Talk:Preferred number/Archive 1

Resistor series

After calculating some numbers on my own, i must say some of these values seem messed up, for example in the capacitor / resistor E24 series, the 12th value is 30, but when calculating it (10*10^11/24), you get 28,7, rounded 29.

Are these values incorrectly defined by ISO / IEC, or are they just wrong here in the Wikipedia? --Abdull 15:23, 19 Jun 2005 (UTC)

I work in the industry and the numbers are right for E24 5% resistors. Your math is also right. Annoying isn't it? You should be able to re-create the series with a simple formula but you cannot. --Ee79 01:19, 13 October 2005 (UTC)

The 30 value is easily explained. The values in the E24 series *should* come from geometric averaging to fill values between the E12 series values. The new value between 27 and 33 from E12 should be (27*33)^^1/2) = 29.8, which is rounded to 30. That still leaves unexplained the mystery of where the "33" and "47" magic numbers came from. --75.41.34.231 18:59, 30 May 2007 (UTC)

It would be nice to have a link to a short computer program that explains the exact algorithm (including rounding strategy, etc.) that the numbers were generated with. I've once reverse engineered the algorithm behind the ISO 216 geometric series in a brief moment of procrastination. Markus Kuhn 12:45, 31 May 2007 (UTC)

I've reverse engineered one way to get the numbers (given a few "magic values" from E6).

Renard numbers are rounded results of the formula

$R(i,b)=10^{\frac {i}{b}}$

One consequence of this is that, for any 3 consecutive values a b c in any Renard series, the middle value c is the geometric average of its neighbors: $b={\sqrt {a*c}}$.

Start with a "R6" series, rounded to 2 decimal places:

 10 15 22 32 46 68 100.

Nudge 2 numbers (why? perhaps a typo in the original standard?) to match the E6 series

 10 15 22 33 47 68 100 (where does the "33" and "47" magic numbers come from?).

Then the E12 series can be derived from E6. Fill in the gaps between each 2 consecutive values a and c with a new middle value, rounded to 2 decimal digits: $b=\operatorname {round} ({\sqrt {a*c}})$. The E24 series can likewise be generated by filling in the gaps between each value of the E12 series with the (rounded) geometric average.

I see the E48 series does *not* come from filling in the gaps in the E24 series. The E48 is easily calculated (without reference to the E6 magic numbers) by the Renard formula rounded to 3 decimal digits

$E48(i)=\operatorname {round} (100*R(i,48))=\operatorname {round} (100*10^{\frac {n}{48}})$

--68.0.120.35 04:43, 11 June 2007 (UTC)

The above explanation by 68.0.120.35 seems to suggest that six fixed points are needed to begin with. However, would the algorithm really need that many fixed points? The graph of E₂₄(i)-10*10^i/24 versus E₂₄(i) looks like a piecewise straight line with only three joints: E=10, E=33, and E=56. It turns out that the following formula provides the actual E24 values:

E₂₄(i) = Round(A * B^i/24) where

if 10<E<33 then A=10, B=10.9

if 33<E<56 then A=33, B= 8.9

if 56<E<100 then A=56, B=10.2

As 10, 33, and 56 belong to the E4-series, I would guess that an early manufacturer of resistors provided 4 values per decade: 10, 18, 33, and 56. The IEC63 commission chose to maintain these established values in the new series E6, E12 and E24, and interpolated the rest.

This guess would imply that E24, which contains multiple deviations from 10*10^i/24, is derived from E4, which contains just a single deviation, being 33. Ceinturion 13:13, 25 August 2007 (UTC)

The E192 series contains another interesting deviation from the formula Round(100*10^i/192). E₁₉₂(185) is actually 920 instead of 919 from the formula. The reason is that E₁₉₂(185) apparently has been defined as the geometric mean of its neighbors. In this case the mean is different from the formula, due to rounding. Ceinturion 14:13, 25 August 2007 (UTC)

IEC 63 versus IEC 60063

According to one source, The E-series of preferred numbers are standardized in IEC 63. I'm not sure if the source is accurate, or of the standard has been updated and got a new number. http://www.elfa.se/se/fakta.pdf, page 43. --HelgeStenstrom 12:20, 20 November 2006 (UTC)

The big +60000 renumbering of IEC standards in 1997 is explained in the article List of IEC standards. Markus Kuhn 12:50, 31 May 2007 (UTC)

Powers of ten versus divisibility by three

An anonymous contributor added at the end of the "Buildings" section first

However, since 3 and 6 are not divisors of any power of ten, that is, since their multiples never coincide with numbers of the kind 10ⁿ (the numbers that were chosen as the basis of the metric system due to our using a decimal numeral system), no integer multiple of 300 mm or 600 mm ever results in a round metric unit. This has the practical inconvenience that, for example, one cannot evenly divide 1 metre, 10 metres, 100 metres nor 1 kilometre into 300 mm or 600 mm modules, nor into modules that are an integer multiple of those (such as 1200 mm). So one has to choose either to go with round metric units (1 m, 1 km, etc.) with the practical drawback of their relatively poor divisibility, or else to go with highly-divisible 300 mm or 600 mm modules which do not really fit into a system of units based on powers of ten.

and later a shortened version

However, 3 and 6 are not divisors of any power of ten, which means no integer multiple of 300 mm or 600 mm ever results in a round metric unit, so one cannot evenly divide 1 metre, 10 metres or 100 metres into 300 mm or 600 mm modules.

This is clearly an off-topic remark in this section, because neither of these three lengths was chosen from the modular coordination scheme described in that section. All that remark says is "If you do not use this technique, you may not achieve its benefits", which sounds not remarkable to me. I am therefore removing it again.

In addition, I would like to point out that how "round" a number is depends on how many digits (other than leading or trailing zeros) are needed to write it down. In that sense (and I don't know any other), 1.2 meters or 24 meters are all wonderfully "round" metric dimensions, simply because they can conveniently be expressed with merely two digits! For comparison, 1.609344 km or 0.45359237 kg are not nice round metric dimensions, because they have lots of digits.

Uh, no. A number is round if it looks as though it may have been rounded. That's why they call it that. Whether 1.2 m is as "round" as 24 m depends on whether you think people are equally likely to round to the nearest 10 cm as to the nearest meter. --Doradus 02:38, 5 June 2007 (UTC)

The comment is reminiscent of a logical fallacy that I have seen applied before as a discussion tactic by anti-metrication campaigners. The goal is to create the impression that there is something fundamentally forbidden about using a metric unit to chose a quantity that can easily be divided by three. This is of course nonsense and has nothing to do with common sense, well-established industry practice, and formal standards, with the latter two being the focus of this article.

The SI prefixes were chosen as powers of ten because that is the only choice that ensures that any mental arithmetic needed for converting between different units remains trivial, namely reduced to shifting the decimal dot. This is entirely independent of the question of divisibility by any factor other than the base of the number system that is normally used to write numbers (which is typically 10 for contemporary humans). If you want a dimension easily divisible by 3, then just make it divisible by 3. If you want it easily divisible by 4, 5, 6, 7, then make it so. This has nothing to do with the unit used. If you want a length in yards to be divisible by 3 or 6, then chose a length that is a multiple of these factors, say 1800 yards. Relying on a larger unit to automatically insert a factor of three is not common practice anywhere; for example 1 mile (1760 yards) is not divisible by 3 or 6 into an integer number of yards either: 1/3 mile = 586.666... yards. Does this make the yard and mile unsuitable to express lengths whose third can be expressed as a nice round number? Of course not. A third of 1.2 miles (2112 yards) is 704 yards. (I'm sure you can quickly find even rounder examples.) Markus Kuhn 16:12, 3 June 2007 (UTC)

No, you are completely misrepresenting the point. The numbers 300 and 600 are decimal-incompatible, because they contain prime factors not in common with those of ten. It is not a matter of how many significant digits one uses, it is because of their very mathematical nature: numbers like 300, 600, 12, 24, 60, etc. are simply not in line with decimality. Choosing them instead of decimal-compatible numbers like 200, 250 or 500 is akin to choosing 240 pence to the pound instead of 100, 12 inches to the foot instead of 10, 24 hours to the day instead of 20 or 10, and 60 minutes to the hour instead of 100, and in fact the reason why 300 mm is a preferred measure for building blocks is exactly the same why 240, 12, 24 and 60 were chosen for those other measures: their high divisibility, at the cost of their non-decimality. Another example: what would you think if they minted 30-cent coins instead of 25-cent coins? No matter how many 30-cent coins you used, you would never be able to get a round dollar quantity with only 30-cent coins; but according to what you have just argued the choice of number is not related to the number base and one is supposedly free to choose whatever number one sees fit disregarding the decimality of the measuring unit, so it wouldn't matter to have a preferred number of 30 cents instead of 25 cents so as to have more divisibility, right? Most people obviously do not think like you.

Decimality is not just about how one represents a number, it implies choosing and preferring certain numbers over others; in particular, in implies preferring numbers whose prime factorization only includes 2 and/or 5, and not 3 or 7 or 11. In decimal the rational number 1⁄3 is all to be avoided, because it is non-representable (one cannot write, store nor operate with an infinite amount of digits, so 1⁄3 has to be rounded to the nearest decimal number, which introduces inexactitude unlike when dealing with 1⁄2 or 1⁄5); same with 1⁄6, 1⁄7, etc. Similarly, multiples of 3 or 6 never result in a round decimal number like 100 or 1000, which means they are unfit for a system based around powers of ten. It is not "a logical fallacy" to say that 300 is not in line with a decimal-based system, it is a mathematical fact because of the prime factorization of 300 which is incompatible with that of ten. The numbers 300 and 600 cannot be preferred numbers in a truly decimal system; the decimal-preferred numbers in that range are 200, 250 and 500 (and to a lesser degree, 400 and 625). Going with your reasoning that the choice of number base supposedly does not imply preference for certain numbers, then there would be nothing "un-decimal" with having a day of 24 hours and an hour of 60 minutes. But certainly that's not what the pro-decimal people say, who are all oposed to having to deal with such non-decimal-compatible quantities as 24 and 60 and want to replace them with a day of 20 or 10 hours of 100 minutes each.

Even though decimal metric system measures like the millimetre have been adopted nominally, the fact is that in practice they are frequently used in non-decimal ways to cope with the problem of their poor divisibility, and the choice of these preferred numbers 300 and 600 is a prime example of this. The fact that in the construction industry they have chosen to go with such non-decimal-compatible numbers, instead of with measures like 250 or 500 mm that would be in line with the decimality of the metric system, is a de facto acknowledgement of the limitations of a system of measures based on ten and of its inadequacy to the many practical purposes which require high divisibility. Going decimal comes at a cost, and that cost is poor divisibility. Choosing preferred numbers like 300 with metric units is defeating the purpose of the decimality of the system: either you go truly decimal and put up with the poor divisibility that comes with number ten, or you leave aside decimality in order to get high divisibility; you cannot have both.

Besides, regarding "The SI prefixes were chosen as powers of ten because that is the only choice that ensures that any mental arithmetic needed for converting between different units remains trivial, namely reduced to shifting the decimal dot", sorry but the same kind of trivial conversion by shifting the dot could have been achieved just as easily by switching to another number base instead of switching to decimal prefixes. The shifting-the-dot thing is certainly not a magical property of number ten and works the same for multiplying/dividing by the radix in whatever radix one chooses. Also, it is a highly debatable issue that trivial unit conversion should have been the primary concern and number one design feature of the system, especially when it was achieved at the cost of divisibility which for many practical purposes is far more important than unit conversion. Moreover, not every human culture has been decimal (even in European cultures, the remnants of former vigesimal systems are still visible in several of the modern languages and as late as by the end of the 18th century it was still common to count by the score in English, as in the famous opening of the Gettysburg Address). There is no reason why it would not have been possible to adapt the number system to a highly-divisible radix to harmonize it with the kind of high divisibility that for practical reasons was common with traditional measures, instead of adapting the system of measures to decimal-based numbers and having to cope with their poor divisibility. 213.37.6.106 23:47, 4 June 2007 (UTC)

It's very difficult to follow what you've just said due to your lack of paragraphs. However, I think the point Markus was making is that these problems (lack of divisiblity by 3 as the prominent example) apply equally to imperial units. 1 mile does not divide by 3 into an integer number of yards. Nor a gallon into pints, nor a pint into fluid oz, nor a ton into stones, nor a stone into pounds, nor a pound into oz. I could go on. Examples that actually work are fairly rare in the imperial system. Therefore the "divide by 3 is easier in imperial" is a fallacy, that was his point. About the only good counterexample is 60 minutes in an hour, which of course divides by 2, 3, 4, 5, 6, 10, 12, 15, and 20.

And whilst it's true that the "shifting the dot" property occurs in any number base, it's a fact that as a vast majority, the human race expresses numerical quantities in base 10; not base 3, 6 or 12, regardless of whether we're talking about imperial or metric. Therefore, the only relevant base for "shifting the dot" is base 10. Oli Filth 01:06, 5 June 2007 (UTC)

There, divided into paragraphs. But who on earth claimed or talked about "dividing by 3 is easier in imperial"? Not me, that's for sure (although it is certainly easier in some particular cases like dividing 1 foot or 1 troy ounce by 3, while it is never easy to divide decimal metric system units by 3). Why do you assume that pointing out the practical inconveniences of the decimality of the metric system implies support for the imperial system? Do you think the alternatives are reduced to going with a system based on ten or else going with a system not based on any particular radix? Have you considered that it would be perfectly possible to base a metric-like system on some number other than ten? Particularly, on some highly composite number like six, twelve or sixty, so as to have the benefit of easy conversion between units that comes with a system of measures coherently based on one radix, while avoiding the practical shortcoming of low divisibility that inevitably comes with the choice of ten as a radix. There exist at least two independent proposals for metric-like systems coherently based on twelve instead of ten (if interested, you can have a look at them: here for the Universal Unit System proposed by Takashi Suga and here for the TGM system proposed by Tom Pendlebury).

What I talked about is not that "imperial is better", what I talked about is that choosing a number like 300 for a module unit is not in line with choosing a decimal-based system of measures, just like choosing a number like 3600 for the number of seconds in an hour is not in line with a decimal system of time measurement. One can perfectly choose to go decimal, but then one should be coherent with that choice and put up with its all practical consequences, both the good ones and the bad ones. That simple. A decimal-based system mandates a preference for numbers like 10, 25, 50, 100, 125, 200, 250, 400, 500, 625, 1000 and the like, which happen not to be precisely very convenient when divisibility is important, but that's a consequence of choosing to go decimal. If one instead selects other, non-decimal-compatible quantities like 300 and 600 as preferred numbers to take advantage of their higher divisibility, then one is de facto renouncing to the decimality of the system. With a 300 mm module what you actually have is a length unit divided into 300 subunits, which clearly is not a decimal-based subdivision of that unit, just like a day divided into 86400 seconds is not decimally subdivided. The whole purpose of basing the metric system on powers of ten was to make unit conversion trivial since if numbers are expressed in the decimal numeral system, then if you have 1375 millimetres you know just by shifting the dot that it corresponds to 1.375 metres or to 137.5 centimetres. But if you choose to base your construction block lengths on 300-millimetre modules so as to have better divisibility of that module unit, you have to make non-trivial calculations to know how many of those units correspond to 1375 millimetres (four 300-mm unit-modules and seven 25-mm twelfth-submodules, incidentally). And there is a high chance (2 in 3) that you will get infinite recurring decimals while doing conversions with a 300-mm unit (1375 mm ÷ 300 mm/module = 4.583333333... modules), whereas you never get recurring digits when doing calculations with decimal-compatible numbers like 250. Therefore by choosing 300 mm as the preferred number for building modules one is defeating the purpose of using a decimal-based system of measures. 213.37.6.106 10:08, 5 June 2007 (UTC)

My mistake, you didn't claim that "imperial is better" (it's just often, "decimal has flaws" and "imperial is better" come as a pair).

I'm indeed aware that one could construct "metric" systems in other number bases. The problem is, of course, that for them to be beneficial requires one to use non-decimal notation (in the cases you cited, duodecimal notation). Whilst some people are used to such a concept (I work in hexadecimal and binary every day), there's no way that the population as a whole could ever be persuaded to adopt such a system. So even if it became commonplace amongst scientists, there would be immense headaches when it came to converting back to "lay" notation.

The argument for the adopting (for example) duodecimal notation because it makes working with divison easier (in some circumstances) is like arguing for widespread adoption of hexadecimal because it makes working with computers easier (in some circumstances). Both arguments have their merits, and would make certain groups of people's lives easier, but both are completely impractical. Oli Filth 18:46, 5 June 2007 (UTC)

On the one hand, the advantages of duodecimal are manifold and are not reduced to making division easier. They are more far-reaching and comprehensive than that, from the pedagogic aspect of making numbers and mathematics more interesting to young children (as Alexander Aitken put it: "The duodecimal tables are easy to master, easier than the decimal ones; and in elementary teaching they would be so much more interesting, since young children would find more fascinating things to do with twelve rods or blocks than with ten. Anyone having these tables at command will do these calculations more than one-and-a-half times as fast in the duodecimal scale as in the decimal." [...] "I come to the conclusion that the efficiency of the decimal system might be rated at about 65 or less, if we assign 100 to the duodecimal."), to facilitating commerce by making packaging units like the 6-packs, the dozens and their multiples and divisors (which are commonplace because of their inherent geometric and divisibility properties) in line with the numeral system, or to improving research in mathematical fields like number theory since many patterns become more obvious in duodecimal, etc. By comparison, switching to hexadecimal would make little sense, since mathematically it offers very few advantages when compared to decimal (sixteen has only one prime factor, which means recurring digits are even more common with hexadecimal than with decimal) and almost none when compared to duodecimal. Computers are meant to be an aid to humans and should adapt to our needs, and not the other way round. Moreover, switching to duodecimal would in fact make dealing with hexadecimal easier than it is with decimal, because duodecimal is more compatible with hex than decimal since twelve is divisible by four while ten is not. Sixteen is a divisor of the second duodecimal power (one gross or 144), whereas ten thousand is the lowest power of ten of which sixteen is a divisor. Fraction 1⁄16 (hex 0.1) corresponds to duodecimal 0.09 (144 ÷ 16 = 9) but to decimal 0.0625 (10000 ÷ 16 = 625), and fraction 1⁄256 (hex 0.01) corresponds to duodecimal 0.0069 (four significant digits) but to decimal 0.00390625 (eight significant digits). Repeated division of the unit by two quickly leads to a long chain of decimals, with one more significant digit needed for every iteration (0.5, 0.25, 0.125, 0.0625, 0.03125, 0.015625, 0.0078125, 0.00390625, etc.), while in duodecimal it only adds one more significant digit every two iterations (0.6, 0.3, 0.16, 0.09, 0.046, 0.023, 0.0116, 0.0069, etc.), thus making dyadic fractions easier to handle in duodecimal (although of course a binary-like base like hexadecimal is the optimal choice for this one particular purpose).

On the other hand, you say it would be impractical to switch to duodecimal, but there was the exact same problem with adopting the originally completely new and alien metric system. People were used to their non-decimal highly-divisible measures and posed a great resistance against converting to metric, a change which offered few if any practical advantages for their everyday needs in commerce. In the very France where the metric system was first introduced, the opposition to metric from the part of the lay people was so strong and they felt so uncomfortable with it that not long after its introduction Napoleon (who was himself not very content with the disadvantages of having gone decimal) introduced the système usuel which permitted non-decimal subdivisions of metric units (e.g., a pied usuel that was 1⁄3 of a metre or 33.33333... cm, and a pouce usuelle which was 1⁄12 of that or 2.77777... cm), so as to let people have what they considered more appropriate for their common everyday needs. Switching to the metric system has always been an imposition by law from the part of government bureaucrats; it was never the people who asked for it to be pushed down their throats. They just had to adapt to the change overnight without having a say, since not once a referendum was held anywhere regarding an impending metrication or decimalization plan. People finally adapted to metric, of course, because they had to, just as people adapted to the decimalization of the pound or to the recent change of currency in the European Union (which cost hundreds of millions of dollars, the phasing out of all previous circulating money in half the continent, and the adaptation of all vending machines and commercial and financial software to the new currency, including the introduction and international standardization of a new typographic symbol to computer fonts, which had to be updated worldwide to support the new character, as well as ubiquitous government- and corporations-sponsored campaigns to educate some two hundred million Europeans in their new currency). Similarly, people would adapt to another numeral system if they had to. In all probability it wouldn't be an easy step to take, especially for the economic costs of adapting existing devices (comparable to those of adapting devices to the euro) and the need to reeducate people to think in terms of dozens instead of in terms of tens (comparable to reeducating them to think in terms of metric measures instead of in terms of traditional measures); but just as it wasn't precisely easy to make any other similar changes like metrication, the decimalization of the pound or the switch to the euro, the enormous costs and efforts of which were assumed because of their perceived long-time benefits. In fact, the switch to a different number base, and in particular the switch from decimal to duodecimal, is an already documented phenomenon: the speakers of the Nimbia dialect of Gwandara, in the Nigerian Middle Belt, first switched from decimal to duodecimal long ago (we know they switched because all other dialects and languages belonging to the Chadic family, spoken in far lying areas, are decimal and because of the etymology of the duodecimal name for eleven as "one more"). The switch happened because duodecimal was the common number base in their surrounding cultures (yes, in that particular area duodecimal and not decimal was the cultural standard, just as in ancient Mesoamerican cultures the standard number base was vigesimal). Recently, the languages in that traditionally duodecimal area have begun to switch to decimal because of the powerful influence from the Western world and the introduction of our alphabet and our Hindu-Arabic decimal numerals, and according to reports people haven't felt the change of number base as something terribly traumatic (they put it in words of the kind "we used to count in the old numbers, now we count in the new numbers"). Anyway, I've started to drift far into off-topic. 213.37.6.106 10:53, 6 June 2007 (UTC)

Sorry, 213.37.6.106's point about 125 being "preferred with the decimal system", but 600 not being so, simply suggests that (s)he has not read the article. Both are surely perfectly valid decimal numbers, and 600 is a much rounder decimal number than 125, because it has only a single-digit mantissa. There are a range of different schemes in use to chose preferred numbers with the decimal system, and this article simply tries to introduce the most important ones of these. Some of these preferred-number schemes include 125, others include 600, and both have their pros and cons. That's why there are several different preferred-number schemes, from which you can pick the one that is best suited for your particular application needs. Changing to a base-12 system would not change the range of different and incompatible application needs in any way. For example, sqrt(2) (a very convenient paper-size aspect ratio) would still not be a very round number in a base-12 number system. Likewise, it would be difficult in a base-12 system to refine a nice set of preferred wine-bottle sizes that are related to each other by simple factors while remaining compatible with the most commonly used existing sizes.

Your lengthy argument is primarily an attempt to construct a problem with the decimal system that I do not think is a big problem, and certainly not one that could be fixed by changing the base. In any case, should we ever have any opportunity to change the choice of main number base used by humanity (extremely unlikely), there can in my view not be the slightest doubt that it will be a power of two, and certainly never 12 or 60. Digital technology has become just too important to even consider anything else, and the rounding errors introduced by conversion between decimal and binary numbers can be quite a pain in practice. Neither 1/10 nor 1/12 or 1/60 have a finite representation as a binary number! If anything, we should remove factors from our number base, rather than add more. But that is all utopic and well outside the scope of this article anyway, which merely introduces and explains existing practice that has been standardized and widely used for more than 1/3 century. Markus Kuhn 11:13, 6 June 2007 (UTC)

Rather than me having not read the article, it seems it's you who haven't read anything of what I have said. Sure 600 has a single-digit mantissa, but that's irrelevant as to its decimal-compatibility. Looking "round" doesn't mean it is convenient or compatible to use with a decimal-based system. The prime factorization contains 3, a prime factor not in common with 10, which means 600 is not a factor of any power of ten, while 125 is exactly 1⁄8 of 1000. With eight 125-mm modules you have already gotten to a round decimal power, i.e. you can cleanly divide 1 metre into eight 125-mm modules, whereas no matter how many 600-mm modules you use, you'll never reach a round decimal power, that is, you'll never reach a higher metric system unit, and unit conversion using 600-mm units completely defeats the purpose of basing a system on decimal powers as I have already explained. Dividing by 600 results in the nuisance of infinite recurring decimals unless the dividend happens to be divisible by 3 (of which there is a chance of only 1 in 3), while this never happens when dividing by a decimal-compatible number like 125 no matter what the divisibility of the dividend. Why do you think coins are minted in 25-cents instead of 30-cents, which is a "rounder" number because it has a single-digit mantissa? 213.37.6.106 15:20, 6 June 2007 (UTC)

If you read the article, you will have learned that most currencies follow a 1-2-5 series, and that the 1-2.5-5 series or the U.S. quarter is very much the exception rather than the rule. Markus Kuhn 16:09, 6 June 2007 (UTC)

So? If you re-read my comment you'll see I was comparing 25 to 30 regarding the fact that 30 is not a decimal-compatible number unlike 25, because it contains the prime factor 3 which is not in common with those of ten, and I did also mention that 20 and 50 are decimal-compatible numbers like 25 and unlike 30. Both the 1-2-5 (10-20-50) and the 1-2.5-5 (10-25-50) series are decimal-compatible, and everyone is familiar with them, no need to read the article to learn about their existence. Whereas the 300 and 600 modules are not decimal-compatible just like the 60-minute hour or the 24-hour day. 213.37.6.106 18:26, 6 June 2007 (UTC)

Why do you think people give more prominence to 25th anniversaries (quarter-of-a-century anniversaries) than to 30th anniversaries? I'm not going to repeat myself to explain again all the details of why 600 and 300 are not compatible with decimal, read my previous comments. And my "lengthy argument" is not "an attempt to construct a problem with the decimal system"; the problem with prime factor 3 is mathematically inherent in decimal, it's not a matter of personal opinion or personal perceptions, and certain I haven't been the first one to point out this obvious and well-known limitation of decimal. Ten is not divisible by 3 and that's it, it's a mathematical fact, not an opinion. Besides, you say you don't see a problem with the non-divisibility of the powers of ten by 3, but then why do you think the people in the building industry have chosen 300 mm as a preferred number instead of a power of ten in the first place? They did it precisely to have a module that is divisible by 3 (and 6, 12, etc.), because divisibility by 3 is very useful for many practical purposes. 213.37.6.106 15:20, 6 June 2007 (UTC)

There are very different criteria according to which units and modules are chosen, and I suspect you do not yet appreciate the difference between a unit and a module and think of them as the same thing. For a module, having a high number of prime factors is beneficial. For a unit, it is utterly irrelevant (unless your brain is somehow culturally locked into deeply believing that units and modules must be the same thing). Markus Kuhn 16:09, 6 June 2007 (UTC)

Please, don't add "citation needed" tags to comments in a talk page, will you? I could then start to add that tag to comments of yours such as the one below regarding the supposed lack of interest of factors 3 and 5, or to your comment here that divisibility is supposedly "utterly irrelevant" for a unit (if so, why on earth a good deal of the traditional measures all around the world have been based on highly divisible numbers like twelve and not on poorly divisible ten?). So you want some reference for the prominence of 25th anniversaries over 30th ones? The 25th anniversary is called Silver Jubilee while the 30th anniversary has not been given a comparably prominent name. And no, the notion that a module is a unit of measurement is definitely not the product of "not yet appreciating their difference" or being "culturally locked into some belief" (comments which I find patronizing and rather insulting, so I would ask you not to go on with that kind of remarks). Check the dictionary before making such assertions: module (from the diminutive of Latin modus "measure") 1. A standard or unit of measurement 2. Architecture The dimensions of a structural component, such as the base of a column, used as a unit of measurement or standard for determining the proportions of the rest of the construction. 3. A standardized, often interchangeable component of a system or construction that is designed for easy assembly or flexible use: a sofa consisting of two end modules. (from the American Heritage). 213.37.6.106 18:26, 6 June 2007 (UTC)

On the other hand, regarding duodecimal vs. hexadecimal, you have not presented any respectable argument to support the convenience of a power-of-two base over 12 or 60. The computer thing is completely irrelevant, because computers are meant to serve us and not the other way round, so it's them who should adapt to our requirements and not us who should renounce to our practical needs in order to adapt to the many limitations of computers. Power-of-two bases not only do not offer significant advantages over decimal (except for the very particular case of representing dyadic fractions), they are even worse and no serious mathematician would attempt make a case for them to be preferrable, while there is a wide consensus that duodecimal and sexagesimal are mathematically superior to decimal, because 12 and 60 are superior highly composite numbers, that is, they represent relative maximums in divisibility, which means no other number of their size has as many factors, plus their divisibility includes all three of the most elemental fractions, namely 1⁄2, 1⁄3 and 1⁄4 (sexagesimal adds 1⁄5 and both include 1⁄6). This makes them the optimal choices for a radix of their size. Binary, for its part, is used in computers only because it facilitates the implementation of its arithmetic in electronic circuitry, but not for any other good reason and certainly not because it offers any significant mathematical advantage over decimal, let alone over duodecimal or sexagesimal. Hexadecimal is good as a shorthand way of representing binary for human readability, since binary has a very simple arithmetic but is a complete nuisance for humans because of the overly lengthy representations of even small quantities that make errors in writing down, calculating or remembering binary numbers all too easy. But since hex is only compatible with powers of two, its multiplication tables are harder to master than the decimal ones, displaying less obvious patterns (just have a look at them). Thus, hexadecimal not only does not aid in multiplication and division compared to decimal, it makes them ever harder; whereas, as the great mathematician and mental calculator Alexander Aitken pointed out based on his own experience, duodecimal multiplication tables are easier to memorize (they display more obvious patterns) and arithmetic is about one and a half times faster for humans to perform in duodecimal than in decimal. Switching to octal or hexadecimal would be a completely backwards move which would introduce even more practical problems than we have with decimal. Not only would it not solve the practical inconvenience of the non-divisibility by 3 and by all its multiples, it would further reduce divisibility to a minimum by leaving only divisibility by 2 and its powers, while with decimal we already have that plus the bonus of divisibility by 5. All rational numbers that are finitely representable in binary or hexadecimal are finitely representable in decimal and in duodecimal, while only a fraction of those finitely representable in the latter ones are finitely representable in the power-of-two bases. That is, the set of finitely representable numbers in binary, octal and hexadecimal is only a subset of those finitely representable in both decimal and duodecimal, and having a bigger set of finitely representable numbers is a big practical benefit of both decimal and duodecimal over the power-of-two bases because it reduces the nuisance of having to deal with recurring digits. If you really want to avoid the rounding errors that occur when doing calculations with decimal numbers in computers because of this limitation and disadvantage of their binary representation, just use BCD (binary-coded decimal) which is what they do for well-designed financial software. Plus we don't know what computers will be like in the future, so it would be crazy to surrender the practical benefits of better divisibility just because current computer technology uses the inferior binary. 213.37.6.106 15:20, 6 June 2007 (UTC)

Sorry, I have already made my case and will not repeat myself. This is anyway getting far too off-topic from the subject of improving the "Preferred number" article, which is what this page discussion is about. For the advantages of base-2 arithmetic in electronic calculations, I can only refer to any introductory computer architecture textbook. BCD etc. are workable hacks, but I doubt we would ever have chosen the base-10 Hindu-Arabic numeral system if we had known at the time of Fibonacci already about digital electronics. (Octal numbers would also have been compatible with the fingers of your hand and leave the thumbs as carry bits. :-) Neither the factor 3 nor the factor 5 are particularly interesting, and one is certainly not more interesting than the other IMHO. If you like superior highly composite numbers, note that 2 is the smallest of these! Markus Kuhn 16:09, 6 June 2007 (UTC)

Actually you haven't made your case, since you haven't provided counter-arguments nor even commented on my detailed argumentation showing how choosing 300 and 600 as preferred numbers defeats the purpose of using a system of measures based on powers of ten. Besides, your assertion that factors 3 and 5 are "not particularly interesting" is not supported in any way neither by mathematical theory where 3 is ubiquitous in for example the most elemental geometry (just think of the triangle), with 5 being also important among other things for its relation to the golden ratio, nor by practical facts such as the extensive use of 3 and its multiples in commerce (packaging, etc.) as well as in traditional, practical-minded measures. Also, factor 3 is more elemental than factor 5, so they are not equal in interest and 3 should be given preference to 5 when only one of them can be chosen (having factor 5 is very nice once factors 3 and 4 have been covered, as in sexagesimal, just as factor 3 is very nice only once factor 2 has already been covered, but unfortunately sexagesimal has the practical inconvenience of being too large for comfortable use by humans). Besides, I know 2 is the lowest of the superior highly composites, which is why binary is the best choice only if you must use an extremely small radix, as in the case of implementing arithmetic in electronic circuitry, and certainly binary is better than similarly-small-sized ternary (which has the important drawback of not supporting the most elemental fraction of all, i.e. 1⁄2, and consequently neither 1⁄4 nor the other dyadic fractions). But humans fortunately do not have the limitations of computers and work comfortably with a radix of a size around ten or twelve. I have already commented on the many drawbacks of binary for human use, and in any case binary is not superior to duodecimal but the other way round, because it offers less divisibility with its only one distinct prime factor. Even senary is superior to binary and octal. 213.37.6.106 17:42, 6 June 2007 (UTC)

dozenal preferred numbers

Getting back to the article topic of "preferred numbers": Imagine a hypothetical world where we all converted to dozenal. What would be the preferred series of numbers? It would be nice if:

• the ratio of consecutive values is approximately equal (equally spaced on logarithmic scale)
• doesn't include "too many" values between repeats
• preferred values are small integers
• repeats every power of 12

(Yes, this is speculation and original research, so it's not appropriate for the Wikipedia article. Is there a better wiki for this?)

(Alternately, we could replace the last criteria with "exactly 12 values before it repeats", use the E12 scale and twelve-tone equal temperament).

Analogous to the Renard numbers:

$R_{dozenal}(i,b)=12^{\frac {i}{b}}$

$R_{dozenal}(i,7)=12^{\frac {i}{7}}$, rounded to the nearest integer (in decimal), gives:

 1
 1.4 ( sqrt(2) )
 2
 3
 4
 6
 8
 12 (1 dozen)
 17 (1 and a half dozen - 1; sqrt(2) * 1 dozen)
 24 (2 dozen)
 35 (3 dozen - 1)
 50 (4 dozen + 2)
 71 (6 dozen - 1)
 101 (8 and a half dozen - 1)
 144 (one gross).
 205 (one and a half gross - 11; sqrt(2) * 1 gross + 1).

This has the advantage of being obvious how to extend to higher precision -- go to $R_{dozenal}(i,14),R_{dozenal}(i,28),$ etc.

Alternating multiplication by 3/2 and 4/3:

 1
 1.5 (1 and a half)
 2
 3
 4
 6
 8
 12 (1 dozen)
 18 (1 and a half dozen)
 24 (2 dozen)
 36 (3 dozen)
 48 (4 dozen)
 72 (6 dozen)
 96 (8 dozen)
 144 (1 gross).
 216 (1 and a half gross).

This has the advantage of always being rounded to exactly 1 dozenal digit of precision (except for the "one and a half"). But how would one extend it to higher precision? --68.0.120.35 04:43, 11 June 2007 (UTC)

I'm not sure how you arrived at some of those values. Anyway, the Renard series for the decimal metric system are based on the powers of the fifth, tenth, twentieth, etc. roots of ten:

$({\sqrt[{10}]{10}})^{0}$ = $({\sqrt[{5}]{10}})^{0}$ = $({\sqrt[{2}]{10}})^{0}$ = 1

$({\sqrt[{10}]{10}})^{1}$ = 1.25892541179416728...

$({\sqrt[{10}]{10}})^{2}$ = $({\sqrt[{5}]{10}})^{1}$ = 1.58489319246111366...

$({\sqrt[{10}]{10}})^{3}$ = 1.99526231496887995...

$({\sqrt[{10}]{10}})^{4}$ = $({\sqrt[{5}]{10}})^{2}$ = 2.51188643150958068...

$({\sqrt[{10}]{10}})^{5}$ = $({\sqrt[{2}]{10}})^{1}$ = 3.16227766016838041...

$({\sqrt[{10}]{10}})^{6}$ = $({\sqrt[{5}]{10}})^{3}$ = 3.98107170553497402...

$({\sqrt[{10}]{10}})^{7}$ = 5.01187233627272466...

$({\sqrt[{10}]{10}})^{8}$ = $({\sqrt[{5}]{10}})^{5}$ = 6.30957344480193535...

$({\sqrt[{10}]{10}})^{9}$ = 7.94328234724281934...

$({\sqrt[{10}]{10}})^{10}$ = $({\sqrt[{5}]{10}})^{5}$ = $({\sqrt[{2}]{10}})^{2}$ = 10

The intermediate values for the powers of the 20th-root of ten are:

1.122018454301963342...

1.412537544622753938...

1.778279410038922093...

2.238721138568338098...

2.818382931264451496...

3.548133892335751494...

4.466835921509626495...

5.623413251903484067...

7.079457843841368891...

8.912509381337441638...

Those are irrational numbers (they are surds), and as such have no terminating representation, so they have to be rounded for practical purposes. There is no uniform rounding criterion for the values given for the Renard series in the article (most are rounded to the nearest tenth, but others to the nearest hundredth). Rounding consistently to the nearest hundredth, we get the logarithmic series:

dec R1   1.00                                                                                                10.00
dec R2   1.00                                              3.16                                              10.00
dec R5   1.00                1.58                2.51                3.98                6.31                10.00
dec R10  1.00      1.26      1.58      2.00      2.51      3.16      3.98      5.01      6.31      7.94      10.00
dec R20  1.00 1.12 1.26 1.41 1.58 1.78 2.00 2.24 2.51 2.82 3.16 3.55 3.98 4.47 5.01 5.62 6.31 7.08 7.94 8.91 10.00
etc.

There is no series (compatible with the decimal R10 series) that offers two intermediate values (nor three or four), which is why —as the article points out— if the decimal R5 series provides too fine a division (and the decimal R2 series too gross a division), one has to resort to some other scheme outside these Renard series, such as the 1-2-5-10 series. This is a consequence of the reduced divisibility of ten.

Now, for the dozenal equivalent of the Renard logarithmic series we have to look at the twelfth root of twelve, which is 1.230075505577971295...₁₀ (or 1.2916A19411A8586...₁₂ if expressed in dozenal), and look at the powers and roots of this twelfth root:

$({\sqrt[{12}]{12}})_{10}^{0}$ = $({\sqrt[{6}]{12}})_{10}^{0}$ = $({\sqrt[{4}]{12}})_{10}^{0}$ = $({\sqrt[{3}]{12}})_{10}^{0}$ = $({\sqrt[{2}]{12}})_{10}^{0}$ = 1

$({\sqrt[{12}]{12}})_{10}^{1}$ = 1.230075505577971295...₁₀

$({\sqrt[{12}]{12}})_{10}^{2}$ = $({\sqrt[{6}]{12}})_{10}^{1}$ = 1.513085749422901691...₁₀

$({\sqrt[{12}]{12}})_{10}^{3}$ = $({\sqrt[{4}]{12}})_{10}^{1}$ = 1.861209718204198449...₁₀

$({\sqrt[{12}]{12}})_{10}^{4}$ = $({\sqrt[{6}]{12}})_{10}^{2}$ = $({\sqrt[{3}]{12}})_{10}^{1}$ = 2.289428485106664046...₁₀

$({\sqrt[{12}]{12}})_{10}^{5}$ = 2.81616990130218614...₁₀

$({\sqrt[{12}]{12}})_{10}^{6}$ = $({\sqrt[{6}]{12}})_{10}^{3}$ = $({\sqrt[{4}]{12}})_{10}^{2}$ = $({\sqrt[{2}]{12}})_{10}^{1}$ = 3.464101615137751278...₁₀

$({\sqrt[{12}]{12}})_{10}^{7}$ = 4.261106545614035568...₁₀

$({\sqrt[{12}]{12}})_{10}^{8}$ = $({\sqrt[{6}]{12}})_{10}^{4}$ = $({\sqrt[{3}]{12}})_{10}^{2}$ = 5.241482788417794634...₁₀

$({\sqrt[{12}]{12}})_{10}^{9}$ = $({\sqrt[{4}]{12}})_{10}^{3}$ = 6.447419590941242618...₁₀

$({\sqrt[{12}]{12}})_{10}^{10}$ = $({\sqrt[{6}]{12}})_{10}^{5}$ = 7.930812913000363729...₁₀

$({\sqrt[{12}]{12}})_{10}^{11}$ = 9.755498703603224797...₁₀

$({\sqrt[{12}]{12}})_{10}^{12}$ = $({\sqrt[{6}]{12}})_{10}^{6}$ = $({\sqrt[{4}]{12}})_{10}^{4}$ = $({\sqrt[{3}]{12}})_{10}^{3}$ = $({\sqrt[{2}]{12}})_{10}^{2}$ = 12₁₀

Which, expressed in dozenal itself, is:

$({\sqrt[{10}]{10}})_{12}^{0}$ = $({\sqrt[{6}]{10}})_{12}^{0}$ = $({\sqrt[{4}]{10}})_{12}^{0}$ = $({\sqrt[{3}]{10}})_{12}^{0}$ = $({\sqrt[{2}]{10}})_{12}^{0}$ = 1

$({\sqrt[{10}]{10}})_{12}^{1}$ = 1.2916A19411A8586...₁₂

$({\sqrt[{10}]{10}})_{12}^{2}$ = $({\sqrt[{6}]{10}})_{12}^{1}$ = 1.61A741A0891B37...₁₂

$({\sqrt[{10}]{10}})_{12}^{3}$ = $({\sqrt[{4}]{10}})_{12}^{1}$ = 1.A40206532B36A64...₁₂

$({\sqrt[{10}]{10}})_{12}^{4}$ = $({\sqrt[{6}]{10}})_{12}^{2}$ =$({\sqrt[{3}]{10}})_{12}^{1}$ = 2.3581709AA9196B6...₁₂

$({\sqrt[{10}]{10}})_{12}^{5}$ = 2.996412324771588...₁₂

$({\sqrt[{10}]{10}})_{12}^{6}$ = $({\sqrt[{6}]{10}})_{12}^{3}$ =$({\sqrt[{4}]{10}})_{12}^{2}$ = $({\sqrt[{2}]{10}})_{12}^{1}$ = 3.569B73BB715207...₁₂

$({\sqrt[{10}]{10}})_{12}^{7}$ = 4.317237B73A07A6...₁₂

$({\sqrt[{10}]{10}})_{12}^{8}$ = $({\sqrt[{6}]{10}})_{12}^{4}$ = $({\sqrt[{3}]{10}})_{12}^{2}$ = 5.241482788417792358...₁₂

$({\sqrt[{10}]{10}})_{12}^{9}$ = $({\sqrt[{4}]{10}})_{12}^{3}$ = 6.5451838A653189...₁₂

$({\sqrt[{10}]{10}})_{12}^{A}$ = $({\sqrt[{6}]{10}})_{12}^{5}$ = 7.B2054056BA7909...₁₂

$({\sqrt[{10}]{10}})_{12}^{B}$ = 9.90960305A99166...₁₂

$({\sqrt[{10}]{10}})_{12}^{10}$ = $({\sqrt[{6}]{10}})_{12}^{6}$ = $({\sqrt[{4}]{10}})_{12}^{4}$ = $({\sqrt[{3}]{10}})_{12}^{3}$ = $({\sqrt[{2}]{10}})_{12}^{2}$ = 10₁₂

Then, for finer subdivisions we have to look at the square root of the twelfth root of twelve (that is, its 24th root, giving the dozenal R24 series), then for its cubic root (dozenal R36 series), its fourth root (dozenal R48), etc. The intermediate values for the powers of the 24th root of twelve, expressed in decimal, are:

1.10908769066200...₁₀

1.36426160182136...₁₀

1.67814477960102...₁₀

2.06424478820076...₁₀

2.53917695148275...₁₀

3.12337937234707...₁₀

3.84199246055163...₁₀

4.72594081833980...₁₀

5.81326404145091...₁₀

7.15075370484596...₁₀

8.79596697875195...₁₀

10.8197035284354...₁₀

Which expressed in dozenal are:

1.138606122B0A82...₁₂

1.44553B394095B1...₁₂

1.817A0156571B25...₁₂

2.093021AABA62BA...₁₂

2.65784590061881...₁₂

3.1592489B93413B...₁₂

3.A12B680225A077...₁₂

4.886513803226A9...₁₂

5.9913A14BA096B6...₁₂

7.19860419830949...₁₂

8.9675207B597B35...₁₂

A.9A05457545A042...₁₂

Again, these values are irrational numbers, so they have to be rounded. Rounding to the nearest hundredth:

doz R1   1.00                                                                                                                     12.00
doz R2   1.00                                                        3.46                                                         12.00
doz R3   1.00                                    2.29                                    5.24                                     12.00
doz R4   1.00                          1.86                          3.46                          6.45                           12.00
doz R6   1.00                1.51                2.29                3.46                5.24                7.93                 12.00
doz R12  1.00      1.23      1.51      1.86      2.29      2.82      3.46      4.26      5.24      6.45      7.93      9.76       12.00
doz R24  1.00 1.11 1.23 1.36 1.51 1.68 1.86 2.06 2.29 2.54 2.82 3.12 3.46 3.84 4.26 4.73 5.24 5.81 6.45 7.15 7.93 8.80 9.76 10.82 12.00
etc.

And expressed in dozenal, rounded to the nearest grossth:

doz R1   1.00                                                                                                                    10.00
doz R2   1.00                                                        3.57                                                        10.00
doz R3   1.00                                    2.36                                    5.24                                    10.00
doz R4   1.00                          1.A4                          3.57                          6.54                          10.00
doz R6   1.00                1.62                2.36                3.57                5.24                7.B2                10.00
doz R10  1.00      1.29      1.62      1.A4      2.36      2.9A      3.57      4.32      5.24      6.54      7.B2      9.91      10.00
doz R20  1.00 1.14 1.29 1.44 1.62 1.82 1.A4 2.09 2.36 2.66 2.9A 3.16 3.57 3.A1 4.32 4.88 5.24 5.99 6.54 7.1A 7.B2 8.97 9.91 A.9A 10.00
etc.

On the other hand, as for preferred numbers for modules and the like, we have to look at the dozenal-compatible numbers. In particular, at those that are small enough to make them easy to handle; that is, numbers that are factors of one dozen or one gross. The factors of twelve are: 1×12, 2×6, 3×4, and the factors of one gross are: 1×144, 2×72, 3×48, 4×36, 6×24, 8×18, 9×16, 12×12 (which expressed in dozenal are: 1×10, 2×6, 3×4, and 1×100, 2×60, 3×40, 4×30, 6×20, 8×16, 9×14, 10×10). Getting used to these factors and their reciprocal relations is a basic skill for proficiency in dozenal, just as getting used to the factors of ten and one hundred (1×10, 2×5, and 1×100, 2×50, 4×25, 5×20, 10×10) is a basic skill for proficiency in decimal. The basic set of preferred numbers in dozenal are those based in the above factors, which can be summed up as:

• the powers of twelve: ..., 1728, 144, 12, 1, 0.083, 0.00694, 0.000578703, etc... expressed in decimal (..., 1000, 100, 10, 1, 0.1, 0.01, 0.001, etc... expressed in dozenal)

• four basic reciprocal pairs: 2×6, 3×4, 8×18, 9×16 expressed in decimal (2×6, 3×4, 8×16, 9×14 expressed in dozenal)

• the results of multiplying a member of those reciprocal pairs by a power of twelve; for example: 0.16, 0.25, 0.3, 0.5, 0.6, 0.75, 1.3, 1.5, 24, 36, 48, 72, 96, 108, 192, 216, 288, 432, etc. expressed in decimal (0.2, 0.3, 0.4, 0.6, 0.8, 0.9, 1.4, 1.6, 20, 30, 40, 60, 80, 90, 140, 160, 200, 300, etc. expressed in dozenal)

The values, from one great-gross to one great-grossth, of the series of preferred numbers for a system based on twelve is, expressed in decimal:

1728, 1276, 1152, 864, 576, 432, 288, 216, 192, 144, 108, 96, 72, 48, 36, 24, 18, 16, 12, 9, 8, 6, 4, 3, 2, 1.5, 1.3, 1, 0.75, 0.6, 0.5, 0.3, 0.25, 0.16, 0.125, 0.1, 0.083, 0.0625, 0.05, 0.0416, 0.027, 0.02083, 0.0138, 0.010416, 0.00925, 0.00694, 0.0052083, 0.004629, 0.003472, 0.0023148, 0.0017361, 0.00115740, 0.00086805, 0.0007716049382, 0.000578703

And expressed in dozenal:

1000, 900, 800, 600, 400, 300, 200, 160, 140, 100, 90, 80, 60, 40, 30, 20, 16, 14, 10, 9, 8, 6, 4, 3, 2, 1.6, 1.4, 1, 0.9, 0.8, 0.6, 0.4, 0.3, 0.2, 0.16, 0.14, 0.1, 0.09, 0.08, 0.06, 0.04, 0.03, 0.02, 0.016, 0.014, 0.01, 0.009, 0.008, 0.006, 0.004, 0.003, 0.002, 0.0016, 0.0014, 0.001

For certain purposes, we might want to restrict the series to values derived only from factors of twelve, which expressed in decimal would be:

1728, 864, 576, 432, 288, 144, 72, 48, 36, 24, 12, 6, 4, 3, 2, 1, 0.5, 0.3, 0.25, 0.16, 0.083, 0.0416, 0.027, 0.02083, 0.0138, 0.00694, 0.003472, 0.0023148, 0.0017361, 0.00115740, 0.000578703

And expressed in dozenal:

1000, 600, 400, 300, 200, 100, 60, 40, 30, 20, 10, 6, 4, 3, 2, 1, 0.6, 0.4, 0.3, 0.2, 0.1, 0.06, 0.04, 0.03, 0.02, 0.01, 0.006, 0.004, 0.003, 0.002, 0.001

Similarly, the equivalent series of preferred numbers appropriate for use in a system based on ten can be obtained from the factors of ten and one hundred. Its values between one thousand and one thousandth are:

1000, 500, 400, 250, 200, 100, 50, 40, 25, 20, 10, 5, 4, 2.5, 2, 1, 0.5, 0.4, 0.25, 0.2, 0.1, 0.05, 0.04, 0.025, 0.02, 0.01, 0.005, 0.004, 0.0025, 0.002, 0.001

Since the possibilities are reduced compared to those obtained for dozenal using the same method (another consequence of the poorer divisibility of ten), we might want to expand it by including also the factors of one thousand (that is, adding the values derived from the reciprocal pair 8×125), obtaining the series:

1000, 800, 500, 400, 250, 200, 125, 100, 80, 50, 40, 25, 20, 12.5, 10, 8, 5, 4, 2.5, 2, 1.25, 1, 0.8, 0.5, 0.4, 0.25, 0.2, 0.125, 0.1, 0.08, 0.05, 0.04, 0.025, 0.02, 0.0125, 0.01, 0.008, 0.005, 0.004, 0.0025, 0.002, 0.00125, 0.001

The series could be further expanded using the factors of ten thousand, of one hundred thousand, etc. (that is, the values derived from the reciprocal pairs 16×625, 32×3125, etc.), but the practical value of such additions decreases as the smallest power of ten they are a factor of increases. The series expanded with factors of up to one hundred thousand is:

1000, 800, 625, 500, 400, 320, 312.5, 250, 200, 160, 125, 100, 80, 62.5, 50, 40, 32, 31.25, 25, 20, 16, 12.5, 10, 8, 6.25, 5, 4, 3.2, 3.125, 2.5, 2, 1.6, 1.25, 1, 0.8, 0.625, 0.5, 0.4, 0.32, 0.3125, 0.25, 0.2, 0.16, 0.125, 0.1, 0.08, 0.0625, 0.05, 0.04, 0.032, 0.03125, 0.025, 0.02, 0.016, 0.0125, 0.01, 0.008, 0.00625, 0.005, 0.004, 0.0032, 0.003125, 0.0025, 0.002, 0.0016, 0.00125, 0.001

If we want or need to restrict the series to those values based only on factors of ten, we get:

1000, 500, 200, 100, 50, 20, 10, 5, 2, 1, 0.5, 0.2, 0.1, 0.05, 0.02, 0.01, 0.005, 0.002, 0.001

There are other possibilities, such as restricting the values of the series to those that are factors of their next higher power of the base. In decimal, that series would include the values derived from 125 and 25, apart from the values derived from the factors of ten, but would exclude the values derived from 4 and 8:

1000, 500, 250, 200, 125, 100, 50, 25, 20, 12.5, 10, 5, 2.5, 2, 1.25, 1, 0.5, 0.25, 0.2, 0.125, 0.1, 0.05, 0.025, 0.02, 0.0125, 0.01, 0.005, 0.0025, 0.002, 0.00125, 0.001

In dozenal, it would include the values derived from 16 and 18 (dozenal 14 and 16) but exclude those derived from 8 and 9. Expressed in decimal:

1728, 864, 576, 432, 288, 216, 192, 144, 72, 48, 36, 24, 18, 16, 12, 6, 4, 3, 2, 1.5, 1.3, 1, 0.5, 0.3, 0.25, 0.16, 0.125, 0.1, 0.083, 0.0416, 0.027, 0.02083, 0.0138, 0.010416, 0.00925, 0.00694, 0.003472, 0.0023148, 0.0017361, 0.00115740, 0.00086805, 0.0007716049382, 0.000578703

And expressed in dozenal:

1000, 600, 400, 300, 200, 160, 140, 100, 60, 40, 30, 20, 16, 14, 10, 6, 4, 3, 2, 1.6, 1.4, 1, 0.6, 0.4, 0.3, 0.2, 0.16, 0.14, 0.1, 0.06, 0.04, 0.03, 0.02, 0.016, 0.014, 0.01, 0.006, 0.004, 0.003, 0.002, 0.0016, 0.0014, 0.001

Another interesting possibility is to restrict the chosen series to numbers that are mutually compatible, that is, that each higher value has all the lower values as factors. Examples for decimal would be series like 1-2-10, 1-2-10-50-100, 1-4-20-100, 1-5-10, 1-5-20-100, 1-5-25-50-100, 1-5-25-125-1000, etc. Examples for dozenal would be, expressed in decimal: 1-2-4-12, 1-3-6-12, 1-3-18-36-72-144, 1-4-12, 1-4-16-48-144, 1-6-12, 1-6-36-216-1728, etc. (expressed in dozenal: 1-2-4-10, 1-3-6-10, 1-3-16-30-60-100, 1-4-10, 1-4-14-40-100, 1-6-10, 1-6-30-160-1000, etc.).

Hope these comments helped clarify the question. 213.37.6.106 20:13, 11 June 2007 (UTC)

resistors in practice

I've been trying to add some more information on how the E series are used but i've come across a bit of a stubling block where reality seems to defy logic. My understanding (Unfortunately I don't have access to the standard so I can't confirm if it actually says this) is the intention of the E series is that the manufacturer/vendor is supposed to choose a series such that the maximum error from choosing a preferred value is the same as the tolerance. However the practice seems a total mess.

Based on a quick look at Farnell's (one of the major suppliers of electronic components in the uk) site I see the following.

20% resistors seem to virtually not exist anymore (farnell listed a total of 4 20% resistors and none of them looked like regular resistors). 10% resistors seem to be availilable in E12 values as one would expect (plus the odd resistor of a value that doesn't fit the series at all). 5% resistors seem to be available in E24 values one would expect (plus the odd resistor of a value that doesn't fit the series at all). So far logical enough but then 2% resistors only seem to be availiable in E12 values 1% resistors are available in both E24 AND E96 values 0.5% resistors seem to come in E24 and E48 values 0.25% resistors seem virtualy nonexistant. 0.1% resistors seem to come in E96 values.

Is this sort of messyness common? If so how best to explain things. Plugwash (talk) 03:08, 20 May 2010 (UTC)

It's all about keeping the number of different units down, what you describe is many 1000 wares, I'd bet that with just that limited selection they have boxes that have collected dust for years because no one bought that specific product. Extending the stock by introducing 0.25% resistors would cost too much to make them actually cheaper than their 0.1% counterparts. The choice of what series to carry is based partly on what customers request, and partly on the availability and price of replacement products. The cheapest modern production technologies are too exact to justify labelling any resistor 20%, and 10% resistors are gradually fading from market since 5% cost almost the same. —Preceding unsigned comment added by EBusiness (talk • contribs) 12:20, 7 July 2010 (UTC)

Compatibility

The article states: Using [preferred numbers] increases the probability that other designers will make exactly the same choice. This is particularly useful where the chosen dimension affects compatibility. For example, if the inner diameters of cooking pots or the distances between screws in wall fixtures are chosen from a series of preferred numbers, then it will be more likely that old pot lids and wall-plug holes can be reused when the original product is replaced.

This statement is unsourced and appears counter-intuitive. It's in manufacturers' interests not to standardise their parts. That way, consumers have to buy replacement parts from the original manufacturer. Look at razor blades, ink cartridges, digital cameras, phone chargers. Each new model frequently introduces new connectors that are incompatible with previous models, nevermind with models from their competitors. —Preceding unsigned comment added by 84.14.112.200 (talk) 17:16, 12 October 2010 (UTC)

It's interesting, because both you and the original writer have valid points. It's a multifaceted topic. It is true that preferred numbers are used by engineers and CAD detailers in industrial settings to increase the chances of things matching up at assembly time or in later field maintenance. But it is also true, as you point out, that manufacturers of consumer products have a financial interest in lack of compatibility and in planned obsolescence. The difference between the instances boils down to financial motives. People work toward compatibility when it is in their own (or their employer's) financial interest (eg, an engineer thinking ahead to reduce future factory maintenance or changeover costs). People also work against compatibility when it is in their own (or their employer's) financial interest (eg, planned obsolescence; trying to force spares to be OEM; etc). I'll go see about working this into the article. — ¾-10 02:04, 14 October 2010 (UTC)

Peculiar units or arithmetic error?

In the section subtitled Buildings: "For example, a multiple of 600 mm (6 M) can always be..." If 6 M is supposed to mean six metres, this should be denoted as "6000 mm (6 m)." If, however, as I'm tempted to guess, 6 M is construction jargon for six decimetres, this needs a demystifying note in parentheses. I'll leave it to someone who knows for sure to do the edit.192.197.54.32 (talk) 15:23, 16 January 2012 (UTC)

The notation "M" for "basic module" (1 M = 100 mm) is already explained in the paragraph preceding the passage you quoted. Indefatigable (talk) 13:30, 17 January 2012 (UTC)

Dead link report

• 75/106/EEC [2], European Union Directive on volume of liquids. 1975.

This is a dead link (http://europa.eu/eur-lex/en/consleg/pdf/1975/en_1975L0106_do_001.pdf). — Preceding unsigned comment added by Wikispaghetti (talk • contribs) 14:48, 7 August 2012 (UTC) — Sorry, I forgot to sign. Wikispaghetti (talk) 18:25, 12 August 2012 (UTC)

I shall delete the whole "References" section. It contains two redirects to the current page, one which directs to the list of ISO standards, one dead link and six redlinks. The Yeti 12:19, 26 June 2015 (UTC)

Music: Metronome markings

The preferred metronome markings in bpm are another example.Rjh82 (talk) 13:19, 18 June 2014 (UTC)

Those are thankfully not very decimal preferred numbers. ;-) Common ones are: (base 10) 60, 63, 66, 69, 72, 76, 80, 84, 88, 92, 96, 104, 108, 112, 120, 126, 132, 138, 144, 156, 168, 180, 192, 200, 208; (base 12) 50, 53, 56, 59, 60, 64, 68, 70, 74, 78, 80, 88, 90, 94, X0, X6, E0, E6, 100, 110, 120, 130, 140, 148, 154. They're practically duodecimal round numbers! d(^_^)b Double sharp (talk) 16:23, 15 April 2015 (UTC)