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Is there an actual grassman number or is it symbolically represented by generators?

Well, the answer depends upon what you mean by an "actual" number!

If for you an "actual number" is one which can be added and multiplied, then Grassmann numbers are 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.

If for you an "actual number" is one which has in some suitable linear algebra-inspired sense only one "component", then a Grassmann number is not an actual number.

i was not sure where to put this topic since I don't know which subject of math grassmann math constitutes.

The natural place for a question about Grassmann numbers would be the "Linear and Algebra" subforum.

Grassmann numbers are elements of a

Desmond Fearnsley-Sander, "A Royal Road to Geometry",

Those interested in "flat two-dimensional Galilean spacetime" (one of the nine planar homogeneous geometries found by Klein) should note that this can be identified with the Grassmann algebra on one generator (a trivial example!) considered as a two-dimensional real algebra.

Someone--- I guess me--- should say that a Grassmann algebra is simply a real linear associative algebra generated by n elements (which we can think of as "vectors" in R^n)

[tex]\vec{e}_1, \; \vec{e}_2, \dots \vec{e}_n[/tex]

where we have a

[tex]\vec{e}_j \wedge \vec{e}_k = -\vec{e}_k \wedge \vec{e}_j[/tex]

which implies

[tex]\vec{e}_j \wedge \vec{e}_j = 0[/tex]

Then wedge products of k distinct generators are "k-multivectors" in R^n; we can think of a k-multivector as a k-dimensional rhombus in R^n,

[tex]\omega = \vec{e}_1 \wedge \vec{e}_2 \wedge \dots \vec{e}_n[/tex]

Here, summing the binomial coefficients gives

[tex]1+n + \left( \begin{matrix} n \\ 2 \end{matrix} \right)

+ \left( \begin{matrix} n \\ 3 \end{matrix} \right) + \dots

+ \left( \begin{matrix} n \\ n-1 \end{matrix} \right) + 1 = 2^n[/tex]

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