I am reading the last chapter of Murphy's $C^*$-algebras and operator theory. He defines $$P[A]=\bigcup_{n=1}^\infty\{p\in M_n(A):\text {$p$ is a projection} \}$$ and construct the Grothendieck group $K_0$ on it. But he doesn't explain why $P[A]$, the set of projections, is of particular interest in studying $K$-theory. What is the motivation of studying $P[A]$ in operator K-theory? Why is the set of projection matrices so important to us?
Thanks in advance!