The Metric Red Herring

This began as a companion to my post about improving our units, which discusses some problems with the metric system, SI and related ideas. This post was a stand-alone argument that advocates of metric are often doing it for the wrong reasons.

Both are now part of a larger collection of suggestions to improve and simplify measurement, notation, numeracy, etc.

The Powers-of-Ten Red-Herring

Most discussions about metric versus imperial units seem to dwell on one specific aspect of metric: that units are related by powers of 10. For example:

However, this is a red herring: those examples have the same units on each side! They only differ in how they factor the multiple; solving these equations reveals the definition of those factors:

The first tells us that c (“centi”) is a hundredth:

100cm = 1m
100c  = 1
   c  = 1/100

The second tells us that m (“milli”) is a thousandth:

10mm = 1cm
10m  = 1c
10m  = 1/100
  m  = 1/1000

The third tells us that k (“kilo”) is a thousand:

1kN = 1000N
 k  = 1000

Notice that these are just statements about numbers: the units we wrote in each case are irrelevant, since they immediately cancelled-out! We could have done the same with inches, pounds, etc. There’s also nothing special about ten: there are convenient factors for many bases, e.g. 3Kim = 3072m.

Metric is Minimal

Focusing on “conversions” like 1km = 1000m obscures a much more important feature of metric: there is only one unit of distance. Likewise there is only one unit of force, one unit of pressure, and so on for each distinct dimension.

Quantities ‘expressed in kilometres’ are still expressed in metres, since kilometres are just multiples of metres. A distance like “5m” is expressed in metres (five of them, since there’s a “5” which means five). A distance like “7km” is also expressed in metres (seven thousand of them, since there’s a “7” which means seven and a “k” which means thousand). “centi”, “milli”, “femto”, “kilo”, “giga”, etc. are just generic ways to abbreviate large and small numbers.

If we use a multiple which isn’t base 10, like “two dozen metres”, that’s still a metric distance; we haven’t invented a new system of units with base 12; “dozen” is just a generic linguistic device meaning “twelves”. Compare this to a foot containing a dozen inches: if we ask a baker for a dozen buns we’ll get 12 buns; if we ask for a foot of buns we’ll a line of buns about a third of a metre long. This is because “foot” is not a generic multiplier: it is specific to distance, and it is independent of other distance units (e.g. we don’t need to say “a foot of inches”). The same applies to “inch”, “mile”, etc. hence they are all separate units, not just multipliers. In contrast, “metres” and “dozen metres” are not separate units, and neither are “metres” and “centimetres”.

(Note that there is a slight wrinkle here, since “kilo” on its own is taken to mean “kilograms”; this is not too important for this claim, but I explain why the kilogram is problematic in the “Problems with SI” section of the companion post!)

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!

Metric is Coherent

In a ‘coherent’ system of units, we can combine our units using multiplication and division, and alway 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:

Quantity Inches Feet Miles Calories per pound Calories per ounce
1 inch 1 0.083 2x10-5 0.03 0.43
1 foot 12 1 2x10-4 0.32 5.18
1 mile 6x104 5280 1 1711 3x104
1 calorie per pound 37.03 3.09 6x10-4 1 16
1 calorie per ounce 2.31 0.19 4x10-5 0.06 1

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:

Quantity Metres Joules per Newton
1 metre 1 1
1 joule per newton 1 1

The conversion factors here are all 1. This is not a coincidence, it is by design! 1 Joule is defined as 1 Newton metre. Likewise:

1 Watt = 1 Joule per second
       = 1 Hertz Joule
       = 1 Hertz Newton metre
       = 1 Hertz Pascal cubic metre
       = 1 Hertz square Coulomb per Farad
       = 1 Newton Coulomb per Tesla Farad metre
       = 1 Newton Joule per Tesla Coulomb metre
       = 1 Newton Volt per Tesla metre
       = 1 Amp Volt
       = 1 Watt

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!

In particular, these conversions are based off known scientific laws. For example Newton’s second law of motion, usually written F = ma, tells us that multiplying a mass by an acceleration results in a force. This is actually a statement of proportionality, e.g. doubling the mass will double the force; to get an equation we need a “constant of proportionality”, which is an arbitrary scaling factor which we usually write as k, so the general form of Newton’s second law should really be expressed as F = kma. Metric units are defined such that these scaling factors are 1, which gives us simple equations without having to remember a bunch of proportionality constants (i.e. those shown in the tables above).

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 differs a little from the metric units that are commonly used day-to-day. In particular, metric often uses extra, redundant units which are not part of SI, including the litre, tonne and hour.

Next, I claimed above that prefixing a unit changes the associated number rather than the unit, e.g. “2km” is 2000 in the unit of metres, rather than 2 in the unit of kilometres. In fact, the SI definition explicitly states that prefixing a unit with a multiplier, like “kilometre”, gives us a new, “derived” unit. This is important for resolving otherwise ambiguous quantities like “3cm3”: according to SI, this is 3(cm)3 = 0.000003m3, whereas treating prefices in the way I describe would give 3c(m3) = 0.03m3. Similar problems arise when dividing, e.g. “per kilometre”.

Whilst the SI method is well-defined, it still places a mental burden on the user. What’s especially annoying is that the SI rules for units are opposite to the usual algebraic rules for multiplication and exponentiation, where ab3 = a(b3)

I cover these in more depth in the companion post, but I think the best thing to do in these situations is stick to the base unit for that dimension (e.g. “cubic metre” or “per metre”), apply prefix multipliers if they are unambiguous; or otherwise add explicit parentheses.

Take Aways

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.