I started with Euler's identity and manipulated it

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Question from anonymous, a student:

I started with Euler's identity and manipulated it
e^i*pi=-1
e^-i*pi=(-1)^-1
e^-i*pi=-1
e^-i*i*pi=(-1)^i
e^--pi=(-1)^i
e^pi=(-1)^i
type it in in a calculator and you get e^pi=23.1406926... and
(-1)^i=0.0432139183... What did I do wrong?

Nothing - you just explored some interesting math & got a surprise!

Complex exponentials are not single-valued; 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!
RD

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