Definition of 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?

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You can also read a short exact sequence $0 \to A \to B \to C \to 0$ as "$B$ is an extension of $C$ by $A$", which can help to justify the additivity requirement explained in Pete L. Clarks answer. –  Alexander Thumm Aug 5 '12 at 19:47
@Alexander: I didn't explain anything; I only hinted. :) –  Pete L. Clark Aug 5 '12 at 19:53
@PeteL.Clark: Which makes it an even better explanation. :D –  Alexander Thumm Aug 5 '12 at 20:01
Consider the short exact sequences $0 \to A \to A \oplus C \to C \to 0$... –  Qiaochu Yuan Aug 5 '12 at 20:20

$0 \rightarrow V_1 \rightarrow V_2 \rightarrow V_3 \rightarrow 0$
$\mathrm{dim} V_1 \leq \mathrm{dim} V_2$ and $\mathrm{dim} V_2 \geq \mathrm{dim} V_3$? –  Matt N. Aug 5 '12 at 19:28
@Clark: I'm looking for an equality of the form $\pm \dim V_1 \pm \dim V_2 \pm \dim V_3 = 0$, with the precise signs to be found by you. I claim that understanding this is necessary and sufficient to answer your question! –  Pete L. Clark Aug 5 '12 at 19:52