The indecomposable sets are those which cannot be expressed as a Minkowski sum in any nontrivial manner. In this paper we concentrate on the indecomposability problem for sets in the domain of binary images. We show that, it is possible to express a binary image as a hypercomplex algebraic number. More interestingly, if we restrict our domain of binary images then Minkowski addition (direct sum) (also called dilation) turns out to be the addition of two such hypercomplex numbers. In that process the indecomposability problem is transformed into a number theoretic problem. As a by-product our treatment of the problem produces an efficient algorithm for computing Minkowski addition of two binary images.