since In other words, a Grassmann variable is a non-zero square-root of zero. Formal definition Formally, let be an -dimensional complex vector space with basis . The Grassmann algebra whose Grassmann variables are is defined to be the exterior algebra of , namely where is the exterior product and is the direct sum. The individual elements of this algebra are then called Grassmann numbers. It is standard to omit the wedge symbol when writing a Grassmann number once the definition is established. A general Grassmann number can be written as where are strictly increasing -tuples with , and the are complex, completely antisymmetric tensors of rank . Again, the , and the (subject to ), and larger finite products, can be seen here to be playing the role of a basis vectors of subspaces of . The Grassmann algebra generated by linearly independent Grassmann variables has dimension ; this follows from the binomial theorem applied to the above sum, and the fact that the -fold product of variables must vanish, by the anti-commutation relations, above. The dimension of is given by choose , the binomial coefficient. The special case of is called a dual number, and was introduced by William Clifford in 1873. In case is infinite-dimensional, the above series does not terminate and one defines The general element is now where is sometimes referred to as the body and as the soul of the supernumber . Properties In the finite-dimensional case (using the same terminology) the soul is nilpotent, i.e. but this is not necessarily so in the infinite-dimensional case. If is finite-dimensional, then and if is infinite-dimensional Finite vs. countable sets of generators Two distinct kinds of supernumbers commonly appear in the literature: those with a finite number of generators, typically = 1, 2, 3 or 4, and those with a countably-infinite number of generators. These two situations are not as unrelated as they may seem at first. First, in the definition of a supermanifold, one variant uses a countably-infinite number of generators, but then employs a topology that effectively reduces the dimension to a small finite number. In the other case, one may start with a finite number of generators, but in the course of second quantization, a need for an infinite number of generators arises: one each for every possible momentum that a fermion might carry. Involution, choice of field The complex numbers are usually chosen