Improving our Units

Posted on by Chris Warburton

This is a companion to my post about the metric red-herring. That post is a simple claim that the main benefit of the metric system is not the “multiply by 10” argument that comes up over and over in ‘metric versus imperial’ discussions. Instead, the two main benefits are that:

This post discusses some problems with metric, the SI system, and our systems of counting more generally; proposes some alternatives; and picks a few of those that would be reasonably practical to employ.

Problems with Metric

The metric system, as it’s used in day-to-day life, has a few problems. The following problems are solved by sticking to SI units:

Problems with SI

The SI system avoids the extraneous units of metric like litres and km/h, and unnecessary degrees of freedom like the zero-point of Celsius. The most obvious problem remaining with SI is that its base unit of mass is the “kilogram”. Whilst a kilogram equals one thousand grams, as we would hope, the definition is backwards: the gram is defined as one thousandth of a kilogram, AKA a millikilogram.

Within the dimension of mass this is merely silly; the real problem arises when we start combining dimensions. In the companion post I use the example of Newton’s second law of motion F = kma where the constant of proportionality k is equal to 1 in SI, due to the way its base units are defined. This is true and good, but the base unit for m (mass) is the kilogram. If we want to use grams, to avoid the sillyness of a prefixed base unit, we end up needing a constant of proportionality k = 1/1000. This is why I strategically chose to avoid using mass in most of my examples!

This is purely a naming issue, since we can just rename the “kilogram” to something without an ambiguous prefix; for example, it used to be called the “grave”. This would rename grams to milligraves, which explains where the factor of 1/1000 comes from.

Note that there’s an alternative approach, where we treat the gram as the base unit and hence the kilogram’s prefix works properly. Such a system was widely used, and is now know as the “CGS” system. Unfortunately that system uses the “centimetre” as its base unit of length, and hence suffers the same problem, except in the length dimension rather than mass.

There is another problem with SI, that I mention in the companion post, which is that units like “cm3” may be misinterpreted. SI resolves this by giving prefices higher precedence than exponents (e.g. the “c” applies first, then the “3” is applied after), but this is opposite to the usual rules of multiplication and exponents, e.g. algebraically we would say that abc = a(bc).

This also allows derived units (e.g. cubic centimetres) to creep into our calculations. My preferred solution to any potential ambiguity is to use explicit parentheses, hence:

Another way to reduce ambiguity and ugliness is to give explicit names to our units, like calling the cubic metre a “kuub” (as mentioned above). These two quantities could then be called 3 microkuub and 3 centikuub, respectively.

Symbolising the Kuub

We can’t symbolise kuub with a “k” or a “c”, since those clash with “kilo” and “centi”. One obvious symbol would be a cube, which we can draw isometrically as a ‘hexagon with spokes’. The closest Unicode I can find for this is Ⓨ (the letter Y inside a circle), so these quantities would be 3µⓎ and 3cⓎ.

Alternatively we could have a “power-agnostic” symbol, e.g. writing 3µ𒑳 and 3c𒑳. Here metres are written as 𒀹 and multiple arrows get “stacked”. This allows inverse metres to be written as 𒀹, so up and down arrows cancel out. We could also use the “forwards” (rightwards) direction for time, writing seconds as 𒀹 and inverse seconds as 𒀹. For example, the acceleration at Earth’s surface g = 9.81𒀹𒃵.

Problems with Time

Time is tricky. The base unit in SI is the second, but it’s common for metric to use hours, e.g. km/h or “kilometres per hour”, where:

1 hour = 60 minutes
       = 60×60 seconds
       = 3600 seconds

One notable attempt at making time metric was the decimal time briefly adopted by France; however, as I note in the companion post, the main advantages of metric don’t come from using base 10. The base 60 relationships between hours, minutes and seconds are perfectly reasonable, and I would argue preferable to base 10. The main problem is that we seem to have three units of time (seconds, minutes and hours), when we would prefer a single base unit and some multipliers (whether base 10 or otherwise).

In fact, the current system of hours/minutes/seconds already provides this! The hour is the base unit of time, and we define the minute as one sixtieth of an hour: originally known as a “minuta” or “pars minuta prima” (“fraction”, “small part” or “first small part”). Dividing by sixty again gives a “pars minuta secunda” (“second small part”, or “second”). This system goes on, with “minuta tertia” (“thirds”), etc. The naming is nice and logical, akin to the “billion”, “trillion”, “quadrillion”, etc. used for large numbers. Despite mostly being used for time, these multipliers are actually generic, since they’re also used for angles, with one degree containing 60 arcminutes, each of which contains 60 arcseconds.

The main problem with time is that SI uses the second as its base unit rather than the hour, so attempting to change that would either require conversion factors of 3600 all over the place, or require redefinition of almost every other unit. I think it’s easier to keep the other units as they are and have the second as our base unit of time. In particular, speeds should be expressed in (prefixed multiples of) metres per second, and so on.

The minute and hour should be replaced by sexagesimal multiples of the second (for sexagesimal prefices see “Problems with Prefices” below). If we want to avoid the historical baggage and numerical ambiguity of the name “second” we could shorten it to “sec”, which is already commonly understood. The plural “secs” invokes the idea of sixes (akin to “sex”, as in “sexagesimal”), which is an interesting side effect.

Problems with Angles

Since I’ve now mentioned angles, it’s worth giving them some thought here. Angles are dimensionless, so choosing a base angle is the same as choosing how many divisions should go into a full turn. An obvious choice is one, which lets us use units like Hz for angular velocity.

SI makes a different choice: the radian, whose sine and cosine functions have slopes oscillating between ±1. Larger divisions, like full turns, give a steeper slope; smaller divisions, like degrees, give a shallower slope. Sine and cosine are derivatives of each other (modulo a minus sign), scaled by this maximum slope; hence this slope acts as a constant of proportionality (like those discussed in the section “Metric Conversions Multiply By One” of the companion post). Forcing this slope to be 1 gives us the radian as our unit of angle. (Radians can be defined in other ways, e.g. the angle subtended by an arc whose length equals its radius; I just think this sine/cosine relationship fits the ‘constant of proportionality’ template nicely). Using radians as our unit for angles gives derived units like radians per second (AKA rad/s) for angular velocity.

There are a little over six radians in a full turn; the exact ratio is irrational, around 6.283…. This number is so ubiquitous in geometry, and mathematics more broadly, that we denote it with the symbol τ; hence one turn equals τ radians. If we use radians as our base angle, certain formulae become quite simple (e.g. the length of an arc is the angle in radians multiplied by the radius), but others require τ (e.g. angular frequency in Hertz is the angular velocity in radians per second divided by τ). If we use turns as our base angle, these sets of formulae switch around, e.g. converting turns to Hertz requires no conversion factor, but the length of an arc is the angle in turns multiplied by the radius multiplied by τ.

I’m undecided as to which is preferable as a base/default, since each have their merits. Current practice is to use radians by default, and use a factor of τ when talking about turns, e.g. “3τ” is three turns. This makes sense, but this irrational multiplier makes many “everyday” situations more complicated than if we use turns (e.g. we can get surprisingly far without irrationals!). This, along with the 1-to-1 conversion between units like the Hertz and Becquerel, makes me favour turns as our unit of angle.

As for other approaches:

Problems with Units

There are alternatives to SI called “natural units”. Whilst the metric and SI systems take their derived units from physical equations like F = ma, natural units go one step further and take their base units from physical constants like the speed of light (in a vacuum). Using natural units can make physical calculations much easier, for example the equation E=mc2 becomes E=m if the speed of light is a base unit.

There are three problems with switching to natural units:

Problems with Prefices

10 isn’t a particularly good base for a number system. It’s only divisible by 1, 2, 5 and 10, which obscures many patterns (i.e. those which don’t have period 2 or 5) and overly-complicates the representation of otherwise ‘simple’ numbers (like the 1Mim example in the companion post). The binary prefices (“kibi”, “mibi”, etc.) are preferable in this regard, although their exponents (10, 20, …) are rather arbitrary; we only use them since they’re close to more familiar powers of 10.

Commonly proposed alternatives to base 10 are base 2 (binary), base 12 (dozenal) and base 60 (sexagesimal). These are superior highly composite numbers (i.e. they have many factors), and hence they can show patterns with more periods. For example, sequences differing by 2, 3, 4 and 6 will show a pattern in their digits (“dozits”?) when written in dozenal. Dozenal also makes it easy to count using one’s fingers. Binary is also easy to count on one’s fingers, but the dearth of digits (“bits”) can lead to very long, unwieldy numbers. Hexadecimal (base 16) is a common way to avoid this problem of binary, whilst still following similar patterns.

I’m not aware of any sexagesimal prefices (powers of 60), but they would be useful e.g. for abbreviating times to begin with, and maybe spreading their highly composite nature into other areas. The existing system of “minutes”, “seconds”, “thirds”, etc. is useful for dividing by 60, although it would be preferable to use it in prefix rather than postfix position for consistency (e.g. “minutearc” rather than “arcminute”). Metric prefices traditionally used Latin for smaller parts (e.g. “centi” and “milli”) and Greek for larger parts (e.g. “kilo” and “mega”). The minutes/seconds/thirds system likewise uses Latin for the smaller parts, so we should use Greek for our sexagesimal multiples: “prota” would be sixties, “defter” would be three thousand six hundreds, “trito” would be two hundred and sixteen thousands, and so on.

Another problem with current unit prefices is that they only increase in fixed multiples (of 1000), e.g. 1Gm = 1000Mm = 1000000km = 1000000000m. Such prefices act in a similar way to unary (or perhaps Roman numerals). This doesn’t scale, requiring the rapid invention of new names.

This problem is likely inherited from the naming system used for large numbers, where “million” is a thousand thousands, “billion” is a thousand millions, “trillion” is a thousand billions, etc. Just like place-value numerals make more efficient use of digits than unary, we can make more efficient use of names by introducing them logarithmically. In base ten we invent a new name for ten tens: the “hundred”. With this new name we can count up to 99,9,9 (“ninety nine hundred and ninety nine”), but our next name (the “thousand”) appears after only 9,9,9; far too early! If the thousand were a hundred hundreds, we could then count up to 9999,99,9,9 (“ninety nine hundred and ninety nine thousand, ninety nine hundred and ninety nine”) before needing a new name (e.g. the “million”); the million currently appears after 99,99,9,9. The next name (“billion”) currently appears after 9,9999,99,9,9, but we only need it once we reach 99999999,9999,99,9,9 (a million millions, using our redefinition of million); the trillion would only be needed once we reach a billion billions, and so on. Note that each name doubles the number of digits we can reach, rather than merely adding three (or six). Such names have a clear relation to binary (powers of two), and also remind me of factoradic numbers. If this were to be adopted, it would be a much better idea to invent new names rather than trying to redefine the current terms!

Closing Thoughts

Some of the above is fanciful, and other parts are purposefully provocative. Whilst I don’t think we’ll see speedometers denoting logarithmically-scaled dozenal divisions of lightspeed any time soon, I think there are some definite steps we can take to improve our use of units.

More long-term I think that switching to dozenal would be nice, and/or hexadecimal (helped by the ubiquity of binary computers), but attempts to do so have struggled for decades. In particular, any advantages of teaching and learning in these systems would be countered by the difficulty of applying them in the “real world” with its prevalence of base 10.