Abstract A 2-knot is a surface in R 4 that is homeomorphic to S 2 , the standard sphere in 3-space. A ribbon 2-knot is a 2-knot obtained from m 2-spheres in R 4 by connecting them with m − 1 annuli. Let K 2 be a ribbon 2-knot. The ribbon crossing number, denoted by r- c r ( K 2 ) is a numerical invariant of the ribbon 2-knot K 2 . It is known that the degree of the Alexander polynomial of K 2 is less than or equal to r- c r ( K 2 ) . In this paper, we show that r- c r ( K 2 ) is estimated by coefficients in the Alexander polynomial of K 2 . Furthermore, applying this fact, for a classical knot k 1 , we also estimate the crossing number, denoted by c r ( k 1 ) .