In industrial design, 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. A guideline aims to streamline particular processes according to a set routine or sound practice. Guidelines may be issued by and used by any organization (governmental or pri ...

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 unit ...

. 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 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 ...

planned obsolescence In economics and industrial design, planned obsolescence (also called built-in obsolescence or premature obsolescence) is a policy of planning or designing a good (economics), product with an artificially limited Product lifetime, useful life o ...

) # 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) 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 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 e ...

. 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 e ...

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 (IEC) began work on a new international standard in 1948. The first version of this

IEC 63 The E series is a system of preferred numbers (also called preferred values) derived for use in electronic components. It consists of the E3, E6, E12, E24, E48, E96 and E192 series, where the number after the 'E' designates the quantity of ...

(renamed into

IEC 60063 The E series is a system of preferred numbers (also called preferred values) derived for use in electronic components. It consists of the E3, E6, E12, E24, E48, E96 and E192 series, where the number after the 'E' designates the quantity of ...

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. 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 An oscilloscope (informally a scope) is a type of electronic test instrument that graphically displays varying electrical voltages as a two-dimensional plot of one or more signals as a function of time. The main purposes are to display repetiti ...

. The denominations of most modern

currencies A currency, "in circulation", from la, currens, -entis, literally meaning "running" or "traversing" is a standardization of money in any form, in use or circulation as a medium of exchange, for example banknotes and coins. A more general def ...

, 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 (island), Saba, and Sint Eustatius. It is subdivided into 100 ''cents'' (Dutch ...

Surinamese dollar The Surinamese dollar (ISO 4217 code ''SRD'') has been the currency of Suriname since 2004. It is divided into 100 ''cent''. The Surinamese dollar is normally abbreviated with the dollar sign ''$'', or alternatively ''Sr$'' to distinguish it from ...

), 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 pias ...

, Syrian pound), and the

Seychellois rupee The rupee is the currency of the Seychelles. It is subdivided into 100 ''cents''. In the local Seychellois Creole (Seselwa) language, it is called the ''roupi'' and roupie in French. The ISO code is SCR. The abbreviation SR is sometimes used for ...

. 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 upright=1, The Smiley, smiley face in the top left corner is a raster image. When enlarged, individual pixels appear as squares. Enlarging further, each pixel can be analyzed, with their colors constructed through combination of the values for ...

are preferred to be multiples of 16, as many compression algorithms ( JPEG,

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). 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-stop In optics, the f-number of an optical system such as a camera lens is the ratio of the system's focal length to the diameter of the entrance pupil ("clear aperture").Smith, Warren ''Modern Optical Engineering'', 4th Ed., 2007 McGraw-Hill ...

s: , , , , 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 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 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.