The aim of this paper is to give for the representable K-theory of a unital \(\sigma -C^*\)-algebra A the formulas analogous with that of the representable K-theory of a topological space; that is, the author produces \(\sigma -C^*\)-algebras P and \(U_{nc}\), equipped with the appropriate analog of an H-group structure, such that there are natural isomorphisms of abelian groups \(RK_ 0(A)\simeq [P,A]_ 1\) and \(RK_ 1(A)\simeq [U_{nc},A]_ 1\) where \([A,B]_ 1\) denotes the set of unital homotopy classes of *-homomorphisms between the unital \(\sigma - C^*\)-algebras A and B.