Preferred number

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.

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 E3 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, notably the euro and British pound, follow a 1–2–5 series. The United States and Canada follow the approximate 1–2–5 series 1, 5, 10, 25, 50, 100 (cents), $1, $2, $5, $10, $20, $50, $100. The ​¹⁄₄–​¹⁄₂–1 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.

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]

The E series is another system of preferred numbers. It consists of the E1, 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).

ISO 266, Acoustics—Preferred frequencies, defines two different series of audio frequencies for use in acoustical measurements. Both series are referred to the standard reference frequency of 1000 Hz, and use the R10 Renard series from ISO 3, with one using powers of 10, and the other related to the definition of the octave as the frequency ratio 1:2.[5]

For example, a set of nominal center frequencies for use in audio tests and audio test equipment is:

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.

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.

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. Camera lens settings are often set to gaps of successive 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 spacing is referred to as "one-third of a stop".

The film speed is a measure of the film's sensitivity to light. It is expressed as ISO values such as "ISO 100". An earlier standard, occasionally still in use, uses the term "ASA" rather than "ISO", referring to the (former) American Standards Association. 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 lens is open to receive 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.

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[7]). 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.

• Convenient number
• Nominal impedance
• Nominal size
• Preferred metric sizes

• Hirshfeld, Clarence Floyd; Berry, C. H. (1922-12-04). "Size Standardization by Preferred Numbers". Mechanical Engineering. New York, USA: The American Society of Mechanical Engineers. 44 (12): 791–.
• Hazeltine, Louis Alan (January 1927) [December 1926]. "Preferred Numbers". Proceedings of the Institute of Radio Engineers. Institute of Radio Engineers (IRE). 14 (4): 785–787. doi:10.1109/JRPROC.1926.221089. ISSN 0731-5996.
• Van Dyck, Arthur F. (February 1936). "Preferred Numbers". Proceedings of the Institute of Radio Engineers. Institute of Radio Engineers (IRE). 24 (2): 159–179. doi:10.1109/JRPROC.1936.228053. ISSN 0731-5996. […] choice of series is influenced by the fact that these units are sold with different standard tolerances, namely five, ten and twenty per cent, and there is a desire to have every unit manufactured, regardless of what its value may be, fall into some standard size and tolerance […]
• Buttner, Harold H.; Kohlhaas, H. T., eds. (1943). Reference Data for Radio Engineers (1 ed.). Federal Telephone and Radio Corporation (FTR). pp. 37–38. Retrieved 2020-01-03. (NB. This 1943 publication already shows a list of new "preferred values of resistance" following what was adopted by the IEC for standardization since 1948 and standardized as the E series of preferred numbers in IEC 63:1952. For comparison, it also lists "old standard resistance values" as follows: 50, 75, 100, 150, 200, 250, 300, 350, 400, 450, 500, 600, 750, 1000, 1200, 1500, 2000, 2500, 3000, 3500, 4000, 5000, 7500, 10000, 12000, 15000, 20000, 25000, 30000, 40000, 50000, 60000, 75000, 100000, 120000, 150000, 200000, 250000, 300000, 400000, 500000, 600000, 750000, 1 Meg, 1.5 Meg, 2.0 Meg, 3.0 Meg, 4.0 Meg, 5.0 Meg, 6.0 Meg, 7.0 Meg, 8.0 Meg, 9.0 Meg, 10.00 Meg.)
• Buttner, Harold H.; Kohlhaas, H. T.; Mann, F. J., eds. (1946). Reference Data for Radio Engineers (PDF) (2 ed.). Federal Telephone and Radio Corporation (FTR). pp. 53–54. Archived (PDF) from the original on 2018-05-16. Retrieved 2020-01-03. (NB. Shows a list of "old standard resistance values" vs. new "preferred values of resistance" following the later standardized E series of preferred numbers.)
• Van Dyck, Arthur F. (March 1951) [February 1951]. "Preferred Numbers". Proceedings of the Institute of Radio Engineers. Institute of Radio Engineers (IRE). 39 (2): 115. doi:10.1109/JRPROC.1951.230759. ISSN 0096-8390. […] For example, some years ago, the Radio-Television Manufacturers Association found it desirable to standardize the values of resistors. The ASA Preferred Numbers Standard was considered, but judged not to suit the manufacturing conditions and the buying practices of the resistor field at the moment, whereas a special series of numbers suited better. The special series was adopted and, since it was an official RTMA list, it has been utilized by later RTMA committees for other applications than resistors, although adopted originally because of seeming advantages for resistors. Ironically, the original advantages have largely disappeared through changes in resistor manufacturing conditions. But the irregular standard remains… […]
• ISO 17:1973-04 - Guide to the use of preferred numbers and of 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 R17-1956 - Preferred Numbers - Guide to the Use of Preferred Numbers and of Series of Preferred Numbers. 1956. (1955) and ISO R17/A1-1966 - Amendment 1 to ISO Recommendation R17-1955. 1966.)
• ISO 497:1973-05 - Guide to the choice of series of preferred numbers and of series containing more rounded values of preferred numbers. International Standards Organization (ISO). May 1973. Archived from the original on 2017-11-02. Retrieved 2017-11-02. (Replaced: ISO Recommendation R497-1966 - Preferred Numbers - Guide to the Choice of Series of Preferred Numbers and of Series Containing More Rounded Values of Preferred Numbers. 1966.)
• ANSI Z17.1-1973 - American National Standard for Preferred Numbers. American National Standards Institute (ANSI). 1973-09-05. (9 pages) (Replaced: ASA Z17.1-1958 - American National Standard for Preferred Numbers. 1958. Reaffirmed as USASI Z17.1-1958 in 1966 and named ANSI Z17.1-1958 since 1969.)
• Paulin, Eugen (2007-09-01). Logarithmen, Normzahlen, Dezibel, Neper, Phon - natürlich verwandt! [Logarithms, preferred numbers, decibel, neper, phon - naturally related!] (PDF) (in German). Archived (PDF) from the original on 2016-12-18. Retrieved 2016-12-18.
• Kienzle, Otto Helmut (2013-10-04) [1950]. Written at Hannover, Germany. Normungszahlen [Preferred numbers]. Wissenschaftliche Normung (in German). 2 (reprint of 1st ed.). Berlin / Göttingen / Heidelberg, Germany: Springer-Verlag OHG. ISBN 978-3-642-99831-7. Retrieved 2017-11-01. (340 pages)
• Bergtold, Fritz (1965). Mathematik für Radiotechniker und Elektroniker [Mathematics for Radio and Electronics Technicians] (in German) (3 ed.). München, Germany: Franzis-Verlag.
• Bauer, Horst, ed. (1995). Kraftfahrtechnisches Taschenbuch (in German) (22 ed.). Düsseldorf, Germany: Bosch, VDI-Verlag. ISBN 3-18419122-2.
• Ries, Clemens (1962). Normung nach Normzahlen [Standardization by preferred numbers] (in German) (1 ed.). Berlin, Germany: Duncker & Humblot Verlag. ISBN 3-42801242-9. (135 pages)
• Berg, Siegfried (1949). Angewandte Normzahl - Gesammelte Aufsätze [Applied preferred number - Collected papers] (in German). Berlin / Köln, Germany: Beuth-Vertrieb GmbH. Retrieved 2017-11-01. (191 pages)
• Tuffentsammer, Karl; Schumacher, P. (1953). "Normzahlen – die einstellige Logarithmentafel des Ingenieurs" [Preferred numbers - the engineer's single-digit logarithm table]. Werkstattechnik und Maschinenbau (in German). 43 (4): 156.
• Tuffentsammer, Karl (1956). "Das Dezilog, eine Brücke zwischen Logarithmen, Dezibel, Neper und Normzahlen" [The decilog, a bridge between logarithms, decibel, neper and preferred numbers]. VDI-Zeitschrift (in German). 98: 267–274.
• Strahringer, Wilhelm (1952). Zauberwelt der Normzahlen [Magic world of preferred numbers] (in German). Frankfurt a. Main, Germany: Verlags- und Wirtschaftsgesellschaft der Elektrizitätswerke m.b.H. (VWEW). (95 pages)