What are "Super Numbers"?

Question

I'm reading Hyperspace by Michio Kaku and in the chapter on SuperGravity "Super Numbers" are mentioned and are described as a number system where for any super number $a$, $a*a=-a*a$. I was wondering if anyone knew anything more about these numbers or could point me to another reference about them. Thanks!

Answer 1

OK, since there's no answer provided, I'll make my comment one:

As you can read here, Grassmann numbers are numbers built up from Grassmann variables $\theta_1,\theta_2,\ldots,\theta_n$ with the special property that they anticommute:

$$\{\theta_i,\theta_j\}=\theta_i\theta_j+\theta_j\theta_i = 0 \; .$$

In particular, $\theta_i^2=0$.

You can then study numbers which are linear combinations of ordinary numbers and Grassmann numbers. For instance, if we only take one Grassmann variable $\theta$, we can make the number $1+\theta$ or $5+2\theta$ and then you can add them or multiply them by using the introduced rules:

$$(1+\theta)+(5+2\theta) = 6+3\theta$$

and

$$(1+\theta)\cdot(5+2\theta) = 5+5\theta+2\theta+2\theta^2 = 5+7\theta \; .$$

These numbers are then probably called super numbers since they appear in the context of supersymmetric field theories in physics. The importance of the Grassmann variables there is that they allow to define path integrals for fermion particles. In fact, even outside supersymmetry, in other contexts like theoretical condensed matter theory where fermions are studied by using path integrals, these Grassmann variables will be used.