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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:
- 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.
- 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.
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.
Preferred dimensions for 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 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
Preferred dimensions for paper documents, envelopes, and drawing pens
Standard paper sizes use factors of sqrt(2), sqrt(sqrt(2)), or sqrt(sqrt(sqrt(2))) as factors between neighbor dimensions (Lichtenberg series, ISO 216). The sqrt(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.
Preferred numbers in computer engineering
When dimensioning computer components, the powers of two are frequently used as preferred numbers:
1 2 4 8 16 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 8 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.
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