This is something that I've pondered for a while and I can't see a logical explanation for. I'll go ahead and demonstrate that it is in fact a real number before you guys think I haven't done my homework. :tongue2:(adsbygoogle = window.adsbygoogle || []).push({});

Starting with the Euler formula:

(1) [tex]e^{ix}=\cos(x)+i\sin(x)[/tex],

and using [itex]\frac{\pi}{2}[/itex] for x, it follows that

(2) [tex]i=e^{i\frac{\pi}{2}}[/tex].

Now using that identity, [itex]i^i[/itex] can be expressed as:

(3) [tex]i^i=(e^{i\frac{\pi}{2}})^i[/tex]

which is equivalent to:

(4) [tex]e^{i^2\frac{\pi}{2}}=e^{-\frac{\pi}{2}}[/tex].

This was all borrowed from Mathworld's site, thanks to them. http://mathworld.wolfram.com/i.html"

Now, can anyone provide an analytic explanation for this? I admit I haven't taken a course on complex analysis, so my skills with complex numbers are limited, but any insight would be appreciated.

Many thanks,

Jameson

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Why is i^i a real number?

**Physics Forums | Science Articles, Homework Help, Discussion**