In industrial design, product developers must choose numerous lengths, distances, diameters, volumes, and other characteristic quantities. While all of these choices are constrained by considerations of functionality, usability, compatibility, safety or cost, there usually remains considerable leeway in the exact choice for many dimensions. Preferred numbers (also called preferred values) are standard guidelines for choosing exact product dimensions within such constraints. Industrial Design is an applied art whereby the aesthetics and usability of products may be improved. ... Quantity is a general term used to refer to any type of quantitative property or attribute, such as mass, length, or time. ...

They serve two purposes:

1. 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.
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 sequence. 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. Charles Renard (1847-1905), French military engineer, started to work after the Franco-Prussian War of 1870/71 on the design of air ships at the French army aeronautical department. ... // Events and Trends Technology The invention of the telephone (1876) by Alexander Graham Bell. ... The International Organization for Standardization (ISO) is an international standard-setting body composed of representatives from national standards bodies. ... In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ...

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

 R40: 1.00 1.25 1.60 2.00 2.50 3.15 4.00 5.00 6.30 8.00 1.06 1.32 1.70 2.12 2.65 3.35 4.25 5.30 6.70 8.50 1.12 1.40 1.80 2.24 2.80 3.55 4.50 5.60 7.10 9.00 1.18 1.50 1.90 2.36 3.00 3.75 4.75 6.00 7.50 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 1.25 1.6 2 2.5 3.2 4 5 6.3 8 1.1 1.4 1.8 2.2 2.8 3.6 4.5 5.6 7.1 9 

 R20": 1 1.2 1.6 2 2.5 3 4 5 6 8 1.1 1.4 1.8 2.2 2.8 3.5 4.5 5.5 7 9 

 R40': 1 1.25 1.6 2 2.5 3.2 4 5 6.3 8 1.05 1.3 1.7 2.1 2.6 3.4 4.2 5.3 6.7 8.5 1.1 1.4 1.8 2.2 2.8 3.6 4.5 5.6 7.1 9 1.2 1.5 1.9 2.4 3 3.8 4.8 6 7.5 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. Cover of brochure The International System of Units. ... The metre, or meter, is a measure of length, approximately equal to 3. ... A kilometre (American spelling: kilometer), symbol: km is a unit of length in the metric system equal to 1000 metres (from the Greek words χίλια (khilia) = thousand and μέτρο (metro) = count/measure). ... This article is about the unit of measure known as the yard. ... A mile is any of a number of units of distance, each in the magnitude of 1–10 km. ...

Renard numbers are rounded results of the formula

,

where b is the selected series value (for example b = 40 for the R40 series), and i is the i-th element of this series (starting with i = 0).

Capacitors and resistors

Some E12 resistor values, which follow a geometric series

International standard IEC 60063 defines another preferred number series that is commonly used for electronic components, especially resistors and capacitors. It works similarly 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. Image File history File links Download high resolution version (1258x770, 29 KB) Summary This is a graph of E12 resistor values, from 1 to 82. ... Image File history File links Download high resolution version (1258x770, 29 KB) Summary This is a graph of E12 resistor values, from 1 to 82. ... The initials IEC can stand for: Independent Electoral Commission Industrial Emergency Council Inertial electrostatic confinement (in fusion energy) 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... Resistor symbols (US and Japan) Resistor symbols (Europe) A pack of resistors A resistor is a two-terminal electrical or electronic component that resists an electric current by producing a voltage drop between its terminals in accordance with Ohms law. ... A capacitor is a device that stores energy in the electric field created between a pair of conductors on which electric charges of equal magnitude, but opposite sign, have been placed. ...

Use of the E series is mostly restricted to resistors and capacitors. Commonly produced dimensions for other types of electrical components are either chosen from the Renard series instead (for example fuses) or are defined in relevant product standards (for example wires). 200 A Industrial fuse. ... IEC 60228 is the International Electrotechnical Commissions international standard on conductors of insulated cables. ...

The IEC 60063 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 

 E96 ( 1%): 100 102 105 107 110 113 115 118 121 124 127 130 133 137 140 143 147 150 154 158 162 165 169 174 178 182 187 191 196 200 205 210 215 221 226 232 237 243 249 255 261 267 274 280 287 294 301 309 316 324 332 340 348 357 365 374 383 392 402 412 422 432 442 453 464 475 487 499 511 523 536 549 562 576 590 604 619 634 649 665 681 698 715 732 750 768 787 806 825 845 866 887 909 931 953 976 

 E192 (0.5%) 100 101 102 104 105 106 107 109 110 111 113 114 115 117 118 120 121 123 124 126 127 129 130 132 133 135 137 138 140 142 143 145 147 149 150 152 154 156 158 160 162 164 165 167 169 172 174 176 178 180 182 184 187 189 191 193 196 198 200 203 205 208 210 213 215 218 221 223 226 229 232 234 237 240 243 246 249 252 255 258 261 264 267 271 274 277 280 284 287 291 294 298 301 305 309 312 316 320 324 328 332 336 340 344 348 352 357 361 365 370 374 379 383 388 392 397 402 407 412 417 422 427 432 437 442 448 453 459 464 470 475 481 487 493 499 505 511 517 523 530 536 542 549 556 562 569 576 583 590 597 604 612 619 626 634 642 649 657 665 673 681 690 698 706 715 723 732 741 750 759 768 777 787 796 806 816 825 835 845 856 866 876 887 898 909 919 931 942 953 965 976 988 

The E192 series is also used for 0.25% and 0.1% tolerance resistors.

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". Elementary arithmetic is the most basic kind of mathematics: it concerns the operations of addition, subtraction, multiplication, and division. ...

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, i.e. one decimetre. This size is called the "basic module" and represented by the letter M (instead of "dm"). Preference is given to the multiples of 3 M (= 300 mm) and 6 M (= 600 mm) of the basic module, which has led to the false belief the sizes were based on a "metric foot". 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) The metre, or meter, is a measure of length, approximately equal to 3. ... This article is about a foot as a unit of length. ... The International Organization for Standardization (ISO) is an international standard-setting body composed of representatives from national standards bodies. ... BSI Kite Mark Logo - Made up of the letters B & S British Standards is the new name of the British Standards Institution 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 the square root of two and related numbers (√√√2, √√2, √2, 2, or 2√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. A comparison of different paper sizes There have been many standard sizes of paper at different times and in different countries, but today there are basically only two systems in place: the international standard (A4 and its siblings), and the North American sizes. ... 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 For example, since This example suggests how square roots can arise when solving quadratic equations such as or... Georg Christoph Lichtenberg is an 18th-century German scientist, satirist and anglophile, most famous for his notebooks published posthumously (which he himself called waste books, using the English bookkeeping term). ... 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 two in software, especially with array sizes, may also have disadvantages. In particular, it can dramatically reduce CPU cache efficiency on processors whose cache memory is not fully associative. With direct-mapped and set-mapped cache designs, memory locations whose addresses are a multiple of the cache size (typically a power of two) apart may share the same cache lines. Algorithms that access such memory locations alternatingly may be slowed down by frequent cache collisions (cache interference). [1] Diagram of a CPU memory cache A CPU cache is a cache used by the central processing unit of a computer to reduce the average time to access memory. ...

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. ... In computing, JPEG (pronounced jay-peg) is a commonly used standard method of lossy compression for photographic images. ... The Moving Picture Experts Group or MPEG is a working group of ISO/IEC 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 [2]). It restricts the list of allowed wine-bottle sizes to 0.1, 0.25 (2/8), 0.375 (3/8), 0.5 (4/8), 0.75 (6/8), 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 when feasible. Adjacent package 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.

References

• ISO 3, Preferred numbers — Series of preferred numbers. International Organization for Standardization, 1973.
• ISO 17, Guide to the use of preferred numbers and of series of preferred numbers. 1973.
• ISO 497, Guide to the choice of series of preferred numbers and of series containing more rounded values of preferred numbers. 1973.
• ISO 2848, Building construction — Modular coordination — Principles and rules. 1984.
• ISO/TR 8389, Building construction — Modular coordination — System of preferred numbers defining multimodular sizes. International Organization for Standardization, 1984.
• IEC 60063, Preferred number series for resistors and capacitors. International Electrotechnical Commission, 1963
• BS 2045, Preferred numbers. British Standards Institute, 1965.
• BS 2488, Schedule of preferred numbers for the resistance of resistors and the capacitance of capacitors for telecommunication equipment. 1966.