1, 2, 3, … (It could be argued that it is reasonable when y=0, but that issue is taken up in a different part of the FAQ). This operation has a number of properties, including
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