chriswarbo-net: a39b10207d84dc082ce6126247b207a6ed630d1a
1: ---
2: title: The Metric Red Herring
3: ---
4:
5: <small>Part of [my units pages](/projects/units)</small>
6:
7: This began as a companion to my post about
8: [improving our units](../../blog/2020-05-22-improving_our_units.html), which
9: discusses some problems with the metric system, SI and related ideas. This post
10: was a stand-alone argument that advocates of metric are often doing it for the
11: wrong reasons.
12:
13: Both are now part of a larger [collection of suggestions to improve and simplify
14: measurement, notation, numeracy, etc.](./index.html)
15:
16: ## The Powers-of-Ten Red-Herring ##
17:
18: Most discussions about metric versus imperial units seem to dwell on one
19: specific aspect of metric: that units are related by powers of 10. For example:
20:
21: - 100cm = 1m
22: - 10mm = 1cm
23: - 1kN = 1000N
24:
25: However, this is a red herring: those examples have *the same units* on each
26: side! They only differ in how they factor the multiple; solving these equations
27: reveals the definition of those factors:
28:
29: The first tells us that `c` ("centi") is a hundredth:
30:
31: ```
32: 100cm = 1m
33: 100c = 1
34: c = 1/100
35: ```
36:
37: The second tells us that `m` ("milli") is a thousandth:
38:
39: ```
40: 10mm = 1cm
41: 10m = 1c
42: 10m = 1/100
43: m = 1/1000
44: ```
45:
46: The third tells us that `k` ("kilo") is a thousand:
47:
48: ```
49: 1kN = 1000N
50: k = 1000
51: ```
52:
53: Notice that these are just statements about numbers: the units we wrote in each
54: case are irrelevant, since they immediately cancelled-out! We could have done
55: the same with inches, pounds, etc. There's also nothing special about ten: there
56: are [convenient factors for many bases](prefix_factors.html), e.g. 3Kim = 3072m.
57:
58: ## Metric is Minimal ##
59:
60: Focusing on "conversions" like 1km = 1000m obscures a much more important
61: feature of metric: there is only *one* unit of distance. Likewise there is
62: only *one* unit of force, *one* unit of pressure, and so on for each distinct
63: [dimension](https://en.wikipedia.org/wiki/Dimensional_analysis).
64:
65: Quantities 'expressed in kilometres' are still expressed in metres, since
66: kilometres are just multiples of metres. A distance like "5m" is expressed in
67: metres (five of them, since there's a "5" which means five). A distance like
68: "7km" is *also* expressed in metres (seven thousand of them, since there's a "7"
69: which means seven and a "k" which means thousand). "centi", "milli", "femto",
70: "kilo", "giga", etc. are just generic ways to abbreviate large and small
71: numbers.
72:
73: If we use a multiple which isn't base 10, like "two dozen metres", that's still
74: a metric distance; we haven't invented a new system of units with base 12;
75: "dozen" is just a generic linguistic device meaning "twelves". Compare this to a
76: foot containing a dozen inches: if we ask a baker for a dozen buns we'll get 12
77: buns; if we ask for a foot of buns we'll a line of buns about a third of a metre
78: long. This is because "foot" is *not* a generic multiplier: it is specific to
79: distance, and it is independent of other distance units (e.g. we don't need to
80: say "a foot of inches"). The same applies to "inch", "mile", etc. hence they are
81: all separate units, not just multipliers. In contrast, "metres" and "dozen
82: metres" are not separate units, and neither are "metres" and "centimetres".
83:
84: (Note that there is a slight wrinkle here, since "kilo" on its own is taken to
85: mean "kilograms"; this is not too important for this claim, but I explain why
86: the kilogram is problematic in the "Problems with SI" section of
87: [the companion post](improving_our_units.html)!)
88:
89: I think this is such a profound advantage that many (most?) people, even those
90: born and raised with metric, never grasp it explicitly. After all, why argue
91: that "power-of-ten conversions are easier" when we could go further and say
92: "there's nothing to convert between"; it can't get any easier than that!
93:
94: ## Metric is Coherent ##
95:
96: In [a 'coherent' system of units](coherence.html), we can combine our units
97: using multiplication and division, and alway get another unit of our system. The
98: most obvious example is speed, which is the same as a distance divided by a
99: time. There are imperial units of speed like the knot, but it's more common to
100: use the "distance over time" form like "miles per hour", or "metres per second"
101: in metric. Similarly for pressure, which is often measured in "pounds per square
102: inch" or "Newtons per square metre".
103:
104: We can do the same thing with any dimensions we like. For example, applying a
105: force (e.g. Newtons or pounds) over a distance (metres or feet) takes a certain
106: amount of energy (Joules or calories); hence we can express distances in units
107: of "energy over force". We can think of this as how far we could push with a
108: unit of force, before we use up the given energy.
109:
110: These 'derived units' will be some multiple of the 'normal' units for that
111: dimension, requiring a conversion factor if we want to convert a number between
112: the two forms. Here are some conversion factors for common imperial units:
113:
114: <div class="summarise">
115: <span class="summary">
116: Conversion factors between various imperial distance units.
117: </span>
118:
119: Quantity Inches Feet Miles Calories per pound Calories per ounce
120: ------------------- ------ ------- ------- ------------------ ------------------
121: 1 inch 1 0.083 2x10^-5^ 0.03 0.43
122: 1 foot 12 1 2x10^-4^ 0.32 5.18
123: 1 mile 6x10^4^ 5280 1 1711 3x10^4^
124: 1 calorie per pound 37.03 3.09 6x10^-4^ 1 16
125: 1 calorie per ounce 2.31 0.19 4x10^-5^ 0.06 1
126:
127: </div>
128:
129: There are conversion tables like this for many other combinations of units, e.g.
130: to find how many slug feet per square hour are in a stone. If the order of the
131: units is the same in the rows and columns then the main diagonal will always be
132: 1. For the remaining numbers, we only need to remember one of the 'triangles'
133: (upper-right or lower-left), since we can get the other by 'reflecting' the
134: positions across the diagonal and taking the reciprocal of the numbers. (Or you
135: can [look them up](https://lmgtfy.com/?q=+1+calorie+per+pound+in+miles+!g&s=d)
136: like I did, rather than cluttering your brain with obsolete junk!)
137:
138: The metric equivalent doesn't have the inch/foot/mile or ounce/pound
139: redundancies, so the table is much smaller, and hence easier to memorise:
140:
141: <div class="summarise">
142: <span class="summary">
143: Conversion factors between metric distance units.
144: </span>
145:
146: Quantity Metres Joules per Newton
147: ------------------ ------ -----------------
148: 1 metre 1 1
149: 1 joule per newton 1 1
150:
151: </div>
152:
153: The conversion factors here are all 1. This is not a coincidence, it is *by
154: design*! [1 Joule is *defined as* 1 Newton
155: metre.](https://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf) Likewise:
156:
157: ```
158: 1 Watt = 1 Joule per second
159: = 1 Hertz Joule
160: = 1 Hertz Newton metre
161: = 1 Hertz Pascal cubic metre
162: = 1 Hertz square Coulomb per Farad
163: = 1 Newton Coulomb per Tesla Farad metre
164: = 1 Newton Joule per Tesla Coulomb metre
165: = 1 Newton Volt per Tesla metre
166: = 1 Amp Volt
167: = 1 Watt
168: ```
169:
170: This is a *huge* advantage to using metric, which I rarely/never see brought up
171: in discussions. I'm not sure whether this is because it's subconsciously taken
172: for granted (like the one-unit-per-dimension feature) or whether it's just used
173: less frequently in "real life" (e.g. day-to-day estimating, rather than explicit
174: engineering calculations). Either way, I think this should be celebrated more!
175:
176: In particular, these conversions are based off known scientific laws. For
177: example Newton's second law of motion, usually written `F = ma`, tells us that
178: multiplying a mass by an acceleration results in a force. This is actually a
179: statement of *proportionality*, e.g. doubling the mass will double the force; to
180: get an equation we need a "constant of proportionality", which is an arbitrary
181: scaling factor which we usually write as `k`, so the general form of Newton's
182: second law should really be expressed as `F = kma`. Metric units are *defined
183: such that* these scaling factors are 1, which gives us simple equations without
184: having to remember a bunch of proportionality constants (i.e. those shown in the
185: tables above).
186:
187: ## Exposing My Lies ##
188:
189: I have to admit that the above is slightly inaccurate, for the purposes of
190: getting across my way of thinking. The first major point to clarify is that when
191: I say "metric" I'm actually referring mostly to
192: [the SI system](https://en.wikipedia.org/wiki/International_System_of_Units),
193: which differs a little from the metric units that are commonly used day-to-day.
194: In particular, metric often uses extra, redundant units which are not part of
195: SI, including the litre, tonne and hour.
196:
197: Next, I claimed above that prefixing a unit changes the associated number rather
198: than the unit, e.g. "2km" is 2000 in the unit of metres, rather than 2 in the
199: unit of kilometres. In fact, [the SI
200: definition](https://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf)
201: explicitly states that prefixing a unit with a multiplier, like "kilometre",
202: gives us a new, "derived" unit. This is important for resolving otherwise
203: ambiguous quantities like "3cm^3^": according to SI, this is 3(cm)^3^ =
204: 0.000003m^3^, whereas treating prefices in the way I describe would give
205: 3c(m^3^) = 0.03m^3^. Similar problems arise when dividing, e.g. "per kilometre".
206:
207: Whilst the SI method is well-defined, it still places a mental burden on the
208: user. What's especially annoying is that the SI rules for units are opposite to
209: the usual algebraic rules for multiplication and exponentiation, where
210: ab^3^ = a(b^3^)
211:
212: I cover these in more depth in
213: [the companion post](improving_our_units.html), but I think the best
214: thing to do in these situations is stick to the base unit for that dimension
215: (e.g. "cubic metre" or "per metre"), apply prefix multipliers if they are
216: unambiguous; or otherwise add
217: [explicit parentheses](https://en.wikipedia.org/wiki/S-expression).
218:
219: ## Take Aways ##
220:
221: Whilst the metric (or SI) system
222: [isn't perfect](improving_our_units.html), it's also much better than
223: those hodge-podges of historical baggage known as imperial units. The most
224: obvious argument against imperial units is that [there are so many, related in
225: arbitrary ways](https://www.youtube.com/watch?v=r7x-RGfd0Yk); but
226: that's a bit of a cheap shot, since few people make regular use of barleycorns,
227: fathoms or leagues. Likewise the common argument *against* imperial, that powers
228: of ten make arithmetic easier, is shallow at best, and irrelevant at worst.
229: Instead, the two *real* advantages of metric (or SI) are having one unit per
230: dimension, and requiring no conversion factor when combining dimensions.
231:
232: I'm a firm believer that seemingly-innocuous complications, like those found in
233: imperial units of measurement, are in fact significant risks; they impede
234: learning, potentially turning people away from areas like maths and science; and
235: their [compounding, confounding behaviour on the large
236: scale](https://en.wikipedia.org/wiki/Mars_Climate_Orbiter) constrains what we're
237: capable of achieving as a species.
238:
239: Every small "gotcha" can be pre-empted on its own, but it takes knowledge and
240: experience to do so, and some small effort every time. As such "minor" issues
241: combine together, they can quickly overflow our limited mental capacity,
242: making it naïve and irresponsible to think their individual avoidability can
243: hold in general. (As a programmer, such
244: [seemingly](https://en.wikipedia.org/wiki/Strong_and_weak_typing)
245: [minor](http://wiki.c2.com/?CeeLanguageAndBufferOverflows)
246: [problems](https://en.wikipedia.org/wiki/Code_injection) are quite widespread,
247: and even skilled experts often slip up now and then!)
248:
249: The only way to combat such unnecessary complication is by an aggressive pursuit
250: of simplicity. Irrelevant details, unwanted degrees of freedom and unnecessary
251: asymmetries only act to slow us down and trip us up. Imperial units have
252: infected our collective mind for so long that we're often unable to see the
253: simplicity that metric provides: we used to waste so much effort converting
254: between redundant imperial units that, when confronted with a single metric
255: equivalent, we started treating multiples as if they were different units, just
256: to make it more familiar.
257:
258: The other advantage, of combining dimensions, is alien to many, despite the
259: prevalence of examples like "miles per hour" and "pounds per square inch".
260: Presumably this is due to how horrible it is to convert between imperial units
261: in this way. It might even be the case that quantities like "miles per hour" and
262: "pounds per square inch" are acceptable precisely because there's no expectation
263: that they be convertible to any existing units (other than their constituents,
264: like "miles" and "hours" for "miles per hour"). This mentality might explain why
265: someone thought it was a good idea to invent monstrosities like "kilowatthours",
266: rather than just sticking a "mega" prefix on to the Joule!
267:
268: In any case, we need to embrace the simplicity of metric; grok what it tells us
269: about the nature of measure and dimension; and use the saved mental effort to
270: tackle bigger, harder problems.
Generated by git2html.