industrial design Industrial design is a process of design applied to physical products that are to be manufactured by mass production. It is the creative act of determining and defining a product's form and features, which takes place in advance of the manufactu ...

, preferred numbers (also called preferred values or preferred series) 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 Standardization or standardisation is the process of implementing and developing technical standards based on the consensus of different parties that include firms, users, interest groups, standards organizations and governments. Standardizatio ...

, whether within a company or within an industry, and it is usually desirable in industrial contexts (unless the goal is

vendor lock-in In economics, vendor lock-in, also known as proprietary lock-in or customer lock-in, makes a customer dependent on a vendor for products, unable to use another vendor without substantial switching costs. The use of open standards and alternat ...

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 A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way—typically the largest numbers in the data are hundreds or even thousands of times larger than the smallest numbers. Such a ...

. 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) multiplied by the powers of a convenient basis, usually 10.

Renard numbers

In 1870

Charles Renard Charles Renard (1847–1905) born in Damblain, Vosges, was a French military engineer. Airships After the Franco-Prussian War of 1870-1871 he started work on the design of airships at the French army aeronautical department. Together with A ...

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, leading to the R5, R10, R20 and R40 scales, respectively. The factor between two consecutive numbers in a

Renard series Renard series are a system of preferred numbers dividing an interval from 1 to 10 into 5, 10, 20, or 40 steps. This set of preferred numbers was proposed in 1877 by French army engineer Colonel Charles Renard. His system was adopted by the ISO in ...

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 In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For ...

. This way, the maximum

relative error The approximation error in a data value is the discrepancy between an exact value and some ''approximation'' to it. This error can be expressed as an absolute error (the numerical amount of the discrepancy) or as a relative error (the absolute er ...

is minimized if an arbitrary number is replaced by the nearest Renard number multiplied by the appropriate power of 10. Example: 1.0, 1.6, 2.5, 4.0, 6.3

E series

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 The International Electrotechnical Commission (IEC; in French: ''Commission électrotechnique internationale'') is an international standards organization that prepares and publishes international standards for all electrical, electronic and ...

(IEC) began work on a new international standard in 1948. The first version of this IEC 63 (renamed into IEC 60063 in 2007) was released in 1952. 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).

1–2–5 series

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 A decade () is a period of ten years. Decades may describe any ten-year period, such as those of a person's life, or refer to specific groupings of calendar years. Usage Any period of ten years is a "decade". For example, the statement that "du ...

(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 international standard is a technical standard developed by one or more international standards organization, standards organizations. International standards are available for consideration and use worldwide. The most prominent such organization ...

. 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 The euro ( symbol: €; code: EUR) is the official currency of 19 out of the member states of the European Union (EU). This group of states is known as the eurozone or, officially, the euro area, and includes about 340 million citizens . ...

and sterling, 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 series (... 0.1 0.25 0.5 1 2.5 5 10 ...) is also used by currencies derived from the former

Dutch gulden The guilder ( nl, gulden, ) or florin was the currency of the Netherlands from the 15th century until 2002, when it was replaced by the euro. The Dutch name ''gulden'' was a Middle Dutch adjective meaning "golden", and reflects the fact that, wh ...

(

Aruban florin The florin (; sign: Afl.; code: AWG) or Aruban guilder is the currency of Aruba. It is subdivided into 100 cents. The florin was introduced in 1986, replacing the Netherlands Antillean guilder at par. The Aruban florin is pegged to the United St ...

Netherlands Antillean gulden The Netherlands Antillean guilder ( nl, gulden) is the currency of Curaçao and Sint Maarten, which until 2010 formed the Netherlands Antilles along with Bonaire, Saba, and Sint Eustatius. It is subdivided into 100 ''cents'' (Dutch plural form: ...

, Surinamese dollar), some Middle Eastern currencies ( Iraqi and Jordanian dinars,

Lebanese pound The pound or lira ( ar, ليرة لبنانية ''līra Libnāniyya''; French: ''livre libanaise''; abbreviation: LL in Latin, in Arabic, historically also £L, ISO code: LBP) is the currency of Lebanon. It was formerly divided into 100 pi ...

Syrian pound The Syrian pound or lira ( ar, الليرة السورية, al-līra as-sūriyya; abbreviation: LS or SP in Latin, ل.س in Arabic, historically also £S, and £Syr; ISO code: SYP) is the currency of Syria. It is issued by the Central Ban ...

), and the Seychellois rupee. However, newer notes introduced in Lebanon and Syria due to inflation follow the standard 1–2–5 series instead.

Convenient numbers

In the 1970s the

National Bureau of Standards The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical sci ...

(NBS) defined a set of convenient numbers to ease

metrication in the United States Metrication (or metrification) is the process of introducing the International System of Units, also known as SI units or the metric system, to replace a jurisdiction's traditional measuring units. U.S. customary units have been defined in ...

. 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.

Audio frequencies

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

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

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 3072 ... (×5) 5 10 20 40 80 160 320 640 1280 2560 5120 ... (×7) 7 14 28 56 112 224 448 896 1792 3584 7168 ... In

computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...

, widths and heights of raster images are preferred to be multiples of 16, as many compression algorithms (

JPEG JPEG ( ) is a commonly used method of lossy compression for digital images, particularly for those images produced by digital photography. The degree of compression can be adjusted, allowing a selectable tradeoff between storage size and imag ...

MPEG The Moving Picture Experts Group (MPEG) is an alliance of working groups established jointly by ISO and IEC that sets standards for media coding, including compression coding of audio, video, graphics, and genomic data; and transmission and f ...

) 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.

Paper documents, envelopes, and drawing pens

Standard metric paper sizes use the

square root In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...

of two () as factors between neighbouring dimensions rounded to the nearest mm (

Lichtenberg Lichtenberg () is the eleventh borough of Berlin, Germany. In Berlin's 2001 administrative reform it absorbed the former borough of Hohenschönhausen. Overview The district contains the Tierpark Berlin in Friedrichsfelde, the larger of Berlin ...

series,

ISO 216 ISO 216 is an international standard for paper sizes, used around the world except in North America and parts of Latin America. The standard defines the "A", "B" and "C" series of paper sizes, including A4, the most commonly available paper si ...

). 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 in ISO 9175-1: 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.

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: , , , , etc. Full f-stops are a

square root of 2 The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princi ...

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 Film speed is the measure of a photographic film's sensitivity to light, determined by sensitometry and measured on various numerical scales, the most recent being the ISO system. A closely related ISO system is used to describe the relation ...

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

, 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 In photography, shutter speed or exposure time is the length of time that the film or digital sensor inside the camera is exposed to light (that is, when the camera's shutter is open) when taking a photograph. The amount of light that rea ...

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, , , , , , , , , , of a second.

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). It restricts the list of allowed wine-bottle sizes to 0.1, 0.25 (), 0.375 (), 0.5 (), 0.75 (), 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 or , in some cases even , , or some other ratio of two small integers.