



 
Nothing  you just explored some interesting math & got a surprise! Complex exponentials are not singlevalued; your two values come from different branches. [They are also reciprocals, but this is at least partially a coincidence.] In general, $(1)^i = \exp(\ln(1) \times i);$ and $\ln(1)$ is $\theta i$ where $\theta$ is any angle you can rotate 1 through to get 1; that is, ...$3 \pi, \pi, \pi, 3\pi,$ ... Thus $(1)^i = \exp(\theta)$ which can take any of the values ...$\exp(3 \pi), \exp(\pi), \exp(\pi), \exp(3 \pi),$... with all values real and consecutive values in a ratio of about 1:535. Good Hunting!  


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences. 