We give a description of the closure of the natural affine continuous function representation of
K
0
(
R
)
K_0(R)
for any exchange ring
R
R
. This goal is achieved by extending the results of Goodearl and Handelman, about metric completions of dimension groups, to a more general class of pre-ordered groups, which includes
K
0
K_0
of exchange rings. As a consequence, the results about
K
0
+
K_0^+
of regular rings, which the author gave in an earlier paper, can be extended to a wider class of rings, which includes
C
∗
C^*
-algebras of real rank zero, among others. Also, the framework of pre-ordered groups developed here allows other potential applications.