When developing a product, designers must choose numerous lengths, distances, diameters, volumes, and other characteristic quantities. While all of these choices are constrained by functional, usability, compatibility or safety considerations, there usually remains considerable leeway in the exact choice for many dimensions. Preferred numbers are standard guidelines for choosing exact product dimensions within such constraints.

They serve two purposes:

1. Using preferred numbers increases the probability that other designers will make the exact 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.
2. Preferred numbers are chosen such that when a product is manufactured in many different sizes, these will end up roughly equally spaced on a logarithmic scale. They therefore help to minimize the number of different sizes that need to be manufactured or kept on stock.

Renard numbers

The French army engineer Col. Charles Renard proposed in the 1870s a set of preferred numbers for use with the metric system. His system was adopted in 1952 as international standard ISO 3. Renard's system of preferred numbers divides the interval from 1 to 10 into 5, 10, 20, or 40 steps. The factor between two consecutive numbers in a Renard series is constant (before rounding), namely the 5th, 10th, 20th, or 40th root of 10 (1.58, 1.26, 1.12, and 1.06, respectively), which leads to a geometric series. This way, the maximum relative error is minimized if an arbitrary number is replaced by the nearest Renard number multiplied by the appropriate power of 10. Events and Trends Franco-Prussian War (1870-1871) results in the collapse of the Second French Empire and in the formation of both the French Third Republic and the German Empire. ... Logo of the International Organization for Standardization The International Organization for Standardization (ISO or Iso) is an international standard-setting body made up of representatives from national standards bodies. ...

The most basic R5 series consists of these five rounded numbers:

 R5: 1.00 1.60 2.50 4.00 6.30 

Example: If our design constraints tell us that the two screws in our gadget can be spaced anywhere between 32 mm and 55 mm apart, we make it 40 mm, because 4 is in the R5 series of preferred numbers.

Example: If you want to produce a set of nails with lengths between roughly 15 and 300 mm, then the application of the R5 series would lead to a product repertoire of 16 mm, 25 mm, 40 mm, 63 mm, 100 mm, 160 mm, and 250 mm long nails.

If a finer resolution is needed, another five numbers are added between the R5 numbers, and we end up with the R10 series:

 R10: 1.00 1.25 1.60 2.00 2.50 3.15 4.00 5.00 6.30 8.00 

Where an even finer grading is needed, the R20 and R40 series can be applied:

 R20: 1.00 1.12 1.25 1.40 1.60 1.80 2.00 2.24 2.50 2.80 3.15 3.55 4.00 4.50 5.00 5.60 6.30 7.10 8.00 9.00 

 R40: 1.00 1.06 1.12 1.18 1.25 1.32 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.12 2.24 2.36 2.50 2.65 2.80 3.00 3.15 3.35 3.55 3.75 4.00 4.25 4.50 4.75 5.00 5.30 5.60 6.00 6.30 6.70 7.10 7.50 8.00 8.50 9.00 9.50 

In some applications more rounded values are desirable, either because the numbers from the normal series would imply an unrealistically high accuracy, or because an integer value is needed (e.g., the number of teeth in a gear). For these needs, more rounded versions of the Renard series have been defined in ISO 3:

 R5': 1 1.5 2.5 4 6 R10': 1 1.25 1.6 2 2.5 3.2 4 5 6.3 8 R10": 1 1.2 1.5 2 2.5 3 4 5 6 8 R20': 1 1.1 1.25 1.4 1.6 1.8 2 2.2 2.5 2.8 3.2 3.6 4 4.5 5 5.6 6.3 7.1 8 9 R20": 1 1.1 1.2 1.4 1.6 1.8 2 2.2 2.5 2.8 3 3.5 4 4.5 5 5.5 6 7 8 9 R40': 1 1.05 1.1 1.2 1.25 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.4 2.5 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.5 4.8 5 5.3 5.6 6 6.3 6.7 7.1 7.5 8 8.5 9 9.5 

As the Renard numbers repeat after every 10-fold change of the scale, they are particularly well-suited for use with SI units. It makes no difference whether the Renard numbers are used with metres or kilometres. But one would end up with two incompatible sets of nicely spaced dimensions if they were applied, for instance, with both yards and miles. SI (disambiguation). ... metre or meter, see meter (disambiguation) The metre is the basic unit of length in the International System of Units. ... A kilometre (American spelling: kilometer) (symbol: km) is a unit of length equal to 1000 metres (from the Greek words khilia = thousand and metro = count/measure). ... This article is about the unit of measure known as the yard. ... A mile is any of several units of distance, or, in physics terminology, of length. ...

Capacitors and resistors

International standard IEC 63 defines another preferred number series that is commonly used for electronic components, especially resistors and capacitors. It works similar to the Renard series, except that it subdivides the interval from 1 to 10 into 6, 12, 24, etc. steps. These subdivisions ensure that when some random value is replaced with the nearest preferred number, the maximum error will be in the order of 20%, 10%, 5%, etc. The initials IEC can stand for: Independent Electoral Commission Industrial Emergency Council Institut des Experts-comptables et des Conseils fiscaux Institut dEstudis Catalans, Catalan Studies Institute Interactive Evolutionary Computation International Education Centre International Electrical Congress International Electrotechnical Commission See also IEC connector for IEC cord International Engineering Consortium International...

The IEC 63 numbers are:

 E6 (20%): 10 15 22 33 47 68 

 E12 (10%): 10 12 15 18 22 27 33 39 47 56 68 82 

 E24 ( 5%): 10 11 12 13 15 16 18 20 22 24 27 30 33 36 39 43 47 51 56 62 68 75 82 91 

Buildings

In the construction industry, it was felt that typical dimensions must be easy to use in mental arithmetic. Therefore, rather than using elements of a geometric series, a different system of preferred dimensions has evolved in this area, known as "modular coordination".

Major dimensions (e.g., grid lines on plans, distances between wall centers or surfaces, widths of shelves and kitchen components) are multiples of 100 mm. This size is called the "basic module" and represented by the letter M. Preference is given to the multiples of 3 M (= 300 mm) and 6 M (= 600 mm) of the basic module. For larger dimensions, preference is given to multiples of the modules 12 M (= 1.2 m), 15 M (= 1.5 m), 30 M (= 3 m), and 60 M (= 6 m). For smaller dimensions, the submodular increments 50 mm or 25 mm are used. (ISO 2848, BS 6750) Logo of the International Organization for Standardization The International Organization for Standardization (ISO or Iso) is an international standard-setting body made up of representatives from national standards bodies. ... British Standards is the new name of the British Standards Institute and is part of BSI Group which also includes a testing organisation. ...

Dimensions chosen this way can easily be divided by a large number of factors without ending up with millimetre fractions. For example, a multiple of 600 mm (6 M) can always be divided into 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 25, 30, etc. parts, each of which is again an integral number of millimetres.

Paper documents, envelopes, and drawing pens

Standard paper sizes use factors of the square root of two (√2, √√2, or √√√2) as factors between neighbor dimensions (Lichtenberg series, ISO 216). The √2 factor also appears between the standard pen thicknesses for technical drawings (0.13, 0.18, 0.25, 0.35, 0.50, 0.70, 1.00, 1.40, and 2.00 mm). This way, the right pen size is available to continue a drawing that has been magnified to a different standard paper size. There have been many standard sizes of paper at different times and in different countries. ... In mathematics, the principal square root of a non-negative real number is denoted and represents the non-negative real number whose square (the result of multiplying the number by itself) is . ... Georg Christoph Lichtenberg. ... ISO 216 specifies international standard (ISO) paper sizes, used in most countries in the world today. ...

Computer engineering

When dimensioning computer components, the powers of two are frequently used as preferred numbers:

 1 2 4 8 16 32 64 128 256 512 1024 ... 

Where a finer grading is needed, additional preferred numbers are obtained by multiplying a power of two with a small odd integer:

 3 6 12 24 48 96 192 384 768 1536 ... 5 10 20 40 80 160 320 640 1280 2560 ... 7 14 28 56 112 224 448 896 1792 3584 ... 

These correspond to binary numbers that consist mostly of trailing zero bits, which are particularly easy to add and subtract in hardware.

Software developers should keep in mind, though, that using powers of 2 in software, especially with array sizes, may also have disadvantages, such as reduced CPU cache efficiency.

In computer graphics, widths and heights of raster images are preferred to be multiples of 16, as many compression algorithms (JPEG, MPEG) divide images into square blocks of that size. Computer graphics (CG) is the field of visual computing, where one utilizes computers both to generate visual images synthetically and to integrate or alter visual and spatial information sampled from the real world. ... A photo of a flower compressed with successively higher compression ratios from left to right. ... The Moving Picture Experts Group (MPEG) is a small group charged with the development of video and audio encoding standards. ...

Retail packaging

In some countries, consumer-protection laws restrict the number of different prepackaged sizes in which certain products can be sold, in order to make it easier for consumers to compare prices.

An example of such a regulation is the European Union directive on the volume of certain prepackaged liquids (75/106/EEC [1] (http://europa.eu.int/eur-lex/en/consleg/pdf/1975/en_1975L0106_do_001.pdf)). It restricts the list of allowed wine-bottle sizes to 0.1, 0.25, 0.375, 0.5, 0.75, 1, 1.5, 2, 3, and 5 litres. Similar lists exist for several other types of products. They vary and often deviate significantly from any geometric series in order to accommodate traditional sizes were feasible. Adjacent packages sizes in these lists differ typically by factors 2/3 or 3/4, in some cases even 1/2, 4/5, or some other fraction of two small integers.