Unit conversion - different degrees of reduction to most basic units...

What are the names, conventions, methods, and resources for converting physical units from one form to another, especially into and out of the SI?

For example, car gasoline “mileage” in the US is customarily written as miles per gallon. You could convert this into kilometers per liter. But liters are themselves convertable into cubic meters, so you could also convert car gasoline use into kilometers per cubic meters, or into inverse square meters. Since meters are a fundamental unit of the SI, there isn’t any further conversion you could do that would further reduce this to a basic SI measurement. There is even a conceptual interpretation of this: if some model car has gasoline consumption efficiency of “X per square meter”, then you could have X of these cars all plumbed together to drive along some highway with a one-square-meter trough of gasoline running alongside it, and they would just consume this trough’s contents as they rode along next to it together.

Also, though there are many possible unit combinations to represent a given physical measurement, only one of them would be this furthest reduced one. That means there is exactly one ultimate unit combination for each possible measurement. It is interesting to study the SI and consider alternative systems, because this doesn’t represent some underlying physical truth about measurement, but rather is just a logical consequence of having a consistently defined unit system with designated base units, and there are a few interesting alternative systems laid out with other designated base units.

Is anything I said incorrect or misleading or confusing? What do you call this furthest possible reduction to base SI units? Are there web sites or utility programs that do such a reduction?


Physical units represent a physical quantity. Fuel mileage can be expressed as kilometers per liter, yes, but that means “(kilometers of roadway travelled) per (liters of gasoline).” Or, with some unit conversion, “(meters of roadway travelled) per (cubic meters of gasoline).” Even though you see meters in both the numerator and denominator, a meter of roadway is not the same thing as a meter of gasoline.

It’s unusual (although not completely unheard-of) to “cancel units” because, in this case, the units are not equivalent. A “meter” is not an abstract concept, it’s a measurement of some physical distance. (Contrast the kpl non-cancellation to, say, calculating force on a dam due to pressure. Here the force = pressure * area, the pressure is in pounds per square inch of wall surface, and the dam area is in square inches. The “square inches” in the numerator and denominator measure the same physical quantity, and so can be happily cancelled.)

As an interesting example of non-cancellation, see permeation. In SI units, “the permeability unit is mostly cm[sup]3[/sup].mm/(m[sup]2[/sup].Bar.day), so that the gas flow in cm3/day results when the area is given in m[sup]2[/sup], the thickness in mm, and the pressure difference in Bar.” Here, the flow volume, thickness, and area are all given explicitly in length units, yet are neither cancelled nor combined.

I think what I am getting at is more like “coherent units”. See this from the Bureat Internation des Poids et Mesures", the body that manages the SI:

“All other quantities are derived quantities, which may be written in terms of the base quantities by the equations of physics. The dimensions of the derived quantities are written as products of powers of the dimensions of the base quantities using the equations that relate the derived quantities to the base quantities. In general the dimension of any quantity Q is written in the form of a dimensional product…”

Napier -
I usually find your posts quite logical, but you must have been up late when you wrote this one. I’m not at all clear about what you are trying to accomplish but, that aside, I don’t think that auto gas milage can be converted to liters / m^2.

Fuel consumption is measure by (volume of fuel) / (linear distance traveled). How can you ever change that into (volume of fuel) / (area traveled) - unless you factor in the footprint of the car, which doesn’t seem very helpful?

See dimensional analysis.

There may be more than one “ultimate” unit combination, if you change the fundamental dimensions of your measuring system. As the Wikipedia link suggests, the MLTQ(theta) system is most common - mass, length, time, charge, temperature - but you could use other dimensions as the most fundamental, and then use derived dimensions for things like mass/length/time.

Irritating. It can be done, but for most non-theoretical purposes it’s obfuscatory.

The OP didn’t say “liters / m^2”, it said “inverse square meters”, which kind of makes sense to me. Length divided by volume gives you the reciprocal of area.

Let’s say your car does 30 miles per gallon. (I’m in the UK so I’ll use imperial gallons).

1 mile = 1609m

1 gallon = 4.55 litres, or 0.00454m[sup]3[/sup].

So the car does (30 x 1609)m / 0.00454m[sup]3[/sup] = 10,632,000m[sup]-2[/sup]. :slight_smile:

Read as, “10,632,000 per square meter of (square root of fuel per distance traveled).”

Which is, as has been pointed out, obfuscatory. Also confusing.

More from the parent body of the SI:

“It is important to emphasize that each physical quantity has only one coherent SI unit, even if this unit can be expressed in different forms by using some of the special names and symbols. The inverse, however, is not true: in some cases the same SI unit can be used to express the values of several different quantities”

The kind of thing I am getting at is how to automate the conversion of units that aren’t already in some unit converter somewhere. I sometimes deal with complicated unit strings that aren’t in anybody’s little converter utility or database, and there is a way of automating this for me and others, which among other things can eliminate errors.

I think this is done in the unit facility in Mathematica software. I also think it was done by the HP 28S calculator. But if these tools aren’t available, what’s a scientist to do?

Keep in mind that “of gasoline” is a critical part of the numerator’s units here. “1 gal” does not equal “1 gal of gasoline”. So, you could reduce miles/gallon to inverse square meters (purely geometrical lengths and volumes), but miles/[gallon of gasoline] is not inverse square meters of gasoline. You can manipulate the words in this manner, but it doesn’t make any sense to do so.

Sure, but it is possible to conceptualize, as the OP has shown. 10,632,000 cars of the same efficiency could all concurrently siphon fuel off of one 1m^2 (internal cross-sectional area) pipeline of gasoline and travel as far as the pipeline goes.

Or one car with a (1/10632000) m^2 pipeline.

I think more specifically that there can be wording that describes the physical thing being measured, which could be “driving distance per volume of fuel consumed”, and which can and does have all units extracted out of it, and also there can be units, which can be “miles per gallon” or “inverse square meters”, and also there can be a dimensionless number. And you can express the statement that describes the entire physical situation by saying the physical thing being measured is so many of such and such units; you can equate what the wording describes with the dimensionless number multiplied by the units.

Carrying this idea further, the wording describing the physical thing does completely specify an abstraction without quantifying it at all, and the number and the units taken separately make no statement at all about a physical thing but taken together are the complete quantification.

All this would have to be mathematically tidy. That is, you should be able to work out the math for such a situation either with or without the units, and the one with units would have to be dimensionally consistent.

And yet, the SI unit of resistivity is the ohm-meter, not the ohm-m^2/m, even though it’s a measure of the resistance times cross-sectional area per length.

And there can be deep physical significance in two things having the same units, even if it’s not apparent at first blush. Classically, we talk about action (energy per frequency) and angular momentum (momentum times leverage distance) as completely different things, and yet in quantum mechanics, hbar is the fundamental quantum of both of them. This is only possible because, despite their apparent differences, action and angular momentum really do have the same units.

This is closer to what I am getting at - see Table 4 at:


What this should provide is an unambiguous way to reduce any physical quantity to a unique and, in an SI-specific way, a most generalized form.

Joe Frickin’ Friday had it - dimensional analysis. It’s what physicists use to check the cromulence of their equations as, constants notwithstanding, the base units will always be the same on both sides of the equation if it is correct. Sadly the neper is dimensionless.

Just to confuse the thread further, the density of liquid fuels vary with temperature, which is why motor racing formulas measure fuel in kg (if it matters). You can get a few more horsepower out of an engine by cooling the fuel as you can squirt in more mass for the same limited volume. I remember one F1 team being fined and penalised on points for having fuel that was too cold after one pit stop; it was calculated it gained them about half a second per lap or so.
On the plus side, you can save a little bit on fuel yourself by filling up first thing in the morning when it’s coolest, as garages charge by volume while your vehicle only cares about mass.

OK, I have a better handle on it.

It isn’t dimensional analysis. The point of dimensional analysis is to apply consideration of the dimensions of quantities to test things about their interrelationships, and dimensional analysis doesn’t need to involve units, only dimensions. It cares about length, for example, but not meters per se. What with me being a physicist and all, I often use dimensional analysis to check equations and the like.

No, what I am getting at is the following structure: dimensions can be characterized by units, and you can collect various units having various historical origins, as has been done for thousands of years. You can also define units in a particular way so that their size makes the basic equations that fundamentally relate the dimensions generally turn out to have coefficients of exactly 1, working with a handful of special “base units” and extending this list with other “derived units” defined in terms of the base units. Doing this gives you a unit system that is “coherent”.

Finally, you can list various derived units in your coherent system, with their symbol, and with their representation in base units by name and symbol, for example:

“magnetic field strength”, H, “ampere per metre”, A/m
So, what I specifically am chasing is a list of units in this format, so that any new unit only requires being added to this list, and can then be converted with all other measurements of the same dimension, programatically.

The usual treatment of unit conversions is to create a table for each possible dimensional combination, and enter into the table all the conversion factors between all possible pairings in the table. Each dimensional combination requires another table.

What I want to do is treat all unit conversions on the basis of my one single list, and programatically deliver either a conversion or an error message that the two quantities asked about have different dimensions.

Or (sorry to bump up a thread a few days old, but it just occurred to me), if you have a car with petrol tank with the base formed by a square 1 metre on each side, you’ll be able to drive 10,632,000 times the height of the tank, if you fill it. :slight_smile:

Are you saying you just want to know the conversion factor for each imperial unit to the most basic metric unit? Like, you want to be able to convert miles per hour to meters per second instead of kilometers per hour.