Not only must scientists and engineers contend with a plethora of legacy units used to measure various quantities (for example, “pint”, a unit of volume whose definition depends upon whether you are measuring a liquid or something dry and where and when you happen to be living). There are at least *ten* definitions of “pint”, varying from 250 (Flanders) to 1696 (old Scotland) millilitres), but some units in present-day use just make you say “What?” when you think about them.

Consider the way the Americans measure vehicle fuel economy: “miles per gallon”, where they’re trying to convey how many of their “miles” (as opposed to nautical miles, or Roman miles, or Italian miles, or Chinese miles, *inter alia*) a vehicle can travel on a gallon (U.S., not imperial) of gasoline (whatever that may be). But let’s write out that unit, then convert the components to SI units.

But metre is a unit of length, and litre is a unit of volume, which has dimension of length cubed, with:

So we can simplify:

This lets us cancel units and divide the numbers, yielding:

This is the “*What?” moment. They’re measuring fuel economy in units of *reciprocal area*? What the heck does that mean?

Well, actually, if you think about it, it does have a kind of perverse meaning. Let’s rescale from metres to milimetres to make the number on the right more tractable, yielding 0.42514371/{\rm mm}^2=0.42514371\ {\rm mm}^{-2}. If we take the reciprocal of the unit, we get:

This area, around 2.35 square millimetres, is the area of a tube of gasoline which the vehicle would have to consume as it drove along to replace the fuel it burned at the rate of one gallon per mile.

This is a particularly simple case of which you can, with a little mind-bending, make sense. But what about the unit astronomers and cosmologists use to express the expansion rate of the universe?

When you cancel out the units, this has dimensions of:

or frequency.

*What?*