--- title: The Metric Red Herring --- This began as a companion to my post about [improving our units](../../blog/2020-05-22-improving_our_units.html), 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.](./index.html) ## 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: - 100cm = 1m - 10mm = 1cm - 1kN = 1000N However, this is a red herring! The qualifiers "centi", "kilo", "milli", etc. are not unique to metric. If we wanted to, we could use them with imperial units too. For example it's perfectly correct (if a little unorthodox) to say that there are 100 centifeet in 1 foot and 1000 pounds in 1 kilopound. A listener might find it strange, but they could figure out what you meant. The "powers of 10" in metric come solely from the common usage of these prefices. There are equivalent [prefices for powers of 2](https://en.wikipedia.org/wiki/Binary_prefix), where "kibi" is 2^10^ (symbol "Ki"; roughly a thousand), "mibi" is 2^20^ (symbol "Mi"; roughly a million), etc. These are used almost exclusively for data sizes and rates, like an SSD with a 1TiB capacity (one tibibyte, or 2^40^ bytes). Yet it's perfectly correct (if a little unorthodox) to say that 1Mim = 1,048,576m or that there are one thousand and twenty four gallons in one kibigallon. ## Multiples Aren't Units ## 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](https://en.wikipedia.org/wiki/Dimensional_analysis). 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](improving_our_units.html)!) 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 Conversions Multiply By One ## Metric only has one unit for each dimension, but some combinations of dimension are equivalent to others. 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:
Conversion factors between various imperial distance 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 6x10^4^ 5280 1 1711 3x10^4^ 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](https://lmgtfy.com/?q=+1+calorie+per+pound+in+miles+!g&s=d) 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:
Conversion factors between metric distance units. Quantity Metres Joules per Newton ------------------ ------ ----------------- 1 metre 1 1 1 joule per newton 1 1