======================================= proposal for combinable metric prefixes ======================================= by Andrew Main (Zefram) 2006-03-13 abstract -------- A revised system of prefixes for the SI (metric) system of units is proposed, to bring the benefit of unit prefixes to applications that are not adequately served by the existing prefixes. The revised system provides a name and symbol for every power of ten in the range 10^-178 to 10^+178 inclusive. table of contents ----------------- 0. rationale 1. design 1.0. objectives 1.1. available symbol letters 1.2. development 2. system 3. references 0. rationale ============ The set of unit prefixes [SI-PREFIX] in the metric system, used to create multiple and submultiple units, has been periodically extended by the CIPM [PFX-HIST], in order to provide SI units at scales smaller and larger than were previously possible. This has been consistently done by inventing new individual prefixes. Further scale extension is entirely foreseeable: physicists and astronomers have for decades dealt with masses and lengths the expression of which is not aided by the existing prefixes. The extent of scale extension likely to be required is such that continuing in the same manner, assigning an individual prefix to each power of 10^3, would require an unwieldy number of prefixes. In addition to the prefixes for powers of 10^3, the SI contains prefixes for 10^-1, 10^+1, 10^-2, and 10^+2. These prefixes are very popular outside science and engineering, particularly when applied to the units metre, litre, and second. However, they only reach these particular powers of ten, close to the basic unit. This compromises the scale independence that the SI otherwise has. It encourages the selection of derived units that are of practical size, rather than the strictly coherent unit, so that these prefixes will be useful. This is probably a factor in the popularity of the litre, an incoherent unit of volume, over the stere, the coherent unit of volume. The usefulness of the SI would be improved if intermediate powers of ten could be specified by prefixes at any scale, rather than only near the basic unit. Of course, doing this by individual prefixes would require an impractical number of prefixes. Both of these problems, range and resolution, can be tackled by resurrecting the old concept of combining prefixes. Tools like the prefixes get more powerful when they can be combined, just as the concept of combining prefixes with basic units was a great leap in power for the metric system. However, whereas prefixes were historically combined in fairly arbitrary arrangements, the proposal below has a particular scheme and syntax for the prefix system. Making each power of ten reachable in exactly one way avoids an unwarranted multiplicity of names for the same unit. 1. design ========= 1.0. objectives --------------- With respect to extending the range of prefixes, to satisfy currently foreseeable needs it should be made possible to express the order of magnitude of (a) the Planck volume and (b) the volume of the known universe using prefixes on the stere (stere = cubic metre), and (c) the mass of the electron neutrino and (d) the mass of the known universe using prefixes on the gram. With respect to resolution, it should be possible to express any power of ten within the supported range using prefixes. A strength of the current prefix system is that a substantial range can be covered to a useful resolution using only single prefixes. This capability should remain, and generally there should be no requirement for long combinations of prefixes. Complete backward compatibility must be maintained: no existing prefix may change meaning. 1.1. available symbol letters ----------------------------- In addition to the letters officially used by the current prefix symbols, "u" must be reserved as a substitute for mu, because it is widely so used in situations where Greek letters are unavailable. "l" is unusable as a prefix because it would make "lm" ambiguous. ("lm" currently means lumen. This is one of only two cases currently in SI of one basic unit symbol being a suffix of another. The other, involving the units coulomb and degree Celsius, rules out the degree symbol ever being used as a prefix.) "i" is used as the distinguishing element of the IEC binary-based prefixes [BIN-PREFIX], and is best reserved for this purpose. "K" is used in the IEC binary-based prefix kibi- in place of "k", and is best reserved for compatible uses. Thus 22 of the 52 possible Latin letter prefix symbols are already used, leaving 30. This proposal uses a further 8 of these, leaving 22. all letters: aAbBcCdDeEfFgGhHiIjJkKlLmMnNoOpPqQrRsStTuUvVwWxXyYzZ already used: a c d Ef Gh i kKl mMn pP Tu yYzZ now available: AbB C De Fg H IjJ L NoO qQrRsSt UvVwWxX used below: D t vVwWxX remaining: AbB C e Fg H IjJ L NoO qQrRsS U Note that available case pairs are a scarce resource. 1.2. development ---------------- First consider the resolution issue. Most current prefixes have a value that is a power of 10^3, but we would like to have prefixes that reach powers of ten between these, such as 10^4. We can reach such powers by combining a power-of-10^3 prefix with one of the few existing non-power-of-10^3 prefixes. Combining prefixes freely, we can reach 10^4 with the combination decakilo-, or alternatively with the combination centimega-. These combinations could also be made the other way round, as kilodeca- and megacenti-. To avoid a proliferation of names for the same thing, which would be counterproductive, let us standardise on a single way of forming these combinations. The greatest uniformity is obtained by allowing only deci- and deca- of the small-jump prefixes; centi- and hecto- are redundant with them. Let us also use the principle that the prefix with larger effect should go closer to the basic unit name, so in combinations the small-jump prefix always comes before the larger-jump prefix. Using deci- and deca- together with the power-of-10^3 prefixes allows us to reach any power of ten in the range 10^-25 (deciyocto-) to 10^+25 (decayotta-). A sample: 10^-1 deci-; 10^-2 decamilli-; 10^-3 milli-; 10^-4 decimilli-; 10^-5 decamicro-; 10^-6 micro-; 10^-7 decimicro-. There is a problem with combining the symbols. Using the existing prefix symbols, deciatto- (a combination we wish to allow) would have the symbol "da", but "da" is the existing symbol for deca-. Not allowing the meaning of "da" to change, we are left unable to retain both "d" for deci- and "a" for atto- in the combining context. Let us therefore change "a" to "t" for atto- (though "a" will still mean atto- when not combined with other prefixes). More on symbols: most of the prefix symbols follow a convention that negative powers of ten are indicated by lowercase letters and positive powers of ten by uppercase letters. When prefixes of both types are being combined, as we are now doing, the convention seems particularly useful, and so the few exceptions seem worth rectifying. Let us therefore change "k" to "K" for kilo-, as the IEC binary-based prefix system [BIN-PREFIX] also does, and "da" to "D" for deca-. The latter also removes the exception of the only two-letter prefix. Our prefix symbols are now readily constructed: from 10^-7 to 10^+7 they go "du", "u", "Du", "dm", "m", "Dm", "d", "", "D", "dK", "K", "DK", "dM", "M", "DM". ("u" here stands in for the Greek mu, as it commonly does in ASCII text.) Now let us consider extending the range covered by the prefix system. We want to support scale factors in excess of 10^100, which would require an inordinate number of prefixes if each power of 10^3 required its own prefix. We can avoid that problem by adding a third size of jump provided by prefixes. Suppose that we use the prefix letters "X" and "x" for the first prefix of this type: to avoid any redundancy of names, we want the prefix combinations "DY" and "dyX" (and also "dy" and "DYx") to be one power of ten apart. (One can also think about it with larger granularity: "Y" and "yX" (and "y" and "Yx") must be one power of 10^3 apart.) This means that "X" has the value 10^+51, and "x" 10^-51. Adding this pair of prefixes at once triples the range of powers supported by the prefix system. To extend the range further than this, there are two options. One is to assign more power-of-10^51 prefixes, just like the historical addition of more power-of-10^3 prefixes, increasing the range at a linear rate. The other is to assign prefixes for successively larger jumps, increasing the range at an exponential rate. Although the latter is theoretically appealing, it would result in unwieldy prefix combinations, requiring users to count in a form of balanced ternary notation with some mixed-radix complication in the power-of-10^3 position. Short prefixes are a strength of the current system, and that strength is best maintained by minimising the number of jump sizes represented among the prefixes. Currently-foreseeable needs can be met with a mere two pairs of power-of-10^51 prefixes. To leave a substantial margin beyond currently-foreseeable needs, these three power-of-10^51 prefix pairs are proposed: SYM NAME VALUE SYM NAME VALUE x xalto- 10^-51 X xalla- 10^+51 w wento- 10^-102 W wenna- 10^+102 v vardo- 10^-153 V varra- 10^+153 The symbol letters chosen are simply the next available in sequence following the "z"/"Z" and "y"/"Y" pairs. The names have no particular etymology, but have been invented wholesale to (a) match the chosen symbol letters, (b) follow the letter patterns established by recent prefixes, and (c) be distinctively different in pronunciation from each other and from all the existing prefixes (including in the form foreseeable when they are adapted for use as binary-based prefixes). The values that it was desired to express using the extended system: (a) Planck volume ~ 4.222*10^-105 st = 4.222 mwst (b) volume of observable universe ~ 1.70*10^81 st = 1.70 zWst (c) mass of electron neutrino < 4.5*10^-33 g = 4.5 Exg (d) mass of observable universe ~ 3*10^55 g = 3 DKXg 2. system ========= Note: this scheme is hypothetical and in no way a standard. Experimental use is encouraged. To avoid confusion, every communication that uses this scheme should contain an explanatory reference to this document, unless context or prior arrangement is such that use of the scheme is expected without explanation. Note: the prefix symbol letter "u" below stands for either the actual Latin letter "u" or the Greek letter lowercase mu. Either letter may be used to construct unit symbols. "u" is the recommended substitute for mu in contexts where Greek letters are unavailable, such as ASCII text. Table 0: small-jump prefixes: SYM NAME VALUE SYM NAME VALUE d deci- 10^-1 D deca- 10^+1 Table 1: medium-jump prefixes: SYM NAME VALUE SYM NAME VALUE m milli- 10^-3 K kilo- 10^+3 u micro- 10^-6 M mega- 10^+6 n nano- 10^-9 G giga- 10^+9 p pico- 10^-12 T tera- 10^+12 f femto- 10^-15 P peta- 10^+15 t atto- 10^-18 E exa- 10^+18 z zepto- 10^-21 Z zetta- 10^+21 y yocto- 10^-24 Y yotta- 10^+24 Table 2: large-jump prefixes: SYM NAME VALUE SYM NAME VALUE x xalto- 10^-51 X xalla- 10^+51 w wento- 10^-102 W wenna- 10^+102 v vardo- 10^-153 V varra- 10^+153 Table 3: non-composable prefixes: SYM NAME VALUE SYM NAME VALUE da deca- 10^+1 c centi- 10^-2 h hecto- 10^+2 k kilo- 10^+3 a atto- 10^-18 A unit symbol is made up in either composable-prefix form or non-composable-prefix form. In composable-prefix form the symbol consists of the concatenation of the following four elements in this order: 0. optionally, one small-jump prefix (from table 0) 1. optionally, one medium-jump prefix (from table 1) 2. optionally, one large-jump prefix (from table 2) 3. basic unit symbol In non-composable-prefix form the symbol consists of the concatenation of the following two elements in this order: 0. exactly one non-composable prefix (from table 3) 1. basic unit symbol In each case the unit name is built up in the same way, using the prefix name corresponding to each prefix symbol. In each case the value of the unit is the product of the values of its components. 3. references ============= [BIN-PREFIX] International Electrotechnical Commission, "Letter symbols to be used in electrical technology - Part 2: Telecommunications and electronics", IEC 60027-2, 2005-08-11. [PFX-HIST] metre.info, "Encyclopaedia of all prefixes", . [SI-PREFIX] Bureau International des Poids et Mesures, "SI prefixes", .