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We represent multidimensional operators in a separated form that can be viewed as a numerical generalization of separation of variables. Since we separate directions, it does not preclude operators with high (operator) rank from having low separation rank. We introduce a concept of the condition number of a separated representation (to measure the potential loss of significant digits) and make its control a part of the algorithm for reduction of the separation rank. Structurally, the separated representation is a tensor decomposition. However, since its purpose and use are novel, it brings up a new set of examples and questions regarding the mathematics of tensor decompositions. Some of these questions lead to interesting problems in harmonic analysis. Since many physically significant operators have low separation rank, our basic approach has already found applications in quantum chemistry and fast multiresolution algorithms.