Contents 1 Renard numbers 2 1-2-5 series 3 Convenient numbers 4 E-series 5 Paper documents, envelopes, and drawing pens 6 Computer engineering 7 Retail packaging 8 Photography 9 See also 10 References 11 Further reading

Renard numbers Main article: Renard series In 1870 Charles Renard proposed a set of preferred numbers.[2] His system was adopted in 1952 as international standard ISO 3.[3] Renard's system divides the interval from 1 to 10 into 5, 10, 20, or 40 steps, leading to the R5, R10, R20 and R40 scales, respectively. 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.

Convenient numbers Main article: Convenient numbers In the 1970s the National Bureau of Standards (NBS) defined a set of convenient numbers to ease metrication in the United States. This system of metric values was described as 1-2-5 series in reverse, with assigned preferences for those numbers which are multiples of 5, 2, and 1 (plus their powers of 10), excluding linear dimensions above 100 mm.[1]

E-series Main article: E-series of preferred numbers The E-series is another system of preferred numbers. It consists of the E3, E6, E12, E24, E48, E96 and E192 series. Based on some of the existing manufacturing conventions, the International Electrotechnical Commission (IEC) began work on a new international standard in 1948.[4] The first version of this IEC 63 (renamed into IEC 60063 in 2007) was released in 1952.[4] 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 Main article: Paper size Standard metric paper sizes use the square root of two (√2) 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 √2 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 √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 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 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[5]). 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 In photography, aperture, exposure, and film speed generally follow powers of 2: The aperture size controls how much light enters the camera. It is 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 is 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.

References ^ a b Milton, Hans J. (December 1978). "The Selection of Preferred Metric Values for Design and Construction" (PDF). U.S. Government Printing Office. Washington, USA: The National Bureau of Standards (NBS). NBS Technical Note 990 (Code: NBTNAE). Archived (PDF) from the original on 2017-11-01. Retrieved 2017-11-01.  ^ "preferred numbers". Sizes, Inc. 2014-06-10 [2000]. Archived from the original on 2017-11-01. Retrieved 2017-11-01.  ^ ISO 3:1973-04 - Preferred numbers - Series of preferred numbers. International Standards Organization (ISO). April 1973. Archived from the original on 2017-11-02. Retrieved 2017-11-02.  (Replaced: ISO Recommendation R3-1954 - Preferred Numbers - Series of Preferred Numbers. July 1954.  (1953)) ^ a b IEC 60063:1952 - Series of preferred values and their associated tolerances for resistors and capacitors (1.0 ed.). International Electrotechnical Commission (IEC). 2007 [1952-01-01]. Archived from the original on 2017-11-01. Retrieved 2017-07-11.  ^ "COUNCIL DIRECTIVE of 19 December 1974 on the approximation of the laws of the Member States relating to the making-up by volume of certain prepackaged liquids (75/106/EEC)" (PDF). 2004-05-01 [1974-12-19]. Archived from the original (PDF) on 2013-05-16.

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