TITLE: Definition of Grothendieck group
QUESTION [5 upvotes]: 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?

REPLY [11 votes]: To get a feel for this kind of relation, consider a short exact sequence 
$0 \rightarrow V_1 \rightarrow V_2 \rightarrow V_3 \rightarrow 0$
of finite-dimensional vector spaces over a field.  What is the relation between their dimensions?