In mathematical physics, a Grassmann number, named after Hermann Grassmann (also called an anticommuting number or supernumber), is an element of the exterior algebra of a complex vector space. The special case of a 1-dimensional algebra is known as a dual number. Grassmann numbers saw an early use in physics to express a path integral representation for fermionic fields, although they are now widely used as a foundation for superspace, on which supersymmetry is constructed.

Grassmann numbers are generated by anti-commuting elements or objects. The idea of anti-commuting objects arises in multiple areas of mathematics: they are typically seen in differential geometry, where the differential forms are anti-commuting. Differential forms are normally defined in terms of derivatives on a manifold; however, one can contemplate the situation where one "forgets" or "ignores" the existence of any underlying manifold, and "forgets" or "ignores" that the forms were defined as derivatives, and instead, simply contemplate a situation where one has objects that anti-commute, and have no other pre-defined or presupposed properties. Such objects form an algebra, and specifically the Grassmann algebra or exterior algebra.

The Grassmann numbers are elements of that algebra. The appellation of "number" is justified by the fact that they behave not unlike "ordinary" numbers: they can be added, multiplied and divided: they behave almost like a field. More can be done: one can consider polynomials of Grassmann numbers, leading to the idea of holomorphic functions. One can take derivatives of such functions, and then consider the anti-derivatives as well. Each of these ideas can be carefully defined, and correspond reasonably well to the equivalent concepts from ordinary mathematics. The analogy does not stop there: one has an entire branch of supermathematics, where the analog of Euclidean space is superspace, the analog of a manifold is a supermanifold, the analog of a Lie algebra is a Lie superalgebra and so on. The Grassmann numbers are the underlying construct that make this all possible.

Of course, one could pursue a similar program for any other field, or even ring, and this is indeed widely and commonly done in mathematics. However, supermathematics takes on a special significance in physics, because the anti-commuting behavior can be strongly identified with the quantum-mechanical behavior of fermions: the anti-commutation is that of the Pauli exclusion principle. Thus, the study of Grassmann numbers, and of supermathematics, in general, is strongly driven by their utility in physics.

Specifically, in quantum field theory, or more narrowly, second quantization, one works with ladder operators that create multi-particle quantum states. The ladder operators for fermions create field quanta that must necessarily have anti-symmetric wave functions, as this is forced by the Pauli exclusion principle. In this situation, a Grassmann number corresponds immediately and directly to a wave function that contains some (typically indeterminate) number of fermions.

When the number of fermions is fixed and finite, an explicit relationship between anticommutation relations and spinors is given by means of the spin group. This group can be defined as the subset of unit-length vectors in the Clifford algebra, and naturally factorizes into anti-commuting Weyl spinors. Both the anti-commutation and the expression as spinors arises in a natural fashion for the spin group. In essence, the Grassmann numbers can be thought of as discarding the relationships arising from spin, and keeping only the relationships due to anti-commutation.

Grassmann numbers are individual elements or points of the exterior algebra generated by a set of n Grassmann variables or Grassmann directions or supercharges ${\displaystyle \{\theta _{i}\}}$, with n possibly being infinite. The usage of the term "Grassmann variables" is historic; they are not variables, per se; they are better understood as the basis elements of a unital algebra. The terminology comes from the fact that a primary use is to define integrals, and that the variable of integration is Grassmann-valued, and thus, by abuse of language, is called a Grassmann variable. Similarly, the notion of direction comes from the notion of superspace, where ordinary Euclidean space is extended with additional Grassmann-valued "directions". The appellation of charge comes from the notion of charges in physics, which correspond to the generators of physical symmetries (via Noether's theorem). The perceived symmetry is that multiplication by a single Grassmann variable swaps the ${\displaystyle \mathbb {Z} _{2}}$ grading between fermions and bosons; this is discussed in greater detail below.

The Grassmann variables are the basis vectors of a vector space (of dimension n). They form an algebra over a field, with the field usually being taken to be the complex numbers, although one could contemplate other fields, such as the reals. The algebra is a unital algebra, and the generators are anti-commuting: