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Part of my units pages

This began as a companion to my post about improving our units. Both are now part of a larger collection of suggestions to improve and simplify measurement, notation, numeracy, etc.

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Advocates of the metric system often focus on the “wrong” advantages, the most common being that metric units are related by powers of ten. While convenient, this is a red herring! The prefixes used by each of those supposed “units” (like kilo- and centi-) are just generic numerical multipliers, not fundamental properties of the units themselves.

For example: “10k” means ten thousand, those are just different ways to write the same number. Hence “10km” means “ten thousand metres”. We can write the same quantity in different ways, but the unit hasn’t changed (it’s still in metres).

The red-herring is to instead interpret “10km” as “ten thousand-metres”; to treat the “thousand-metre” as some separate unit, independent of the metre; then try to argue that metric is simple, because translating between units of metres and thousand-metres uses a nice conversion factor of 1000. That’s just unnecessary over-complication!

The same applies for quantities which aren’t powers of ten, e.g. two dozen is twenty four, so two dozen metres is twenty four metres. It’s not a couple of “dozen-metres”; or one “couple-dozen-metre”!

Metric is coherent

When metric advocates focus on this red-herring, they miss a much more important feature of metric: that it is coherent.

This means that metric only has one unit of distance (the metre), one unit of force (the Newton), one unit of pressure (the Pascal), and so on for each distinct dimension.

I think this is such a profound advantage that many (most?) people, even those born and raised with metric, never grasp it explicitly. After all, why argue that “power-of-ten conversions are easier” when we could go further and say “there’s nothing to convert between”; it can’t get any easier than that!

In a ‘coherent’ system of units, we can combine our units using multiplication and division, and always get another unit of our system. The most obvious example is speed, which is the same as a distance divided by a time. There are imperial units of speed like the knot, but it’s more common to use the “distance over time” form like “miles per hour”, or “metres per second” in metric. Similarly for pressure, which is often measured in “pounds per square inch” or “Newtons per square metre”.

We can do the same thing with any dimensions we like. For example, applying a force (e.g. Newtons or pounds) over a distance (metres or feet) takes a certain amount of energy (Joules or calories); hence we can express distances in units of “energy over force”. We can think of this as how far we could push with a unit of force, before we use up the given energy.

These ‘derived units’ will be some multiple of the ‘normal’ units for that dimension, requiring a conversion factor if we want to convert a number between the two forms. Here are some conversion factors for common imperial units:

There are conversion tables like this for many other combinations of units, e.g. to find how many slug feet per square hour are in a stone. If the order of the units is the same in the rows and columns then the main diagonal will always be 1. For the remaining numbers, we only need to remember one of the ‘triangles’ (upper-right or lower-left), since we can get the other by ‘reflecting’ the positions across the diagonal and taking the reciprocal of the numbers. (Or you can look them up like I did, rather than cluttering your brain with obsolete junk!)

The metric equivalent doesn’t have the inch/foot/mile or ounce/pound redundancies, so the table is much smaller, and hence easier to memorise:

The conversion factors here are all 1. This is not a coincidence, it is by design!

This is a huge advantage to using metric, which I rarely/never see brought up in discussions. I’m not sure whether this is because it’s subconsciously taken for granted (like the one-unit-per-dimension feature) or whether it’s just used less frequently in “real life” (e.g. day-to-day estimating, rather than explicit engineering calculations). Either way, I think this should be celebrated more!

Exposing My Lies

I have to admit that the above is slightly inaccurate, for the purposes of getting across my way of thinking. The first major point to clarify is that when I say “metric” I’m actually referring mostly to the SI system, which avoids redundant units (like the litre, tonne and hour).

Another subtlety is that prefix precedence can appear ambiguous in quantities like “cm³”.

Takeaways

Whilst the metric (or SI) system isn’t perfect, it’s also much better than those hodge-podges of historical baggage known as imperial units. The most obvious argument against imperial units is that there are so many, related in arbitrary ways; but that’s a bit of a cheap shot, since few people make regular use of barleycorns, fathoms or leagues. Likewise the common argument against imperial, that powers of ten make arithmetic easier, is shallow at best, and irrelevant at worst. Instead, the two real advantages of metric (or SI) are having one unit per dimension, and requiring no conversion factor when combining dimensions.

I’m a firm believer that seemingly-innocuous complications, like those found in imperial units of measurement, are in fact significant risks; they impede learning, potentially turning people away from areas like maths and science; and their compounding, confounding behaviour on the large scale constrains what we’re capable of achieving as a species.

Every small “gotcha” can be pre-empted on its own, but it takes knowledge and experience to do so, and some small effort every time. As such “minor” issues combine together, they can quickly overflow our limited mental capacity, making it naïve and irresponsible to think their individual avoidability can hold in general. (As a programmer, such seemingly minor problems are quite widespread, and even skilled experts often slip up now and then!)

The only way to combat such unnecessary complication is by an aggressive pursuit of simplicity. Irrelevant details, unwanted degrees of freedom and unnecessary asymmetries only act to slow us down and trip us up. Imperial units have infected our collective mind for so long that we’re often unable to see the simplicity that metric provides: we used to waste so much effort converting between redundant imperial units that, when confronted with a single metric equivalent, we started treating multiples as if they were different units, just to make it more familiar.

The other advantage, of combining dimensions, is alien to many, despite the prevalence of examples like “miles per hour” and “pounds per square inch”. Presumably this is due to how horrible it is to convert between imperial units in this way. It might even be the case that quantities like “miles per hour” and “pounds per square inch” are acceptable precisely because there’s no expectation that they be convertible to any existing units (other than their constituents, like “miles” and “hours” for “miles per hour”). This mentality might explain why someone thought it was a good idea to invent monstrosities like “kilowatthours”, rather than just sticking a “mega” prefix on to the Joule!

In any case, we need to embrace the simplicity of metric; grok what it tells us about the nature of measure and dimension; and use the saved mental effort to tackle bigger, harder problems.

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