Preferred number

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In industrial design, preferred numbers (also called preferred values, preferred series or convenient numbers[1]) 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:

  1. Using them increases the probability of compatibility between objects designed at different times by different people. In other words, it is one tactic among many in standardization, whether within a company or within an industry, and it is usually desirable in industrial contexts (unless the goal is vendor lock-in or planned obsolescence)
  2. They 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 in stock.

Preferred numbers represent preferences of simple numbers (such as 1, 2, and 5) and their powers of a convenient basis, usually 10.[1]

Renard numbers[edit]

The French army engineer Col. Charles Renard proposed in the 1870s a set of preferred numbers.[2] 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 approximately constant (before rounding), namely the 5th, 10th, 20th, or 40th root of 10 (approximately 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 Renard numbers are not always rounded to the closest three-digit number to the theoretical geometric sequence.

The most basic R5 series consists of these five rounded numbers, which are powers of the fifth root of 10, rounded to two digits:

 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. These are rounded to a multiple of 0.05. Where an even finer grading is needed, the R20, R40, and R80 series can be applied. The R20 series is usually rounded to a multiple of 0.05, and the R40 and R80 values interpolate between the R20 values, rather than being powers of the 80th root of 10 rounded correctly. In the table below, the additional R80 values are written to the right of the R40 values in the column named "R80 add'l". The R40 numbers 3.00 and 6.00 are higher than they "should" be by interpolation, in order to give rounder numbers.

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. In the table below, rounded values that differ from their less rounded counterparts are shown in bold.

least rounded
R5 R10 R20 R40 R80 add'l
1.00 1.00 1.00 1.00 1.03
1.06 1.09
1.12 1.12 1.15
1.18 1.22
1.25 1.25 1.25 1.28
1.32 1.36
1.40 1.40 1.45
1.50 1.55
1.60 1.60 1.60 1.60 1.65
1.70 1.75
1.80 1.80 1.85
1.90 1.95
2.00 2.00 2.00 2.06
2.12 2.18
2.24 2.24 2.30
2.36 2.43
2.50 2.50 2.50 2.50 2.58
2.65 2.72
2.80 2.80 2.90
3.00 3.07
3.15 3.15 3.15 3.25
3.35 3.45
3.55 3.55 3.65
3.75 3.87
4.00 4.00 4.00 4.00 4.12
4.25 4.37
4.50 4.50 4.62
4.75 4.87
5.00 5.00 5.00 5.15
5.30 5.45
5.60 5.60 5.75
6.00 6.15
6.30 6.30 6.30 6.30 6.50
6.70 6.90
7.10 7.10 7.30
7.50 7.75
8.00 8.00 8.00 8.25
8.50 8.75
9.00 9.00 9.25
9.50 9.75
10.0 10.0 10.0 10.0 10.3
medium rounded
R'10 R'20 R'40
1.00 1.00 1.00
1.05
1.12 1.12
1.20
1.25 1.25 1.25
1.30
1.40 1.40
1.50
1.60 1.60 1.60
1.70
1.80 1.80
1.90
2.00 2.00 2.00
2.10
2.20 2.20
2.40
2.50 2.50 2.50
2.60
2.80 2.80
3.00
3.20 3.20 3.20
3.40
3.60 3.60
3.80
4.00 4.00 4.00
4.20
4.50 4.50
4.80
5.00 5.00 5.00
5.30
5.60 5.60
6.00
6.30 6.30 6.30
6.70
7.10 7.10
7.50
8.00 8.00 8.00
8.50
9.00 9.00
9.50
10.0 10.0 10.0
most rounded
R′′5 R′′10 R′′20
1.0 1.0 1.0
1.1
1.2 1.2
1.4
1.5 1.5 1.6
1.8
2.0 2.0
2.2
2.5 2.5 2.5
2.8
3.0 3.0
3.5
4.0 4.0 4.0
4.5
5.0 5.0
5.5
6.0 6.0 6.0
7.0
8.0 8.0
9.0
10 10 10

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 millimetres. But one would end up with two incompatible sets of nicely spaced dimensions if they were applied, for instance, with both inches and feet.

Each of the Renard sequences can be reduced to a subset by taking every nth value in a series, which is designated by adding the number n after a slash.[2] For example, “R10′′/3 (1...1000)” designates a series consisting of every third value in the R′′10 series from 1 to 1000, that is, 1, 2, 4, 8, 15, 30, 60, 120, 250, 500, 1000.

1-2-5 series[edit]

In applications for which the R5 series provides a too fine graduation, the 1-2-5 series is sometimes used as a cruder alternative. It is effectively an R3 series rounded to one significant digit:

... 0.1 0.2 0.5 1 2 5 10 20 50 100 200 500 1000 ...

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 1-2-5 series has not been formally adopted as an international standard. However, the Renard series R10 can be used to extend the 1-2-5 series to a finer graduation.

This series is used to define the scales for graphs and for instruments that display in a two-dimensional form with a graticule, such as oscilloscopes.

The denominations of most modern currencies follow a 1-2-5 series. The United States and Canada follow the series 5, 10, 25, 50, 100 (cents), and also $5 and $10 which belong to the same series. However, after that comes $20, not $25. The ¼-½-1 series (... 0.1 0.25 0.5 1 2.5 5 10 ...) is also 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 1-2-5 series instead.

E series[edit]

This graph shows how any value between 1 and 10 is within ±10% of an E12 series value, and its difference from the ideal value in a geometric sequence
Two decades of E12 values, which would give resistor values of 1 Ω to 82 Ω
A decade of the E12 values shown with their electronic color codes on resistors.

In electronics, international standard IEC 60063 defines another preferred number series for resistors, capacitors, inductors and Zener diodes. 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 arbitrary value is replaced with the nearest preferred number, the maximum relative error will be on the order of 20%, 10%, 5%, etc.

Use of the E series is mostly restricted to resistors, capacitors, inductors and Zener diodes. 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 as follows. The E6 series is every other element of the E12 series, which is in turn every other element of the E24 series:

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  12  15  18  22  27  33  39  47  56  68  82
              11  13  16  20  24  30  36  43  51  62  75  91

With the E48 series, a third decimal place is added, and the values are slightly adjusted. Again, the E48 series is every other value of the E96 series, which is every other value of the E192 series:

E48  ( 2%): 100  121  147  178  215  261  316  383  464  562  681  825
             105  127  154  187  226  274  332  402  487  590  715  866
              110  133  162  196  237  287  348  422  511  619  750  909
               115  140  169  205  249  301  365  442  536  649  787  953

E96 (  1%): 100  121  147  178  215  261  316  383  464  562  681  825
             102  124  150  182  221  267  324  392  475  576  698  845
              105  127  154  187  226  274  332  402  487  590  715  866
               107  130  158  191  232  280  340  412  499  604  732  887
                110  133  162  196  237  287  348  422  511  619  750  909
                 113  137  165  200  243  294  357  432  523  634  768  931
                  115  140  169  205  249  301  365  442  536  649  787  953
                   118  143  174  210  255  309  374  453  549  665  806  976
E192 (0.5%) 100  121  147  178  215  261  316  383  464  562  681  825
             101  123  149  180  218  264  320  388  470  569  690  835
              102  124  150  182  221  267  324  392  475  576  698  845
               104  126  152  184  223  271  328  397  481  583  706  856
                105  127  154  187  226  274  332  402  487  590  715  866
                 106  129  156  189  229  277  336  407  493  597  723  876
                  107  130  158  191  232  280  340  412  499  604  732  887
                   109  132  160  193  234  284  344  417  505  612  741  898
                    110  133  162  196  237  287  348  422  511  619  750  909
                     111  135  164  198  240  291  352  427  517  626  759  920
                      113  137  165  200  243  294  357  432  523  634  768  931
                       114  138  167  203  246  298  361  437  530  642  777  942
                        115  140  169  205  249  301  365  442  536  649  787  953
                         117  142  172  208  252  305  370  448  542  657  796  965
                          118  143  174  210  255  309  374  453  549  665  806  976
                           120  145  176  213  258  312  379  459  556  673  816  988

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

1% resistors are available in both the E24 values and the E96 values.

Paper documents, envelopes, and drawing pens[edit]

Main article: Paper size

Standard metric paper sizes use the square root of two and related numbers (√√√2, √√2, √2, 2, or 2√2) as factors between neighbour dimensions (Lichtenberg series, ISO 216). An A4 sheet for example has an aspect ratio very close to √2 and an area very close to 1/16 square metre. A5 is like half an A4, and has the same aspect ratio. 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.

Computer engineering[edit]

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:

     1  2   4   8    16    32    64    128     256     512     1024       ...
(×3)      3   6   12    24    48    96     192     384     768     1536   ...
(×5)         5   10    20    40    80    160     320     640     1280     ...
(×7)           7   14    28    56    112     224     448     896     1792 ...
Preferred aspect ratios
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. Black-and-white JPEG images are divided into 8×8 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.

Retail packaging[edit]

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]). It restricts the list of allowed wine-bottle 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 ratio of two small integers.

Photography[edit]

In photography, aperture, exposure, and film speed generally follow powers of 2:

The aperture size controls how much light enters the camera. It's measured in f-stops: f/1.4, f/2, f/2.8, f/4, etc. Full f-stops are a square root of 2 apart. Digital cameras often subdivide these into thirds, so each f-stop is a sixth root of 2, rounded to two significant digits: 1.0, 1.1, 1.2, 1.4, 1.6, 1.8, 2.0, 2.2, 2.5, 2.8, 3.2, 3.5, 4.0 (i.e. the R′′20 series with 3.2 instead of 3.0).

The film speed is a measure of the film’s sensitivity to light. It's expressed as ISO values such as “ISO 100”. Measured film speeds are rounded to the nearest preferred number from a modified Renard series including 100, 125, 160, 200, 250, 320, 400, 500, 640, 800... This is the same as the R10′ rounded Renard series, except for the use of 6.4 instead of 6.3, and for having more aggressive rounding below ISO 16. Film marketed to amateurs, however, uses a restricted series including only powers of two multiples of ISO 100: 25, 50, 100, 200, 400, 800, 1600 and 3200. Some low-end cameras can only reliably read these values from DX encoded film cartridges because they lack the extra electrical contacts that would be needed to read the complete series. Some digital cameras extend this binary series to values like 12800, 25600, etc. instead of the modified Renard values 12500, 25000, etc.

The shutter speed controls how long the camera records light. These are expressed as fractions of a second, roughly but not exactly based on powers of 2: 1 second, 1/2, 1/4, 1/8, 1/15, 1/30, 1/60, 1/125, 1/250, 1/500, 1/1000 of a second.

See also[edit]

References[edit]

Further reading[edit]

  • ISO 3: "Preferred numbers. Series of preferred numbers." (1973)
  • IEC 60063: "Preferred number series for resistors and capacitors." (2015)