Are you perplexed about the way imaginary numbers are presented in physics, with a quasi-mystic halo?
Do you realize that vector algebra taught in high school has some problematic aspects?
Have you sensed that numbers are hiding something more than a bare magnitude?

You have come to the right place. Here you will find the mathematics we all wanted to learn in high school, because it would have made physics and mathematics much easier to understand. It is called geometric algebra and is the reworking of Clifford’s algebra, developed in the 19th century by Hermann Grassmann, William Rowan Hamilton and William Kingdon Clifford.

The description of physics with vector analysis that you start learning in high school is not – as you might think – the only possible language. In fact, this turns out to be a historical path that established itself at the beginning of the last century.

In recent decades, the geometric algebra approach has experienced a significant revamp, thanks to the efforts of David Hestenes and also to computer visualization techniques, often necessary to approach the subject in the most intuitive way possible.

We hope that these pages will contribute to a step forward in the teaching of mathematics and physics. To encourage intuition, we will limit ourselves to illustrating the main results, leaving the demonstrations to more technical texts (see bibliography).

It is difficult to say at which school level these pages are addressed: the basic concepts are accessible even in middle school, some developments can be successfully taught in high school, but then the application of the GA as a work tool is certainly university level, next to linear algebra.

This presentation is by no means complete and serves as an appetizer only: there is still a lot of work to be done, especially in the creation of specific exercises and cards on the application of GA techniques to mathematics and physics problems. For this, we also hope to find willing collaborators among our audience.

Enjoy the reading!

Thanks for the interesting discussions Andrea Calaon, Alessandro Duci, Carlo Andrea Gonano, Daniele Malesani, Salvatore Mattina, Martino Pani, Enzo Tonti and Sebastià Xambó-Descamps.

Bergamo, January 2022

Paolo Sirtoli

v1.0 – 05/01/2022