elements of the exterior algebra

So, the Grassmann numbers are elements of the exterior algebra, and the Clifford algebra has an isomorphism to the exterior algebra, but the exterior algebra's relation to the orthogonal group and the spin group, used to construct the spin representations, give the exterior algebra a deep geometric significance.

individual elements or points of the exterior algebra generated by a set of Grassmann variables or Grassmann directions or supercharges

Grassmann numbers are individual elements or points of the exterior algebra generated by a set of Grassmann variables or Grassmann directions or supercharges , with possibly being infinite.

with possibly being infinite

Grassmann numbers are individual elements or points of the exterior algebra generated by a set of Grassmann variables or Grassmann directions or supercharges , with possibly being infinite.

generated by anti-commuting elements or objects

Grassmann numbers are generated by anti-commuting elements or objects.

the $c$-numbers for fermionic creation or annihilation operators in the same way as the ordinary commuting $c$-numbers are the numbers for the ordinary bosonic creation or annihilation operators (and many other operators)

The Grassmann numbers are the $c$-numbers for fermionic creation or annihilation operators in the same way as the ordinary commuting $c$-numbers are the numbers for the ordinary bosonic creation or annihilation operators (and many other operators).

individual elements

The individual elements of this algebra are then called Grassmann numbers.

anti-commuting numbers

Grassmann numbers are anti-commuting numbers, so that x times y = –y times x.

the underlying construct that make this all possible

The Grassmann numbers are the underlying construct that make this all possible.

a Grassmann number ()

\phi,\tag{0} $$ where $\phi$ is a Grassmann number (i.e. a Grassmann number anticommutes with other Grassmann numbers).

a supernumber

A single element of the exterior algebra is called a supernumber or Grassmann number.

a path integral representation for fermionic fields

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.

a sort of classical limit of anticommuting operators in Hilbert space

The problem is that if the Grassmann numbers are indeed a sort of classical limit of anticommuting operators in Hilbert space, then the limit $\hbar\rightarrow0$ itself does not make any sense from a physical point of view since in this limit $\hbar\rightarrow0$, the spin observables vanish totally and what we obtain then would be a $0$, which is a trivial theory.

by convention over the complex numbers

Thus, by convention, the Grassmann numbers are usually defined over the complex numbers.

-dimensional generalization

The -dimensional generalization, the Grassmann number, was introduced by Hermann Grassmann in the late 19th century.

the “classical analogues” of anticommuting operators

In quantum field theory, Grassmann numbers are the “classical analogues” of anticommuting operators.

mathematical analysis to be performed, in analogy to analysis on complex numbers

The definition of Grassmann numbers allows mathematical analysis to be performed, in analogy to analysis on complex numbers.

actual numbers

If an 'actual number' is one which has a geometric interpretation (e.g. complex numbers can be interpreted as plane vectors), then Grassmann numbers are actual numbers, although their interpretation is rather different from what you might have in mind if you are only familiar with real and complex numbers.

a unital associative algebra , S subject to the relation K generated by a set

A Grassmann algebra (also known as an exterior algebra) is a unital associative algebra K generated by a set, S subject to the relation χξ+ξχ=0

a mechanism known as Grassmann numbers to describe time as a sheet

Instead, 'an entirely new type of symmetry' called supersymmetry formulated in the 1970's provides a mechanism known as Grassmann numbers to describe time as a sheet. ...

the parameter spaces for linear subspaces, of a given dimension

Grassmannians by definition are the parameter spaces for linear subspaces, of a given dimension, in a given vector space W {\displaystyle W} .