The base 10 expansion of π is the decimal expansion (3.1415926539...), the base 2 expansion is, um, whatever that is in binary, the base 16 expansion is whatever that is in hex, and so on.

Or, formally:

If you can write

π = Σ_n=0^infinity (a_nk^-n)

for some base k, with each a_n an integer between 0 and k-1 inclusive, then that sum is the base-k expansion of π.

(I'm not completely sure about that definition, I'm only an undergraduate; the expansion might actually be when you write it as

a₀.a₁a₂a₃a₄a₅...

but I think the idea's right)

btw, the n=0 and infinity should have the same horizontal position, but you can't do that without invoking MathML or LaTeX or something, and I suppose to be completely general you should include all the coefficients (multiples) of 10ⁿ for n positive; I leave that as an exercise to the reader, as maths textbooks always say ;-)

</maths-geek>