============================= names for powers of the metre ============================= by Andrew Main (Zefram) 2006-04-09 abstract -------- Some of the convenience supplied by unit prefixes in the metric system is lost when a unit is being raised to a power other than positive or negative unity. Among the SI units, this commonly arises with the square metre (m^2) and cubic metre (m^3). The convenience can be regained by giving the compound unit its own name and symbol. The novel name "lamin" and symbol "la" are proposed for the square metre, and the preexisting name "stere" and symbol "st" are endorsed for the cubic metre. table of contents ----------------- 0. general 0.0. problem 0.1. solutions 0.2. recommendations 1. square metre 2. cubic metre 3. references 0. general ========== 0.0. problem ------------ The use of prefixes [SI-PREFIX] with unit names and symbols in the metric system can avoid the need for a large range of numerical values to express measurements. Because there are prefixes at regular intervals of a factor of 10^3, it is possible to express measurements spanning 51 orders of magnitude using numbers that span only three orders of magnitude. For example, it suffices to use the numerical range [1, 10^3) with prefixes: a measurement of 1.234 * 10^17 m can be expressed as "123.4 Pm" (spoken as "one two three point four petametres"). This can be done with any metric unit that has its own name and symbol. This convenience also extends to compound units constructed by multiplying or dividing named units. A prefix can be attached to any of the constituent units, conventionally to the first. Thus a measurement of 1.234 * 10^17 W/sr can be expressed as "123.4 PW/sr" (or, if desired, "123.4 W/fsr"). However, a problem arises when a compound unit is constructed by multiplying a named unit by itself. In this case it is no longer reasonably possible to apply a prefix to exactly one of the factors. With the compound unit expressed as a power of one unit, adding a prefix to that unit means that the prefix is also raised to the power. A measurement of 1.234 * 10^17 m^2 cannot be expressed as "N Xm^2" with any existing prefix X and with 1 <= N < 10^3. The closest that are possible are "0.123 Gm^2" and "123 400 Mm^2". The intervals between the compound units of area are 10^6, so numbers in a range of six orders of magnitude are required. The situation is worse with a cubed unit: the prefixed units are now nine orders of magnitude apart, and 1.234 * 10^17 m^3 must be expressed as "123 400 000 km^3". 0.1. solutions -------------- This problem could in principle be resolved by expressing the compound unit as an ordinary product instead of using power notation. Thus 1.234 * 10^17 m^2 could be expressed as "123.4 Pm m", and 1.234 * 10^17 m^3 as "123.4 Pm m^2". This is inconvenient, and an unnatural way to express the unit, but can in theory be applied in all cases. Another possible solution is to have a name and symbol for the dimensionless value 1, and attach prefixes to that in compound units. For example, if the symbol "U" were used for 1, then 1.234 * 10^17 m^2 could be expressed as "123.4 PU m^2". This is similarly inconvenient and unnatural, but also a universal solution. The problem can solved more conveniently by introducing an individual name and symbol for the compound unit. Prefixes can then be applied to the new name and symbol without the prefix being subjected to an exponentiation operation. In principle this solution applies to any situation where a named unit is raised to any power other than +1 or -1, and so its unlimited application would require an unlimited number of new unit names and symbols. That would be a problem in itself. However, unlimited application is not required. In practice there is only one metric unit that is regularly raised to a non-trivial power and not combined with any other unit: the metre. Also, in practice there are only two powers to which it is regularly raised in this manner: 2 and 3. These compound units, being the SI units of area and volume, have extremely wide application, extending to all areas of science, commerce, and mundane life. Thus quite a lot of convenience could be gained by adopting a mere two more unit names and symbols. 0.2. recommendations -------------------- The recommendations of this paper are: (0) Henceforth individual names for the square metre and cubic metre should be adopted by standards bodies and by unit users for all situations requiring such units. Specific recommendations for names and symbols for these units are given in the following sections. (1) Do not at this time invent individual names for other powers of the metre or for powers of other units. (2) Bear this technique in mind for any future situation where a power of a unit is widely used. 1. square metre =============== There is already a named metric unit of area: the are, adopted by the CIPM in 1879 [SI-WITH]. There are two problems with it. The first is that its customary symbol, "a", causes ambiguities. Not only is "a" also used as a symbol for the year, but also the resulting symbol for the petaare, "Pa", clashes with the symbol for the pascal. Those problems could be resolved by selecting a new symbol for the unit. The second problem is intractable, however: the are is equal to 10^2 m^2 (1 dam^2), and so is not coherent with the SI [SI]. Lacking any existing name for the square metre, I propose the name "lamin", derived from the Latin word "lamnia" meaning a sheet of metal. This Latin word is etymologically responsible for English words such as "lamina" and "laminate", and the French word "laminoir", referring to flat things. I propose that the symbol for the lamin should be "la". This symbol is appropriately short and does not cause ambiguity with any existing metric unit symbol. 2. cubic metre ============== There are already two named metric units of volume. The first is the litre, defined in its current form by the CGPM in 1964 [LITRE] but of much older origin. The litre is in common use already, which is perhaps an indicator of the common desire for a specific name for a unit of volume. The litre is equal to 10^-3 m^3 (1 dm^3), and so is not coherent with the SI [SI]. The other named metric unit of volume is the stere, of similarly old origin but now only rarely used. It is equal to the cubic metre, and so is the right unit to adopt. The name "stere" is derived from the Greek word "stereos" meaning solid. The symbol "st" for the stere was established by the CGPM in 1948 [WRITING]. This symbol is appropriately short and does not cause ambiguity with any other metric unit symbol. I propose that the existing unit name "stere" and symbol "st" for the cubic metre should be adopted for general use. Ultimately, the stere should replace the incoherent litre in all its uses. 3. references ============= [LITRE] Conference Generale des Poids et Mesures, "Litre", Resolution 6 of the 12th CGPM, 1965, . [SI] Bureau International des Poids et Mesures, "SI", . [SI-PREFIX] Bureau International des Poids et Mesures, "SI prefixes", . [SI-WITH] Bureau International des Poids et Mesures, "Units used with the SI", . [WRITING] Conference Generale des Poids et Mesures, "Writing and printing of unit symbols and of numbers", Resolution 7 of the 9th CGPM, 1949, .