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In industrial design, 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:

  1. 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 or planned obsolescence)
  2. 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.[1]

Comparison of preferred numbers of the 1-2-5, Renard and f-stop series on a logarithmic scale divided into 40 equal intervals (blue)

Renard numbers

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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. Example: 1.0, 1.6, 2.5, 4.0, 6.3

E series

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Main article: E series of preferred numbers
Graph of two decades of E12 series resistor values, which gives resistor values from 1 to 82 ohms (Ω)

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).

1–2–5 series

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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 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 1⁄4–1⁄2–1 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.

Convenient numbers

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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]

Audio frequencies

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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 third-octave bands used in audio tests and audio test equipment is: 20, 25, 31.5, 40, 50, 63, 80, 100, 125, 160, 200, 250, 315, 400, 500, 630, 800, 1000, 1250, 1600, 2000, 2500, 3150, 4000, 5000, 6300, 8000, 10000, 12500, 16000, 20000 Hz.[6]

Computer engineering

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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 ...
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.

Paper documents, envelopes, and drawing pens

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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 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

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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". (Rounding is not exact in the cases of f/1.2, f/3.5, f/5.6, f/22, etc.)

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.

Retail packaging

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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.

See also

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References

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Further reading

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Preferred number

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from Grokipedia
Preferred numbers are conventionally rounded values from , including powers of 10, designed to provide standardized increments for dimensions, sizes, and ratings in , , and design applications. These series, such as the , divide each decade (from 1 to 10) into a limited number of steps—typically 5, 10, 20, or 40—to ensure compatibility, simplify production, and minimize inventory variety while covering a wide range of practical needs. The concept originated in 1877 with French military engineer Charles Renard (1847–1905), who reduced over 400 unique lengths of mooring cables for observation balloons to just 17 standardized sizes by applying principles, thereby improving logistics and efficiency. Renard's system was later expanded and adopted internationally, becoming the basis for ISO 3:1973, which defines series of preferred numbers with specific ratios like the fifth root of 10 (approximately 1.585) for the R5 series. This standardization has influenced fields from to , where variants like the E series (e.g., E6 for resistors with 20% tolerance) adapt the principles for component values. Key Renard series include:
  • R5 (coarsest, 5 steps per decade): Values such as 1, 1.6, 2.5, 4.0, 6.3, scaled by powers of 10 (e.g., 10, 16, 25, 40, 63, 100); used for applications like lens focal lengths and capacitors where broader tolerances suffice.
  • R10 (10 steps): Includes 1, 1.25, 1.6, 2.0, 2.5, 3.15, 4.0, 5.0, 6.3, 8.0; common in general engineering for shafts, screws, and electrical ratings.
  • R20 (20 steps, finer increments of about 12%): Starts with 1, 1.12, 1.25, 1.4, 1.6, etc., up to 10; applied in precision tooling and machinery.
  • R40 (40 steps, approximately 6% steps): Provides even closer spacing for high-precision needs, though less common due to increased variety.
Complementary standards like ISO 17:1973 guide their selection, ensuring preferred numbers remain a cornerstone of global and design efficiency.

Fundamentals

Definition and Principles

Preferred numbers are a set of standardized values designed to limit the selection of sizes, dimensions, or other quantitative parameters in , , and design processes. They consist of conventionally rounded terms derived from geometrical series that include integral powers of 10, with specific ratios to ensure a logical and efficient progression of values. This standardization promotes interchangeability among components and products, facilitating compatibility across different systems and manufacturers. The core principles of preferred numbers rely on geometric progressions, where each subsequent value is obtained by multiplying the previous one by a constant ratio, typically approximating 101/k10^{1/k} for kk steps per decade (a factor of 10 in scale). This ratio ensures that the series covers a range from 1 to 10 (and multiples thereof) with evenly spaced intervals on a logarithmic scale, providing a balanced distribution of options. For instance, in a series with k=10k = 10, the common ratio r≈1.2589r \approx 1.2589, yielding values such as 1,r,r2,…1, r, r^2, \dots, up to just below 10, after which the pattern repeats by scaling with powers of 10. Such progressions create simple, memorable grading systems that can be extended indefinitely across scales. The adoption of preferred numbers yields significant benefits, including reduced inventory costs through fewer unique sizes and types required to meet diverse needs, thereby minimizing and production complexity. They simplify design decisions by offering a limited yet sufficient set of choices, enhancing compatibility and international harmonization in technical specifications. Additionally, the logarithmic spacing aligns with human perceptual tendencies for proportional changes, making the series intuitive for evaluating relative differences in dimensions or values.

Historical Development

The concept of preferred numbers originated in the late when French military Charles Renard proposed a system in 1877 to standardize the diameters of ropes and cords used in construction. Facing over 400 different sizes that complicated production and logistics for the French army's aeronautical efforts, Renard reduced them to just 17 standard values based on a , enabling efficient manufacturing while maintaining functional tolerances. This approach was initially adopted within French military and aeronautical applications, where it facilitated and streamlined supply chains in the late 1800s. Following , the need for global standardization to support efficient production and international trade accelerated the formalization of . The (ISO), established in 1947, formed Technical Committee ISO/TC 19 on preferred numbers in 1949, resuming pre-war efforts interrupted by the conflict. The was adopted as ISO Recommendation R3 in 1953, later evolving into the ISO 3, which defined series like R5, R10, R20, and R40 for broad engineering use. Concurrently, the (IEC) published its first edition of Publication 63 in 1952, introducing the E series (initially E3, E6, E12, E24) specifically for electronic components such as resistors and capacitors, adapting to ensure compatibility in tolerances. The systems continued to evolve through the mid-20th century to meet demands for greater precision. These developments reflected post-war priorities for rationalization, reducing manufacturing complexity while fostering interoperability across industries.

Core Series

Renard Series

The Renard series represents a system of preferred numbers developed by French aeronautical engineer Charles Renard in 1877 to standardize mechanical components, initially for military applications such as balloon ropes, by reducing the variety of sizes needed for production and logistics. These series are based on geometric progressions that divide each decade (from 1 to 10) into a fixed number of steps, promoting efficiency in mechanical engineering by limiting options while covering essential ranges. Standardized internationally under ISO 3:1973, the series include R5 (5 values per decade), R10 (10 values), R20 (20 values), and R40 (40 values), each with progressively finer step ratios derived from powers of 10. The step ratios for these series follow the general formula rk=101/(5×2k−1)r_k = 10^{1/(5 \times 2^{k-1})} for k=1,2,…k=1,2,\dots, where the exponent determines the ; for R5 (k=1k=1), the is \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=5&&&citation_type=wikipedia}}{10} \approx 1.58, approximately 60% increments between values, while for R10 (k=2k=2), it is 101/10≈1.2610^{1/10} \approx 1.26, providing about 25% increments. This structure ensures that products and quotients of terms remain within the series, facilitating . The values are conventionally rounded and extend across multiple decades by multiplying by powers of 10. In the R5 series, typical values for the 1–10 range are 1.0, 1.6, 2.5, 4.0, and 6.3, scaling to higher decades as 10, 16, 25, 40, and 63. The R10 series introduces intermediate steps for greater precision, such as 1.0, 1.25, 1.6, 2.0, 2.5, 3.15, 4.0, 5.0, 6.3, and 8.0, continuing up to 10 with similar spacing. Finer series like R20 and R40 follow analogous patterns with ratios of approximately 1.12 and 1.06, respectively, but are used less frequently for coarse applications. These series are primarily applied to standardize dimensions in per ISO guidelines, such as screw threads, pipe diameters, and tool specifications, ensuring interchangeability and without excessive variety. For instance, bolt sizes often align with R10 values like M4, M6, and M10. Unlike the E series, which offers finer, binary-aligned subdivisions for electronic components, the prioritizes coarser mechanical tolerances with ratios starting at 1.58.

E Series

The E series represents a system of preferred numbers standardized by the (IEC) under IEC 60063 for specifying nominal values of electronic components, including resistors and capacitors. It provides logarithmic progressions divided into series from E3 (3 values per decade) to E192 (192 values per decade), enabling efficient selection of component values that align with manufacturing tolerances and minimize inventory needs. The system is particularly suited for due to its binary-subdivided structure, which allows finer granularity compared to mechanical preferred number systems. The values in the E series follow a geometric progression with a base ratio of approximately 101/3≈2.15410^{1/3} \approx 2.154 for the coarsest series, ensuring even coverage across . Higher series are generated by further subdivisions, approximating factors like 2≈1.122\sqrt{2} \approx 1.122
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