Why is $i! = 0.498015668 - 0.154949828i$? While moving my laptop the other day, I ended up mashing the keyboard a little, and by pure chance managed to do a google search for i!.
Curiously, Google's calculator dutifully informed me that $i!$ was, in fact, $0.498015668 - 0.154949828i$.
Why is this?
I know what a factorial is, so what does it actually mean to take the factorial of a complex number? Also, are those parts of the complex answer rational or irrational? Do complex factorials give rise to any interesting geometric shapes/curves on the complex plane?
 A: $$i!=\Gamma(i+1)=\int_0^{\infty}e^{-x} x^{i}dx$$ where $\Gamma(n) $ represents the Gamma Function
Note  $$x^i=e^{i\ln x}=\cos(\ln x)+i\sin(\ln x)$$
A: To answer your last question,

Do complex factorials give rise to any interesting geometric shapes/curves on the complex plane?

There are a couple of Gamma fractals shown on Wolfram's reference article for Gamma under "Neat Examples":

*

*DensityPlot[ Arg[Nest[Gamma, x + I y, 3]], {x, -1.25, -0.6}, {y, -0.25,  0.25}] // Quiet

*ArrayPlot[ Table[c = N[cr + I ci];  Length @NestWhileList[ If[Abs[#] > 20., Indeterminate, Gamma[#/c]] &,  c, (# =!= Indeterminate) &, 1, 20], {ci, -2.5, 2.5,  5/100}, {cr, -2, 2, 4/100}]] // Quiet
See also Christopher Olah's blog post, Gamma Fractals, and from there, Bidimensional zoom in on the Z=Gamma(Z) iteration with display of the arguments, both of which have some nice images.
A: It is sort of an abuse of what is meant by factorial. The usual definition of
$$n! = \prod_{k=1}^n k$$
obviously cannot apply because you can sit and count integers until the end of time and beyond and you'll never find $i$.
However, we can generalise what we mean by factorial by using a property of the gamma function, which is defined to be
$$\Gamma(z) = \int_0^{\infty} e^{-t}t^{z-1}\, dt$$
This has the useful property that, for any $n \in \mathbb{N}$,
$$\Gamma(n) = (n-1)!$$
which has an easy proof by induction on $n$. It also has lots of nice analytical properties which make it a good choice for an extension of the factorial function.
Anyway, since the gamma function can be defined (after analytic continuation; see LVK's comment) on the entire complex plane, minus the non-positive integers, for a general $z \in \mathbb{C} - \{ -1, -2, \cdots \}$ we can put
$$z! \overset{\text{def}}{=} \Gamma(z+1)$$
For this reason we get
$$i! = \Gamma(i+1) = \int_0^{\infty} e^{-t}t^{i}\, dt \approx 0.498015668−0.154949828i$$
See also here and here.
