In a way it makes sense though right? There's a text by Brian Cantwell that introduces the dimensional analysis chapter with,

Any physical relationship must be expressible in dimensionless form... This is because the variables are subject to measurement by an observer in terms of units that are selected at the *arbitrary* discretion of the observer. It is clear that a physical event ... cannot depend on the particular ruler used to measure space, the clock used for time, the scale used to measure mass... depending on the dimensions that appear in the problem.

If you and I run a race on a straight track that is 30 antehiros long, I run at an average pace of vScytale = 3 kilosquattes per hemtistool, and you run at an average pace of vFlatfingers = 49 terribads per naysec, who's going to win?

We don't know because we have no idea of the relative size of the measures we're using here, but we

*do* know the winner depends on only these two quantities. The underlying principle of dimensional analysis is that, in this case, these two quantities can be collapsed into one

*because they have the same dimension*: length per time. That means we can form a dimensionless number, call it the Flatfingers number,

Fa = vScytale / vFlatfingers

And so we can see that the winner of the race depends not on *two* quantities, but on one: the Flatfingers number. I don't know how big a kilosquatte is, and I don't know how long a naysec takes, but I know there are three solutions: if Fa > 1, I win: if Fa < 1, you win. If Fa = 1, it's a tie. In fact, Fa=1 is degenerate ("we both win").

Of course, to determine the Flatfingers number I need to know the conversion factors between the various units cited, so this all seems rather trivial. We've just shifted the problem into different language. But the language makes very clear what it is we should care about:

**it doesn't matter how fast I run, it only matters if I run faster than you.**

If Fa is huge, it means I'm a speed demon

*compared to you*, and if Fa is tiny, it means I'm a dismal failure compared to you.

We could say the Flatfingers number describes the "relative effects" of Scytale vs Flatfingers in the race. This is exactly the same language people use to talk about things like the Reynolds number: it describes the relative effects of inertia vs viscosity. And we know it works

**because** it's dimensionless. If it weren't dimensionless, we couldn't know in general if inertia and viscosity were being compared on the same terms.

This is what makes these numbers so useful: Bond number is relative effects of surface tension vs gravity. Froude number, inertia/gravity, and so on.

Thank goodness for dimensionless numbers! If it weren't for Reynolds number, we couldn't describe

this lab photo in exactly the same terms as

this satellite view . Same relative effects, even though our measuring sticks are different.