I'm reading the Wiki article about the Grothendieck group.
What's the reason we define $[A] - [B] + [C] = 0 $ rather than $[A] + [B] - [C] = 0 $ (or something else) for every exact sequence $0 \to A \to B \to C \to 0$? What is the property we obtain if we define it this way? I suppose it has something to do with exactness at $B$ but what?