How to compute $K_0(S^1)$?

Question

I am taking a course on K-theory and we have been asked to compute $K_0(S^1)$. Now I have a few conceptual problems: I see that $M_\infty(C(S_1))$ corresponds to "matrices of potentially infinite size with coeficients in $C(S^1)$", but how do I identify idempotents therein? My second question would probably be how to identify algebraically equivalent idempotents, but this question might be trivial depending on the answer of the first.
I want to compute this directly, so without using MV-sequence like arguments.