Editing K-theory (section)

=== Example for natural numbers ===
An illustrative example to look at is the Grothendieck completion of <math>\N</math>. We can see that <math>G((\N,+)) = (\Z,+).</math> For any pair <math>(a,b)</math> we can find a minimal representative <math>(a',b')</math> by using the invariance under scaling. For example, we can see from the scaling invariance that

:<math>(4,6) \sim (3,5) \sim (2,4) \sim (1,3) \sim (0,2)</math>

In general, if <math>k := \min\{a,b\}</math> then

:<math>(a,b) \sim (a-k,b-k)</math> which is of the form <math>(c,0)</math> or <math>(0,d).</math>

This shows that we should think of the <math>(a,0)</math> as positive integers and the <math>(0,b)</math> as negative integers.