as the field for the definition of the Grassmann numbers, as opposed to the real numbers, as this avoids some strange behaviors when a conjugation or involution is introduced. It is common to introduce an operator * on the Grassmann numbers such that: when is a generator, and such that One may then consider Grassmann numbers z for which , and term these (super) real, while those that obey are termed (super) imaginary. These definitions carry through just fine, even if the Grassmann numbers use the real numbers as the base field; however, in such a case, many coefficients are forced to vanish if the number of generators is less than 4. Thus, by convention, the Grassmann numbers are usually defined over the complex numbers. Other conventions are possible; the above is sometimes referred to as the DeWitt convention; Rogers employs for the involution. In this convention, the real supernumbers always have real coefficients; whereas in the DeWitt convention, the real supernumbers may have both real and imaginary coefficients. Despite this, it is usually easiest to work with the DeWitt convention. Analysis Products of an odd number of Grassmann variables anti-commute with each other; such a product is often called an a-number. Products of an even number of Grassmann variables commute (with all Grassman numbers); they are often called c-numbers. By abuse of terminology, an a-number is sometimes called an anticommuting c-number. This decomposition into even and odd subspaces provides a grading on the algebra; thus Grassmann algebras are the prototypical examples of supercommutative algebras. Note that the c-numbers form a subalgebra of , but the a-numbers do not (they are a subspace, not a subalgebra). The definition of Grassmann numbers allows mathematical analysis to be performed, in analogy to analysis on complex numbers. That is, one may define superholomorphic functions, define derivatives, as well as defining integrals. Some of the basic concepts are developed in greater detail in the article on dual numbers. As a general rule, it is usually easier to define the super-symmetric analogs of ordinary mathematical entities by working with Grassmann numbers with an infinite number of generators: most definitions become straightforward, and can be taken over from the corresponding bosonic definitions. For example, a single Grassmann number can be thought of as generating a one-dimensional space. A vector space, the -dimensional superspace, then appears as the -fold Cartesian product of these one-dimensional It can be