Quantum states and complex numbers - newbie question

[carmatic]

this wikipedia article http://en.wikipedia.org/wiki/Qubit says

The qubit is described by a quantum state in a two-state quantum-mechanical system, which is formally equivalent to a two-dimensional vector space over the complex numbers.

i am kind of comfortable with the physics of it, but i am totally lost on the thing about vector space over the complex numbers

can someone please lend me a hand? it seems that the more i try to read about it, the less i know

 

[mathman]

In general a vector space consists of vectors with scalar multipliers. In two dimensions vectors are of the form (x,y) where x and y are numbers (scalars). For a real vector space x and y will be real numbers and the scalars are real numbers, while for a complex vector space x and y will be complex numbers and the scalars may be complex.. Vectors can be added or subtracted and multiplied by scalars.

For example Let (x,y) and (u,v) be vectors and a and b scalars, then
a(x,y) + b(u,v) = (ax+bu,ay+bv).