In industrial design, preferred numbers (also called preferred values, preferred series or convenient numbers) 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:
- 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)
- 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.
In 1870 Charles Renard proposed a set of preferred numbers. His system was adopted in 1952 as international standard ISO 3. Renard's system 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.
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.
The E-series is another system of preferred numbers. It consists of the E3, E6, E12, E24, E48, E96 and E192 series. Originally introduced in 1948, it was standardized in the international standard IEC 63 (later renamed into IEC 60063) by the International Electrotechnical Commission (IEC) in 1963.
It works similarly to the Renard series, except that it subdivides the interval from 1 to 10 into 3, 6, 12, 24, 48, 96 or 192 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 40%, 20%, 10%, 5%, etc.
Use of the E series is mostly restricted to electronic parts like resistors, capacitors, inductors and Zener diodes. Commonly produced dimensions for other types of electrical components are either chosen from the Renard series instead or are defined in relevant product standards (for example wires).
Paper documents, envelopes, and drawing pens
Standard metric paper sizes use the square root of two (√) as factors between neighbouring dimensions rounded to the nearest mm (Lichtenberg series, ISO 216). An A4 sheet for example has an aspect ratio very close to √ and an area very close to 1/16 square metre. An A5 is almost exactly half an A4, and has the same aspect ratio. The √ 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:
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 ...
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.
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). 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.
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, etc.
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.