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

The conversion factors here are all 1. This is not a coincidence, it is *by
design*! [1 Joule is *defined as* 1 Newton
metre.](https://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf) 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](https://en.wikipedia.org/wiki/International_System_of_Units),
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](https://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf)
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 "3cm^3^": according to SI, this is 3(cm)^3^ =
0.000003m^3^, whereas treating prefices in the way I describe would give
3c(m^3^) = 0.03m^3^. 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
ab^3^ = a(b^3^)
I cover these in more depth in
[the companion post](2020-05-22-improving_our_units.html), 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](https://en.wikipedia.org/wiki/S-expression).
## Take Aways ##
Whilst the metric (or SI) system
[isn't perfect](2020-05-22-improving_our_units.html), 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](https://www.youtube.com/watch?v=r7x-RGfd0Yk); 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](https://en.wikipedia.org/wiki/Mars_Climate_Orbiter) 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](https://en.wikipedia.org/wiki/Strong_and_weak_typing)
[minor](http://wiki.c2.com/?CeeLanguageAndBufferOverflows)
[problems](https://en.wikipedia.org/wiki/Code_injection) 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.