In physics and engineering, one of our most reliable analytical tools is dimensional analysis. Briefly, by labeling numbers by their physical use (length, time, force, etc.) we can troubleshoot calculations by checking for consistent dimensions (i.e. physics is a “strongly typed” language). Also, we can often gain insight into relationships between quantities by looking for dimensionless combinations. Particularly, when teaching undergraduate students we’re always nagging them about properly labeling units and dimensions.
One thing that’s a little confusing in this regard about GA is that each grade of a multivector has a different dimensionality. This is particularly vexing when trying to parse geometric products for dimensional content.
Does anyone have suggestions on easy ways to track dimensionality in GA? For my own use I keep notes about the grades present in any expressions, but that seems to clunky and tedious to teach to students.