Summary: In non-commutative spaces, it is unknown whether the Pontrjagin class gives integer, as well as, the relation between the instanton number and Pontrjagin class is not clear. Here we define “instanton number” by the size of the matrices \(B_{\alpha }\) in the ADHM construction. We show the analytical derivation of the non-commuatative \(U(1)\) instanton number as an integral of Pontrjagin class (instanton charge) with the Fock space representation. Our approach is for the arbitrary converge non-commutative \(U(1)\) instanton solution, and is based on the anti-self-dual (ASD) equation itself. We give the Stokes’ like theorem for the number operator representation. The theorem shows that instanton charge is given by some boundary sum. Using the ASD conditions, we conclude that the instanton charge is equivalent to the instanton number.