In industrial design, preferred numbers (also called preferred values) are standard guidelines for choosing exact product dimensions within a given set of constraints. 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 serve two purposes:
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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.
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 should be placed 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 to the series, one after each of the original 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, R40, and R80 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
R80: 1.00 1.25 1.60 2.00 2.50 3.15 4.00 5.00 6.30 8.00 1.03 1.28 1.65 2.06 2.58 3.25 4.12 5.15 6.50 8.25 1.06 1.32 1.70 2.12 2.65 3.35 4.25 5.30 6.70 8.50 1.09 1.36 1.75 2.18 2.72 3.45 4.37 5.45 6.90 8.75 1.12 1.40 1.80 2.24 2.80 3.55 4.50 5.60 7.10 9.00 1.15 1.45 1.85 2.30 2.90 3.65 4.62 5.80 7.30 9.25 1.18 1.50 1.90 2.36 3.00 3.75 4.75 6.00 7.50 9.50 1.22 1.55 1.95 2.43 3.07 3.87 4.87 6.15 7.75 9.75
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 10fold change of the scale, they are particularly wellsuited 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.
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 ith element of this series (with i = 0 through i = b).
In applications for which the R5 series provides a too fine graduation, the 125 series is sometimes used as a cruder alternative:
This series covers a decade (1:10 ratio) in three steps. Adjacent values differ by factors 2 or 2.5. Unlike the Renard series, the 125 series has not been formally adopted as an international standard. However, the Renard series R10 can be used to extend the 125 series to a finer graduation.
This series is used to define the scales for graphs and for instruments that display in a twodimensional form with a graticule, such as oscilloscopes.
The denominations of most modern currencies follow a 125 series. An exception are some quartervalue coins, such as those of Canada and the United States (the latter denominated as "quarter dollar" rather than 25 cents). A ¼½1 series (... 0.1 0.25 0.5 1 2.5 5 10 ...) is used by currencies derived from the former Dutch gulden (Aruban florin, Netherlands Antillean gulden, Surinamese dollar), some Middle Eastern currencies (Iraqi and Jordanian dinars, Lebanese pound, Syrian pound), and the Seychellois rupee. However, newer notes introduced in Lebanon and Syria due to inflation follow the standard 125 series instead.
In electronics, international standard IEC 60063 defines another preferred number series for resistors, capacitors, and inductors. 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 on the order of 20%, 10%, 5%, etc.
Use of the E series is mostly restricted to resistors, capacitors and inductors. 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).
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
E48 ( 2%): 100 105 110 115 121 127 133 140 147 154 162 169 178 187 196 205 215 226 237 249 261 274 287 301 316 332 348 365 383 402 422 442 464 487 511 536 562 590 619 649 681 715 750 787 825 866 909 953
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%) 100 101 102 104 105 106 107 109 110 111 113 114 115 117 118 120 121 123 124 126 127 129 130 132 133 135 137 138 140 142 143 145 147 149 150 152 154 156 158 160 162 164 165 167 169 172 174 176 178 180 182 184 187 189 191 193 196 198 200 203 205 208 210 213 215 218 221 223 226 229 232 234 237 240 243 246 249 252 255 258 261 264 267 271 274 277 280 284 287 291 294 298 301 305 309 312 316 320 324 328 332 336 340 344 348 352 357 361 365 370 374 379 383 388 392 397 402 407 412 417 422 427 432 437 442 448 453 459 464 470 475 481 487 493 499 505 511 517 523 530 536 542 549 556 562 569 576 583 590 597 604 612 619 626 634 642 649 657 665 673 681 690 698 706 715 723 732 741 750 759 768 777 787 796 806 816 825 835 845 856 866 876 887 898 909 920 931 942 953 965 976 988
The E192 series is also used for 0.25% and 0.1% tolerance resistors.
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 drawings, 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 in the standards by the letter M). Preference is given to the multiples of 300 mm (3 M) and 600 mm (6 M) of the basic module (see also "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)
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.
Standard metric 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.
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 directmapped and setmapped 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]} Furthermore, if the data structure in question has any other data in addition to the array itself, then the entire structure is less likely to fit in a single cache line, or on a single memory page, if the array size itself is a power of two.
16:  15:  12:  

:8  2:1  3:2  
:9  16:9  5:3  4:3 
:10  8:5  3:2  
:12  4:3  5:4  1:1 
In computer graphics, widths and heights of raster images are preferred to be multiples of 16, as many compression algorithms (JPEG, MPEG) divide color images into square blocks of that size. Blackandwhite JPEG images are divided into 8x8 blocks. Screen resolutions often follow the same principle. Preferred aspect ratios have also an important influence here, e.g. 2:1, 3:2, 4:3, 5:3, 5:4, 8:5, 16:9.
In some countries, consumerprotection 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]). It restricts the list of allowed winebottle sizes to 0.1, 0.25 (1/4), 0.375 (3/8), 0.5 (1/2), 0.75 (3/4), 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.
While some instruments (trombone, theremin, etc.) can play a tone at any arbitrary frequency, other instruments (such as pianos) can only play a limited set of tones. The very popular "twelvetone equal temperament" selects tones from the geometric sequence
where k is typically 440, though other standards have been used in the past. However, other less common tuning systems have also been historically important as preferred audio frequencies.
Since 2^{10}≈10^{3}, 2^{1/12}≈10^{3/120}=10^{1/40}, and the resultant frequency spacing is very similar to the R40 series.
