A note on the Grothendieck group of an additive category

There are two abelian groups which can naturally be associated to an additive category $\mathcal{A}$: the split Grothendieck group of $\mathcal{A}$ and the triangulated Grothendieck group of the homotopy category of (bounded) complexes in $\mathcal{A}$. We prove that these groups are isomorphic. Along the way, we deduce that the ‘Euler characteristic’ of a complex in $\mathcal{A}$ is invariant under homotopy equivalence, a result which has implications for (de)categorification.