TITLE: What Gauss *could* have meant?
QUESTION [29 upvotes]: I was reading the Wikipedia entry on Euler's identity ($e^{i\pi}+1=0$) and I came across this statement:
"The mathematician Carl Friedrich Gauss was reported to have commented that if this formula was not immediately apparent to a student upon being told it, that student would never become a first-class mathematician."
I know that this isn't too important and shouldn't be taken seriously, but I was just wondering what led him to say this. Is it really obvious? Was he assuming that the student should already know the series expansion for $e^x$? Etc.
Note: I am interested in Gauss's statement in particular, not a proof of Euler's identity.
All views on this will be appreciated, thanks.

REPLY [2 votes]: As an electrical engineer's son, the way I first heard the definition of multiplication of complex numbers was that a multiplication $z\mapsto az$ is consists of rotating and dilating.  That immediately tells you that the exponential function of $n$, $\{i^n\}_{n\in\mathbb Z}$ just goes around in circles.
Learning about $e$ is another step along the way.  Once you've understood what's "natural" about $e$ and that the graph of that exponential function is a helix, you're almost there.