A better use of multiples?

To ease the transition to metric measures, straight substitution of units is often used – kg for pounds, metres for yards, km for miles and so on. Ronnie Cohen argues that, as a result, we fail to take advantage of metric’s superiority in dealing with a range of numbers, including the very large (and small).

We often see large numbers used with multiples in the metric system. For example, it is normal to express large distances on earth in thousands of kilometres and large distances across space in millions of kilometres instead of using megametres or gigametres. I suspect that many see the kilometre as the metric counterpart to the mile and are probably unaware that the metric system has bigger multiples of length. The promotion of larger multiples of length, which imperial cannot match, would show one of the strengths of the metric system, namely the ability to express extremes at both ends of the scale.

Similarly, use of gigawatts instead of thousands of megawatts and greater use of kilotonnes and megatonnes would enable more Britons to become more familiar with SI prefixes and make metric more comprehensible for all.

In the UK, it is common to use the rule of 1000 and avoid intermediate multiples and that probably contributes to the image of the metric system as scientific and technical rather than a general-purpose measurement system that is suitable for all purposes. For example, millilitres and litres are commonly used and one is much more likely to see a product label for a drink that shows a volume of several hundred millilitres than a smaller number of centilitres or decilitres. These intermediate units are less frequently used in the UK. Perhaps a label that shows 5 dl or 50 cl would look more comprehensible and human in scale than 500 ml.

The rule of 1000 is commonly followed in DIY stores where far more products are labelled in millimetres than centimetres, probably because architects and builders work in millimetres. I have seen private height restriction signs and heights of buses shown in millimetres and metres but never in centimetres. It seems odd to use millimetres for such purposes when we consider that we would be unlikely to express our height in millimetres. We might do so in centimetres or metres but never in millimetres.

The way that metric units are commonly used in the UK is probably a legacy of the use of imperial where metric units are often seen as equivalents of imperial units. Poor choices in the use of multiples have been exploited by anti-metric campaigners. The users who make those poor choices, not the system itself, are at fault. Sensible use of the appropriate multiples would make it possible to use smaller numbers to express measurements and might promote acceptance of metric in the UK.

Let’s see how better use of multiples can be put into practice with practical examples. On 16 June 2012, The Economist published a special report about the Arctic, which contained some measurements with huge numbers. Here are a few quotes from the report with examples:

  • “Then, in 2007, the sea ice crashed, melting to a summer minimum of 4.3m sq km, close to half the average for the 1960s and 24% below the previous minimum, set in 2005.” 4.3m sq km can be expressed as 4.3 square megametres.
  • “According to a 2008 study by the US Geological Survey, the Arctic may hold 90 billion barrels of oil and 1669 trillion cubic feet of natural gas, respectively 13% and 30% of the world’s estimated undiscovered reserves.” 90 billion barrels of oil can be expressed as 14.3 teralitres of oil. 1669 trillion cubic feet of natural gas can be expressed as 47.26 thousand cubic kilometres of natural gas.
  • “According to an estimate made in 2009, terrestrial permafrost holds about 1.7 trillion tonnes of carbon, roughly twice as much as the atmosphere.” 1.7 trillion tonnes can be expressed as 1.7 teratonnes.

With the imperial system, we are often forced to use extremely large numbers for expressing very large quantities because the imperial system has a limited range at both ends of the scale and struggles when describing the extremely large and extremely small. This practice is often seen with metric units but it is not necessary.

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15 Responses to A better use of multiples?

  1. John Steele says:

    @RonnieC

    Agree completely. The use of "large counting words" (million, billion, trillion, etc) should be avoided completely with the SI. The choices are to use a suitable prefix or use the un-prefixed unit with scientific (power of ten) notation.

    I believe the UK now uses US definitions, but other countries use different definitions of billion, trillion, etc (or the related word in their language). NIST SP 811 (a metric style guide) comments in section 7.10.3 on the unacceptability of ppm, ppb, ppt (parts per million, billion, trillion) but two sentences go further:
    "Because the names of numbers 10^9 and larger are not uniform worldwide, it is best that they be avoided entirely; the preferred way of expressing large numbers is to use powers of 10. . . . Another, and more important one, is that it is inappropriate to use abbreviations that are language dependent together with internationally recognized signs and symbols" The second sentence would rule out "4.3m sq km" (because of the m and sq) in favor of 4.3 Mm², although 4.3 square megameters would be acceptable.

    On this point, SP811 claims to be in agreement with ISO 31-0, but I don't have the latter document and don't know if the wording is identical.

    The US is similar to the UK in using milliliter or liter for volume net contents. In fact the deciliter or centiliter is not permitted under labelling law. (however, the deciliter is widely used by the medical profession as the denominator unit for blood test results).

    I find it remarkably hard to envision a teraliter. I would suggest for amounts over 1000 L, the cubic meter may be a better choice, with prefixes as necessary
    1 ML = 1 dam³
    1 GL = 1 hm³
    1 TL = 1 km³
    SAE metric practice standard TSB003 does not permit prefixes larger than one to be used with the liter.
    NIST SP811 does not disallow it, it merely describes it as "unusual" and discourages it.

    I will note that the Department of Energy totally ignores the advice of their NIST friends in the Department of Commerce, and speaks mostly of quadrillion BTUs of energy (known as quads in DoE-speak).

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  2. philh says:

    I found myself reacting to this article with mixed feelings.

    I certainly agree that metric is poorly represented in the way it is often used and it is a big mistake to project imperial habits onto it to make it seem more palatable e.g. writing 2.5 kg as 2 kg 500 g (I have actually seen this), or expressing 12.5 mm as a one and a quarter cm (in other word fractionalizing the cm instead of using mm) and so on.

    I too would like to see greater use of units such as megametre, gigametre and terametre etc for large terrestrial and astronomical distances.

    Where I part company on some of the suggestions is, for example, "... 47.26 thousand cubic kilometres ... " I think that should be 47 260 cubic kilometres.

    Okay let's not be writing great strings of zeros or using big numbers when we have a much neater way of doing so with a better choice of prefix, but equally let's not fall into the trap of thinking we can't count above a hundred.

    I am quite happy, for example, with expressing personal height in metres or in centimetres (e.g. 1.67 m or 167 cm) but equally I don't think using the mm should be particularly discouraged either (1670 mm is fine).

    I say don't fall into the trap of thinking that just because something is given in mm it has to mean that it is accurate to a mm. Consider a situation where, for instance, you estimate the distance to a bus stop up the road is about 200 metres. Does that make your estimate to be to the nearest metre? No of course not so why do that with the mm? Take it in context!

    I could go on but suffice it to say that, yes, we do have to get it across that the metric system is fundamentally different from traditional units; it is not just a question of different size units and names (if that was all there was to it then the change would be more difficult to justify) but we must not obscure its flexibility by being too prescriptive or inventing rules that may not always work.

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  3. BrianAC says:

    A thought provoking one this. In electronics the multiples of Ohms and sub-division of the Farad are perfect examples of a mix of "what is used v what is available". Kilo, Mega are used, never Deka. GHz is in everyday usage but mostly never understood by those buying computers and mobile phones. The microfarad, nanofarad and picofarad are used for small values but I have never seen 10,000 microfarad listed as 10 millifarad. The use of the Greek alphabet does not help as ASCII use is not predictable.

    I totally agree that the use of those awful billions and trillions should go (1 US billion is 1,000 million, the UK billion is a bi-million, 1,000,000 million; UK Ministers take note, it could prove expensive some day).

    Then to land measure - when do we change from ares to sq km? Surely that should be hectares, kilares and Megares??
    Certainly the use of the Mega meter would be sensible, the Earth being (roughly) 12 Mm in diameter and 39 Mm circumference. The distance across the Atlantic bridge being 4.5 Mm (not to be confused with 4.5 mm).

    Now we use Decades and Centuries for years of time, which often irks me personally when over-used, so the over use of the same in metric would also be seen as irksome to many. A sensible use of multipliers appropriate to the application in hand I guess would be my answer. (Even the spellchecker don’t like most of this!)

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  4. Jake says:

    @BrianAC

    I'm afraid I must disagree with you about the UK billion. I believe it used to be 1,000,000 million many years ago, but today a billion is understood (and used by the media) to mean 1,000 million (as in the USA).

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  5. Martin Clutterbuck says:

    The use of multiples in the metric system in the UK is at best erratic as illustrated by the article and responses. My question would be simply, why?

    A recent visit to a department store on a mission to find some 5 cm wide ribbon for my wife revealed that the assistant did not realise that 50 mm was the same!

    Is this the result of a failed education system, poor 'marketing' by standards bodies or just resistance to change?

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  6. Ronnie Cohen says:

    @philh

    I concede that you have a point about my use of cubic kilometres when I said that, "1669 trillion cubic feet of natural gas can be expressed as 47.26 thousand cubic kilometres of natural gas.". I should have converted cubic feet into litres rather than cubic metres to avoid this unnatural and unintuitive conversion into cubic kilometres.

    It would have been much better to say that "1669 trillion cubic feet of natural gas can be expressed as 47.26 petalitres of natural gas." (1 petalitre = 10 to the power of 15 litres). Again, we avoid the use of ridiculously large numbers by using the appropriate muliples in the metric system. With imperial, the use of enormous numbers with lots of zeroes on the end is often unavoidable.

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  7. BrianAC says:

    @Jake, So you agree with me then? From the Oxford English dictionary (today): -
    In British English, a billion used to be equivalent to a million million (i.e. 1,000,000,000,000), while in American English it has always equated to a thousand million (i.e. 1,000,000,000). British English has now adopted the American figure, though, so that a billion equals a thousand million in both varieties of English.

    I was educated in England in the '40's and '50's, no one has to my knowledge ever re-defined my UK billion (bi-million) except "by common usage". How would that stand up in a court of law if I sued someone to a factor of x1000? Is it defined in any world standard?
    Back on topic, we need a universal system thet the whole world understands, not what the Americans tell us we should use. That is defined quite well by using metric multipliers.

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  8. Jake says:

    @BrianAC. Yes, I think we agree on what a billion is. A thousand million is a very useful number to have a name for, even if in this instance it looks as though the UK media adopted US usage.

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  9. dygituljunky says:

    Before he passed away, Pat Naughtin, the publisher of the Metrication Matters newsletter, observed that using multiples of 1000 actually makes it easier for people to adopt the metric system. When countries use the prefixes that denote anything less than multiple of 1000, it becomes more confusing for people with lower numeracy skills. He observed that when you're able to move the decimal 3 places for every prefix moving between prefixes is less confusing. Pat advocated ignoring any prefixes that are not representatives of multiples of 1000. I don't know if his newsletters and articles are still available online but he clearly laid out case studies of the textile and carpentry industries in Australia and their trouble and success, respectively, in adopting the metric system within those industries. Pat noted that the textile industry use the centimeter as the default prefix/unit combination whereas the carpentry industry used the millimeter as the default prefix/unit. Pat said that the textile industry had difficulty adopting and the carpentry industry successfully changed over. He further clarified that the carpentry industry use the millimeter for all measurements not just fine measurements; all architectural plans, measurements, and cuts on construction sites in Australia are done in millimeters.

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  10. Erithacus says:

    I worry about the nerdish nature of the original article and of some of the subsequent comments.

    It is not helpful to the campaign to popularise the metric system to introduce additional unfamiliar units such as megametres and cubic decametres. It is a struggle to get your average local radio presenter to understand that litres are related to cubic metres without compounding the problem with petalitres. In any case litres are not SI.

    There is also a flaw in the argument. If you want to compare a small quantity with a very large quantity - say, the area of Gibraltar with the area of Europe - it impedes understanding if you change the measurement units rather than the numbers. According to http://en.wikipedia.org/wiki/Area_and_population_of_European_countries , the area of Gibraltar is 6 square kilometres, while that of Europe is 26 680 000 square kilometres . Would it really help to write that as 26.68 square megametres? Surely not (I'm not even sure if I have got the decimal point in the right place!).

    Furthermore, if the number of trailing zeroes really becomes a problem, then it would be better to handle the problem by using scientific notation and sticking to base units. In scientific notation the area of Gibraltar is 6 x 106 m2, whereas the area of Europe is 2.668 x 1013 m2. (No doubt somebody will correct me if I haven't got the powers right!). Obviously,this is not appropriate outside scientific publications.

    As already pointed out, there is an obvious solution to the billion/trillion problem: avoid them. Write thousand million or million million as appropriate. And don't use "m" as an abbreviation for "million".

    I think this issue is strictly for the nerds.

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  11. John Steele says:

    OK, I am a nerd and proud of it.

    On large counting words, I am aware that billion used to have a different definition in the UK. However, the government adopted the 10^9 figure in 1974, yet posters here are unsure whether it is 10^9 or 10^12. http://hansard.millbanksystems.com/written_answers/1974/dec/20/billion-definition#S5CV0883P0_19741220_CWA_439

    There is no better argument for not using "large counting words." By the way, per the OED, you have also adopted the US definition of trillion, 10^12. If there is not agreement on what such words mean, they are useless for communications. I have never heard anyone dispute the definition of a metric prefix. Even if they did, there is an international agreement defining it, not conflicting national agreements. Do your kids not learn metric prefixes in school? American kids do, even if they promptly forget them when out of school. If they were really used in our cultures, kids would remember them.

    The man on the street may not really care about extremely large or small numbers. Whether I describe the annual energy consumption of the US in quadrillions of BTUs or exajoules, it is difficult to have a feeling for the number. Yet scientists, engineers, politicians, environmentalists, and the media need to be able to talk about such numbers. It is a nerdy subject, but if you hope to have a national energy policy, it is not JUST for nerds. Those who need to talk about it need guidelines on how to talk about it correctly. I have little concern if the "man on the street" gets it wrong, but when those who talk about it professionally (this includes the media) get it wrong, they set a bad example for us all.

    The "suggestion of 1000" sounds unbelievably wimpy compared to the "rule of 1000," but it is, after all, just a guideline. It is usually given with the statement that there are exceptions and a few are usually listed. It should be ignored in columns of a table (pick a prefix and use it throughout) or when numbers must be directly comparable. The other part of the "suggestion of 1000" is avoiding prefixes which are not multiples of 1000; however, that is more a recommendation that they not be introduced into new areas, where they are not already entrenched.

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  12. WJG says:

    I have mixed feelings regarding the original article.

    Very large, and very small units, of measure are almost always related to science, or science-related issues. Although I agree, that prefixes such as tera, peta, exa, and pico, femto, and atto should be used in a scientific context, I don't think they should be promoted, in the metric measures that we use, in our daily lives. Here in New Zealand, we officially changed to metric measures almost 40 years ago, and the prefixes above are not often seen or used by "the man in the street".
    The UK hybrid mixed measurement muddle, is a hurdle to the advancement of metric measures, and I understand the need to hunanise metric measure, athough I don't agree with it. Let us KIS, (ie: keep it simple). Metric measures are simple to learn, simple to understand, and above all simple to use. I stress the "simple to use" because it appears to me that some of the commentators here have not used metric measures. The "rule of 1000" and the "whole number rule" are more than just guidelines. They set the way in which we "use" metric measures. They set the way in which we "use" calculations of metric measures. They keep it simple, which it needs to be, to educate the wider population. Its the reason why we see metric measures displayed in millimetres, grams, and millilitres.
    The late Pat Naughtin's site (metricationmatters) has more information on the "rule of 1000" and the "whole number rule" It's very interesting. See http://www.metricationmatters.com/docs/WholeNumberRule.pdf

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  13. philh says:

    At the risk of sounding nerdish I would like to point out that the so called "rule of 1000" is OK most of the time but it can present a problem for area and volume. For then it becomes the rule of a million and thousand million respectively. This leaves us with 9 orders of magnitude between one option and the next in the case of volume. I guess that was the reason for units like the hectare and litre.

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  14. Mårten Hammarstrand says:

    It saddens me to read that you would use megatonne instead of teragram, which would be the correct unit. If people understand mega combined with tonne, the should have no troubles applying it with what comes next after the kilogram, the megagram.

    I do believe that people can easily learn the basic metric prefixes from 10^-9 to 10^9; a non-scientific person does not need more than that. Here's an example:
    The manufacturing size in integrated circuits is often quoted in nanometres, medicine doses are given in micrograms, water bottles comes in millilitres, kilojoule on food packaging, a large wind power plant produces several megawatts and your wifi-router operates in the gigahertz field. The rule of a thousand is a good rule, like Pat Naughtin argues, it is easier to comprehend those jumps. The smaller prefixes centi-deci-deka-hecto gives too much resolution, for instance when have you ever used centigrams, dekavolts or hectowatts?

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  15. philh says:

    Consider a couple of examples:

    Area of a piece of A4 (210 x 297 mm)

    0.0624 square metre
    62 370 square millimetre
    624 square centimetre

    Volume of a fish tank with dimensions 200 x 250 x 400 mm

    0.02 cubic metre
    20 000 000 cubic millimetre
    20 000 cubic centimetre
    20 cubic decimetre (or litre)

    Which of the above options would seem the most user friendly?

    Also consider calculating the volume of the tank. Which way would be easier:

    200 x 250 x 400 (in millimetre)
    or 2 x 2.5 x 4 (in decimetre)?

    I find the decimetre quite handy for reckoning volume in litre.

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