industrial design Industrial design is a process of design applied to physical Product (business), 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 adva ...

, preferred numbers (also called preferred values or preferred series) are standard

guideline A guideline is a statement by which to determine a course of action. It aims to streamline particular processes according to a set routine or sound practice. They may be issued by and used by any organization (governmental or private) to make ...

s for choosing exact product dimensions within a given set of constraints. Product developers must choose numerous lengths, distances, diameters, volumes, and other characteristic

quantities Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a u ...

. 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 (American English) or standardisation (British English) is the process of implementing and developing technical standards based on the consensus of different parties that include firms, users, interest groups, standards organiza ...

, 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 lockin, makes a customer dependent on a vendor for products, unable to use another vendor without substantial switching costs. The use of open standards and alternati ...

planned obsolescence In economics and industrial design, planned obsolescence (also called built-in obsolescence or premature obsolescence) is the concept of policies planning or designing a good (economics), product with an artificially limited Product lifetime, u ...

) # 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 method used to display numerical data that spans a broad range of values, especially when there are significant differences among the magnitudes of the numbers involved. Unlike a linear Scale (measurement) ...

. 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 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 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. 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; ) is an international standards organization that prepares and publishes international standards for all electrical, electronics, electronic and related technologies. IEC standards cover a va ...

(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 (from , , ) is a period of 10 years. Decades may describe any 10-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 ...

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

. 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 (currency symbol, symbol: euro sign, €; ISO 4217, currency code: EUR) is the official currency of 20 of the Member state of the European Union, member states of the European Union. This group of states is officially known as the ...

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 (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 (, ) or florin was the currency of the Netherlands from 1434 until 2002, when it was replaced by the euro. The Dutch name was a Middle Dutch adjective meaning 'golden', and reflects the fact that, when first introduced in 1434, its ...

(

Aruban florin The florin (; abbreviation: Afl.; ISO 4217, 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 Aruba currency exchang ...

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

Lebanese pound The lira or pound is the currency of Lebanon. It was formerly divided into 100 piastres (or ''qirsh'' in Arabic) but, because of high inflation during the Lebanese Civil War (1975–1990), subunits were discontinued. The plural of lira is eithe ...

Syrian pound The Syrian pound or lira (; abbreviation: LS or SP in Latin alphabet, Latin, ل.س in Arabic script, Arabic, historically also Pound sign, £S, and £Syr; ISO 4217, ISO code: SYP) is the currency of Syria. It is issued by the Central Bank of S ...

), 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 sc ...

(NBS) defined a set of convenient numbers to ease

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

. 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 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. 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 and art with the aid of computers. Computer graphics is a core technology in digital photography, film, video games, digital art, cell phone and computer displays, and many specialized applications. ...

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

JPEG JPEG ( , short for Joint Photographic Experts Group and sometimes retroactively referred to as JPEG 1) is a commonly used method of lossy compression for digital images, particularly for those images produced by digital photography. The degr ...

MPEG The Moving Picture Experts Group (MPEG) is an alliance of working groups established jointly by International Organization for Standardization, ISO and International Electrotechnical Commission, IEC that sets standards for media coding, includ ...

) 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 ratio The aspect ratio of a geometry, geometric shape is the ratio of its sizes in different dimensions. For example, the aspect ratio of a rectangle is the ratio of its longer side to its shorter side—the ratio of width to height, when the rectangl ...

s 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 y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...

of two () as factors between neighbouring dimensions rounded to the nearest mm ( Lichtenberg 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, which includes the A4, the most commonly availabl ...

). 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 the positive real number that, when multiplied by itself or squared, equals the number 2. It may be written as \sqrt or 2^. It is an algebraic number, and therefore not a transcendental number. Te ...

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". (Rounding is not exact in the cases of , , , , etc.) 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 introduced in 1974. A closely related system, also known as IS ...

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 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 (photography), shutter is open) when taking a photograph. The am ...

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.