# What are “Super Numbers”?

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!

• I think you're looking for Grassmann numbers. – Raskolnikov Mar 30 '11 at 17:24
• Here's the wikipage for Grassmann numbers. The other page was about their use in supersymmetry, which is probably where the term super number originates from. – Raskolnikov Mar 30 '11 at 17:27
• I don't own a copy of the book, but the table of contents suggests that there is a "References for Suggested Readings" section in Kaku's Hyperspace. That would possibly be a good place to start. – Willie Wong Mar 30 '11 at 17:45
• planetmath.org/supernumber – Jonathan Gleason Jun 12 '14 at 21:36

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.

• As an addendum, it's merely $2 \theta^2_i = 0$ that follows from anticommuting. The conclusion $\theta^2_i = 0$ follows from the additional assumption that $2$ is multiplicatively invertible, which does not hold in all contexts where such such a thing might be of interest. – Hurkyl Jun 18 '17 at 3:39

If you're looking for a reference, the text "Supermanifolds" by Bryce DeWitt provides an excellent pedagogical overview of supernumbers, analysis over supernumbers, supermanifolds, super Lie groups, and supersymmetry.