in terms of N variables such that: where the indices are summed over all permutations so that as a consequence: for some N > 2. These are useful for calculating hyperdeterminants of N-tensors where N > 2 and also for calculating discriminants of polynomials for powers larger than 2. There is also the limiting case as N tends to infinity in which case one can define analytic functions on the numbers. For example, in the case with N = 3 a single Grassmann number can be represented by the matrix: so that . For two Grassmann numbers the matrix would be of size 10×10. For example, the rules for N = 3 with two Grassmann variables imply: so that it can be shown that and so which gives a definition for the hyperdeterminant of a 2×2×2 tensor as See also Grassmannian Hermann Grassmann (linguist and mathematician) Superspace Exterior algebra Notes References Hypercomplex numbers Supersymmetry Quantum field theory