So I have been thinking about dimensional analysis and I have been thinking about quantities with components that have negative and positive exponents in the same expression.
seconds/second, T T-1, also known as time drift. It's the dimension of the leap day, the inaccuracy of clocks (atomic or otherwise), among other things.
meters/meter3, L L-3, also known as fuel efficiency, or how far you can go per volume of fuel.
Now I have some questions. The laws of algebra would say that it is legal to reduce T T-1 into a "dimensionless" (we'll get to why it's in quotes soon) quantity. Which makes sense to me, that change in time over time would not have a dimension, per se.
So that would also mean that L L-3 would reduce to L-2, otherwise known as inverse area or 1/meter2. That is interesting to me. I'm not quite sure how to visualize that, or even if there is a physical representation. But Wolfram Alpha says it's true. So how would I visualize that and what is it's physical representation of fuel efficiency being inverse area? My guesses are probably nowhere near the mark, so I'll refrain.
Also, are there quantities that are not just "dimensionless", but precisely of dimension zero, other than the trivial ones like pi and phi? Since I cannot say that time drift has the zeroth dimension, simply that the two parts are still there, but in that expression they in a sense "cover over" each other. Meaning they cancel each other out, for the sake of making the paper equations simpler, but are still part of the representation in explicit form.