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Renard series are a system of preferred numbers dividing an interval from 1 to 10 into 5, 10, 20, or 40 steps.[1] This set of preferred numbers was proposed ca. 1877 by French army engineer Colonel Charles Renard[2][3][4] and reportedly published in an 1886 instruction for captive balloon troops, thus receiving the current name in 1920s.[5] His system was adopted by the ISO in 1949[6] to form the ISO Recommendation R3, first published in 1953[7] or 1954, which evolved into the international standard ISO 3.[1] 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. One application of the Renard series of numbers is the current rating of electric fuses. Another common use is the voltage rating of capacitors (e.g. 100 V, 160 V, 250 V, 400 V, 630 V).

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

Base series

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The most basic R5 series consists of these five rounded numbers, which are powers of the fifth root of 10, rounded to two digits. The Renard numbers are not always rounded to the closest three-digit number to the theoretical geometric sequence:

R5: 1.00 1.60 2.50 4.00 6.30

Examples

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  • If some design constraints were assumed so that two screws in a gadget should be placed between 32 mm and 55 mm apart, the resulting length would be 40 mm, because 4 is in the R5 series of preferred numbers.
  • If a set of nails with lengths between roughly 15 and 300 mm should be produced, then the application of the R5 series would lead to a product repertoire of 16 mm, 25 mm, 40 mm, 63 mm, 100 mm, 160 mm, and 250 mm long nails.
  • If traditional English wine cask sizes had been metricated, the rundlet (18 gallons, ca 68 liters), barrel (31.5 gal., ca 119 liters), tierce (42 gal., ca 159 liters), hogshead (63 gal., ca 239 liters), puncheon (84 gal., ca 318 liters), butt (126 gal., ca 477 liters) and tun (252 gal., ca 954 liters) could have become 63 (or 60 by R″5), 100, 160 (or 150), 250, 400, 630 (or 600) and 1000 liters, respectively.

Alternative series

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If a finer resolution is needed, another five numbers are added to the series, one after each of the original R5 numbers, and one ends up with the R10 series. These are rounded to a multiple of 0.05. Where an even finer grading is needed, the R20, R40, and R80 series can be applied. The R20 series is usually rounded to a multiple of 0.05, and the R40 and R80 values interpolate between the R20 values, rather than being powers of the 80th root of 10 rounded correctly. In the table below, the additional R80 values are written to the right of the R40 values in the column named "R80 add'l". The R40 numbers 3.00 and 6.00 are higher than they "should" be by interpolation, in order to give rounder numbers.

In some applications more rounded values are desirable, either because the numbers from the normal series would imply an unrealistically high accuracy, or because an integer value is needed (e.g., the number of teeth in a gear). For these needs, more rounded versions of the Renard series have been defined in ISO 3. In the table below, rounded values that differ from their less rounded counterparts are shown in bold.

As the Renard numbers repeat after every 10-fold change of the scale, they are particularly well-suited for use with SI units. It makes no difference whether the Renard numbers are used with metres or millimetres. But one would need to use an appropriate number base to avoid ending up with two incompatible sets of nicely spaced dimensions, if for instance they were applied with both inches and feet. In the case of inches and feet a root of 12 would be desirable, that is, n√12 where n is the desired number of divisions within the major step size of twelve. Similarly, a base of two, eight, or sixteen would fit nicely with the binary units commonly found in computer science.

Each of the Renard sequences can be reduced to a subset by taking every nth value in a series, which is designated by adding the number n after a slash.[4] For example, "R10″/3 (1…1000)" designates a series consisting of every third value in the R″10 series from 1 to 1000, that is, 1, 2, 4, 8, 15, 30, 60, 120, 250, 500, 1000.

See also

[edit]
Preceded by
ISO 2 		Lists of ISOs
ISO 3 		Succeeded by
ISO 4
Preceded by
ISO 16 		Lists of ISOs
ISO 17 		Succeeded by
ISO 18


References

[edit]

Further reading

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Renard series

View on Grokipedia
from Grokipedia
The Renard series is a system of developed in 1877 by French military engineer Charles Renard to standardize the sizes of mooring cables for the , reducing approximately 425 different sizes to just 17 by creating a that divides the logarithmic interval from 1 to 10 into evenly spaced steps. This approach addressed logistical challenges in military supply by minimizing the variety of components needed while ensuring practical coverage of size requirements. The series is based on a common ratio derived from the fifth root of 10 (approximately 1.5849), allowing each subsequent number to increase by this factor, with every fifth step multiplying by 10 to span decades. Renard's innovation was formalized under the (ISO) in 1952 as the Renard preferred number series, denoted by "R" followed by the number of steps per decade (e.g., R5 for five steps, R10 for ten). The ISO standard includes four primary series—R5, R10, R20, and R40—covering progressively finer gradations from coarse (R5, with steps like 1, 1.6, 2.5, 4, 6.3, 10) to precise (R40, with 40 steps), plus an exceptional R80 series for specialized high-precision needs. These series are scalable by powers of 10, enabling application across multiple orders of magnitude, and subsets can be selected for specific tolerances, such as the R5/3 variant. Widely adopted in and , the Renard series facilitates in areas like electrical components (e.g., values in the E12 series, a close relative), mechanical fasteners (e.g., ISO metric bolt sizes such as M4, M6, M10), and optical equipment (e.g., lens focal lengths). By promoting geometric rather than arithmetic progressions, it optimizes production , , and interchangeability while accommodating manufacturing tolerances—ranging from ±24% for R5 to tighter values in denser series. The system's enduring influence is evident in its integration into slide rules, software tools, and modern standards like those for low-voltage equipment ratings.

History and Development

Origins in French Engineering

Charles Renard (1847–1905), a French military engineer and aeronautical pioneer, played a pivotal role in early developments during the late . As an officer in the French army's engineer corps, he focused on advancing captive balloon technology for military observation and signaling purposes in the , including work on structural components for moored balloons and early dirigibles. His expertise extended to constructing the first practical steerable dirigible, La France, in 1884, but his foundational contributions to standardization began earlier with balloon-related projects. The impetus for the Renard series stemmed from practical challenges in French , where the production of diverse component sizes for aeronautical equipment led to inefficiencies. Specifically, mooring cables and ropes for captive required over 400 distinct sizes, complicating , , and while driving up costs for the army. Renard recognized that many sizes fell within overlapping tolerance ranges, allowing for a reduced set of standardized values without compromising functionality, particularly for materials like ropes and wires used in balloon construction. In 1877, Renard introduced the preferred number system in a report commissioned by the French military, proposing a rational approach to select key sizes based on logarithmic progression to cover the range from 1 to 10 effectively. This work, conducted between 1877 and 1879, emphasized simplifying tolerances for practical applications. By around 1880, the system was integrated into French specifications for aeronautical materials, standardizing rope diameters and wire gauges to streamline production for balloon operations and reduce the variety of parts needed. These early implementations marked the Renard series as a tool for enhancing efficiency in an era of rapid aeronautical innovation.

International Adoption and

Following its initial development in France, the Renard series received post-World War I recognition through inclusion in national engineering standards, with early adoption in French norms (NF) during the 1920s to streamline manufacturing specifications. By the late 1940s, as international cooperation in standardization intensified, France proposed the series to the (ISO) in 1947, building on prior discussions within predecessor bodies like the International Federation of the National Standardizing Associations (ISA). The proposal gained momentum when, in July 1949, ISA Bulletin 11 explicitly recommended ISO adopt the R5, R10, R20, and R40 series for , a suggestion accepted by ISO leading to the publication of ISO Recommendation R3 in July 1953. This document established the core Renard series as a global framework for geometric progressions in values, emphasizing rounded terms for practical use across industries. ISO R17, issued in 1955 with amendments in 1966, further provided guidance on applying these series, solidifying their role in international norms. In 1973, ISO Recommendation R3 evolved into the full ISO 3, which specified the series of including integral powers of 10 and defined ratios for R5 through R40, with minor updates to enhance clarity and applicability. National bodies paralleled this progression: integrated the series into DIN standards by the 1950s (e.g., DIN 323 for and DIN 13 for metric threads in the ), the adopted equivalents in during the 1940s and formalized them in BS 350 (1959) for conversions and measurements, and the aligned via ANSI equivalents, such as ANSI B1.1 for unified threads in the and reaffirmation of ANSI B48.1 in 1947 for inch-millimeter relations. The Renard series' influence extended to metric systems by the mid-20th century, notably integrating into ISO/R 261 (1962, updated 1973) for preferred metric screw threads and related tolerances, enabling consistent sizing in ISO 261 and facilitating global in mechanical components. This adoption by ISO Technical Committee 1 on screw threads and other bodies underscored the series' role in reducing manufacturing variants while promoting efficiency.

The Renard Number System

Fundamental Principles

Preferred numbers form a standardized of discrete values designed to approximate a continuous range of measurements within a , typically spanning from 1 to 10, thereby enabling efficient selection in and contexts. This approach limits the variety of sizes or specifications to a manageable set, promoting uniformity while covering essential increments across decades by scaling through powers of 10. The primary objectives of , including the Renard series, are to minimize inventory requirements, reduce the proliferation of manufacturing variants, and streamline production processes by standardizing component sizes. By restricting choices to these predefined values, organizations can achieve cost savings in and assembly without compromising functionality, as the series ensures adequate coverage for most practical needs. A key design feature is that any value can be rounded to the nearest with a maximum relative deviation approximately equal to half the geometric step interval (e.g., ~24% for R5, ~10% for R10), ensuring practical coverage without excessive variety. In the Renard system, these are structured as , where successive terms increase by a constant derived from of 10, ensuring that no arbitrary value exceeds approximately 10% deviation from the closest preferred number in the R10 series (with analogous proportions for other densities). This facilitates logarithmic uniformity, making the series scalable and intuitive for applications requiring proportional scaling. The underlying tolerance philosophy emphasizes economic production by aligning manufacturing precision with the series' spacing; for instance, components in the R10 series are typically produced to a ±10% tolerance, which aligns with the step intervals to avoid unnecessary over-precision while ensuring interchangeability. This balance supports broader standardization efforts, as originally motivated by Charles Renard's work on efficient sizing in 19th-century engineering.

Construction of the Base Series

The base Renard series are constructed by generating a geometric progression of numbers within the decade from 1 to 10, using a constant ratio α=101/n\alpha = 10^{1/n}, where nn specifies the series (5 for R5, 10 for R10, 20 for R20, or 40 for R40). The sequence begins at r0=100/n=1r_0 = 10^{0/n} = 1 and proceeds as rk=10k/nr_k = 10^{k/n} for integer values of kk from 0 to nn, yielding rn=10n/n=10r_n = 10^{n/n} = 10. This logarithmic spacing divides the decade into nn equal intervals on a log scale, providing a foundation for standardized values that minimize variety while covering the range effectively. Following calculation, the terms are rounded to two or three significant digits to form practical , with adjustments made to the nearest suitable value that preserves monotonic increase and approximates uniform logarithmic intervals. The rounding prioritizes usability in contexts, such as component sizing, ensuring the relative ratios between consecutive terms remain close to α\alpha (e.g., approximately 1.58 for R5) and avoiding overlaps or gaps. This step is guided by ISO 3 standards, which specify the final rounded values to limit errors in the series progression. To cover broader scales, the base series (1 to 10) is extended by multiplying each rounded number by powers of 10, such as 10m10^m where mm is any (positive or negative), generating infinite sequences like 0.1, 1, 10, 100, and so on for each base term. This extension maintains the geometric property across decades, facilitating applications in dimensional . An illustrative derivation for the R5 base series (n=5n=5) proceeds as follows: k=0:100/5=1.000→1k=1:101/5≈1.585→1.6k=2:102/5≈2.512→2.5k=3:103/5≈3.981→4k=4:104/5≈6.310→6.3k=5:105/5=10.000→10\begin{align*} k=0: & \quad 10^{0/5} = 1.000 \to 1 \\ k=1: & \quad 10^{1/5} \approx 1.585 \to 1.6 \\ k=2: & \quad 10^{2/5} \approx 2.512 \to 2.5 \\ k=3: & \quad 10^{3/5} \approx 3.981 \to 4 \\ k=4: & \quad 10^{4/5} \approx 6.310 \to 6.3 \\ k=5: & \quad 10^{5/5} = 10.000 \to 10 \end{align*} These rounded values ensure the series steps by roughly 58% per term, aligning with the preferred numbers concept for rationalized production.

Specific Renard Series

R5 and R10 Series

The R5 series represents the coarsest of the Renard preferred number series, consisting of six values (corresponding to five steps) within each that approximate a with a common of approximately 1.58. These values are 1.00, 1.60, 2.50, 4.00, 6.30, and 10.00, extended across decades by multiplying by powers of 10 (e.g., 16, 25, 40, 63, 100, and further to 160, 250, etc.). The series is designed for applications requiring broad tolerances, corresponding to roughly 60% increments between consecutive values, which allows for significant variation in component sizing without compromising functionality in initial design stages.
DecadeR5 Values
1–101.00, 1.60, 2.50, 4.00, 6.30, 10.0
10–10016, 25, 40, 63, 100
100–1000160, 250, 400, 630, 1000
The R10 series provides a slightly finer granularity than R5, with eleven values (corresponding to ten steps) per following a of common approximately 1.26. Its values are 1.00, 1.25, 1.60, 2.00, 2.50, 3.15, 4.00, 5.00, 6.30, 8.00, and 10.00, similarly extended by powers of 10 (e.g., 12.5, 20, 31.5, 50, 80, 125, etc.). This series supports about 25% increments, making it suitable for tolerances where moderate precision is needed but still prioritizing ease of selection over exactness. Notably, the R5 series forms a of the R10 series, with every R5 value appearing in R10, allowing designers to default to R5 for the simplest approximations while escalating to R10 if additional steps are required. Both series emphasize simplicity in heavy engineering contexts, such as selecting nominal sizes for components like fasteners or plate thicknesses, where broad increments (60% for R5 and 25% for R10) accommodate manufacturing variations and reduce inventory complexity during preliminary design phases. They are particularly valued for their , ensuring even coverage across scales without gaps or excessive overlaps in tolerance bands.

R20 and R40 Series

The R20 series represents a finer subdivision of the Renard preferred numbers, dividing each decade from 1 to 10 into 20 steps to provide closer value spacing for applications requiring moderate precision. The series values, rounded for practicality, are: 1.00, 1.12, 1.25, 1.40, 1.60, 1.80, 2.00, 2.24, 2.50, 2.80, 3.15, 3.55, 4.00, 4.50, 5.00, 5.60, 6.30, 7.10, 8.00, 9.00, 10.00. These values extend across decades by multiplying by powers of 10, ensuring logarithmic uniformity. The series is designed for scenarios where a maximum relative deviation of approximately ±6% suffices, allowing without excessive proliferation of sizes. The R40 series further refines this approach, interpolating 40 steps per decade for even tighter spacing, suitable for applications demanding higher precision while maintaining compatibility with coarser series. Its values, also rounded, include: 1.00, 1.06, 1.12, 1.18, 1.25, 1.32, 1.40, 1.50, 1.60, 1.70, 1.80, 1.90, 2.00, 2.12, 2.24, 2.36, 2.50, 2.65, 2.80, 3.00, 3.15, 3.35, 3.55, 3.75, 4.00, 4.25, 4.50, 4.75, 5.00, 5.30, 5.60, 6.00, 6.30, 6.70, 7.10, 7.50, 8.00, 8.50, 9.00, 9.50, 10.00. Like the R20, it scales by powers of 10 and supports a tighter tolerance of about ±3%, reducing the need for custom values in precise designs. All R20 values are subsets of the R40 series, ensuring and seamless integration in multi-series efforts.
SeriesKey CharacteristicsExample Values (1 to 10)Approximate Tolerance
R2020 steps per ; moderate precision for general 1.00, 1.12, 1.25, ..., 9.00, 10.00±6%
R4040 steps per ; finer including R20 values1.00, 1.06, 1.12, ..., 9.50, 10.00±3%
These series prioritize practical over exact geometric ratios, facilitating adoption in where value density balances cost and precision. In contrast to coarser R5 and R10 series, R20 and R40 enable selections for tolerances below 10%, though R20 remains preferred over R40 to avoid over-specification.

Applications in Engineering

Electrical and Mechanical Components

The Renard series provides a standardized framework for selecting preferred values in the design and specification of electrical components, particularly resistors, capacitors, and inductors. These components often adhere to the E-series of preferred numbers, which is derived from the Renard system and defined in IEC 60063 for resistance and capacitance values. For instance, the E24 series, suitable for 5% tolerance levels, incorporates 24 values per decade that align closely with the R20 Renard series, ensuring geometric progression with steps of approximately 1.1 times the previous value to cover a wide range efficiently. This approach allows engineers to select from a limited set of values, such as 10, 11, 12, 13, 15, 16, 18, 20 ohms for resistors in the 10-20 range, minimizing custom fabrication while maintaining functional coverage across circuit requirements. In , the Renard series guides the preferred sizes for shafts, bearings, and to promote interchangeability and ease of assembly. Nominal diameters for shafts and in slide bearings, for example, follow the R20 series, where three shaft diameters per housing size are selected from steps like 10, 12.5, 16 mm to optimize load distribution and precision. Similarly, ISO 286 establishes tolerance grades for holes and shafts using steps that align with R10 progressions, such as fundamental deviations scaled in approximate 1.26 ratios to ensure fits like H7/g6 for precise mechanical interfaces without excessive variation. benefit from Renard-based module selections in standards like ISO 54, where pitch diameters progress in R10 increments to standardize tooling and reduce design iterations. Practical examples illustrate the series' utility in component specification. For power cables, wire diameters conform to the R20 series under IEC 60317-0-1, with preferred values like 0.20, 0.25, 0.315, 0.40 mm for enamelled wires, facilitating consistent electrical conductivity and insulation compatibility. In , speeds are rounded to a modified Renard progression per ISO 5800, yielding standard ISO ratings such as 100, 125, 160, 200 to approximate sensitometric measurements while simplifying camera settings and management. The primary benefit of applying the Renard series lies in reducing the number of stock variants required for components, transitioning from potentially infinite custom sizes to a finite set like 10 per in the R10 series. This rationalization lowers costs, streamlines , and enhances production efficiency, as evidenced in practices where preferred sizes for shafts and electrical values significantly reduce variant proliferation compared to arbitrary selections.

Standardization in Manufacturing

The Renard series facilitates in by providing a framework for preferred sizes and tolerances that streamline production processes, inventory management, and . Incorporated into international standards such as ISO 3, which defines the series of , the Renard system ensures consistent geometric progressions for dimensions across components like bolts and tools. This integration reduces the variety of parts needed, enabling manufacturers to optimize assembly lines and dies for efficiency. In , tolerance intervals based on Renard series, such as ±12% for the R10 series, allow for predictable deviations that align with capabilities without requiring custom specifications. Specific ISO standards embed Renard principles to support practices. For instance, ISO 801 establishes general principles for tolerancing, emphasizing the use of preferred values to simplify drawings and ensure interchangeability in production. ISO 2768 applies these concepts to tolerances, defining classes (fine, medium, coarse, very coarse) that correspond to practical increments derived from preferred number systems for linear and angular dimensions. Similarly, ISO 4759 specifies tolerances for fasteners like bolts and nuts, drawing on Renard-derived sizes (e.g., M4, M6, M10) to standardize product grades A, B, and C, which minimizes tooling variations and supports global supply chains. These standards promote cost reductions in production and logistics by limiting unique sizes to essential steps, as evidenced by applications in bolt where first-preference Renard values cover most needs. In global contexts, Renard series underpin regulatory frameworks, such as EU Machinery Directive 2006/42/EC, which harmonizes with ISO standards to ensure safety and compatibility across member states. In the automotive sector, standards for geometrical dimensioning and tolerancing (GD&T) incorporate preferred number progressions, facilitating efficient part design and assembly in supply chains. Overall, these integrations foster in global manufacturing, reducing waste and enabling seamless .

Comparisons and Alternatives

Relation to E-series

The E-series of preferred numbers originated as an adaptation of the Renard series specifically for electronic components, formalized in the (IEC) standard 60063, first published in 1952 and revised as IEC 60063:2015. This standard drew directly from Charles Renard's geometric progression principles, introduced in the 1870s, to define preferred values for resistors, capacitors, inductors, and Zener diodes by subdividing each decade (from 1 to 10) into a series of logarithmic steps. For instance, the E12 series, suitable for 10% tolerance components, derives from the Renard R10 series by expanding to 12 values per decade (e.g., 10, 12, 15, 18, 22, 27, 33, 39, 47, 56, 68, 82), while the E24 series, for 5% tolerance, builds on the R20 series with 24 values (e.g., adding intermediates like 11, 13, 16). Key differences between the Renard and E-series lie in the number of subdivisions and their alignment with manufacturing tolerances, with the E-series introducing finer granularity beyond the Renard R40 (40 steps). The E48, E96, and E192 series, for example, provide 48, 96, and 192 steps respectively, enabling precise values for 2%, 1%, and 0.5% tolerances, such as the E96 value of 9.53 (versus a pure geometric 9.55 in R40-like progression). While Renard series round values to simplify general applications, the E-series applies exact logarithmic spacing with minor adjustments for practicality, ensuring no gaps larger than the tolerance band—though higher E-series like E96 occasionally deviate slightly (e.g., 9.20 instead of 9.19) to optimize production. Both systems share a logarithmic basis, promoting uniform relative coverage across magnitudes to minimize inventory needs, but the E-series extends Renard principles to higher exponents (up to 10^6 or more) tailored for , whereas Renard remains more general for mechanical and other fields. In early , Renard values were adopted directly for component sizing before the 1952 IEC refinement into the E-series, which standardized global production and reduced variability in tolerances like 1% and 5%.

Other Preferred Number Systems

The 1-2-5 series represents a simple arithmetic-geometric progression of preferred numbers, comprising values such as 1, 2, 5, 10, 20, 50, and their decimal multiples, employed in early U.S. standards for rough approximations in and applications. This series gained prominence during the 1970s U.S. initiatives, where it facilitated the selection of convenient metric values for linear dimensions, force specifications, and building scales. In , standards such as DIN 323 (1974) outlined preferred number series for . These standards aligned with international norms for s. U.S. adaptations, exemplified by ASTM F568 for metric bolts, screws, and studs, incorporate the R10 series for nominal diameters (e.g., M1.6, M2, M2.5, M3, up to M100), while providing imperial equivalents through grade conversions and rounding for compatibility in mixed-unit environments. Unlike the Renard series, these alternative systems often lack even logarithmic spacing, resulting in uneven coverage across ranges; for instance, the 1-2-5 series exhibits larger proportional gaps beyond 5, reducing precision in finer gradations compared to geometrically balanced progressions.

Mathematical Properties

Geometric Progression Basis

The Renard series are constructed as designed to standardize numerical values in by dividing the logarithmic interval from 1 to 10 into a fixed number of steps. Specifically, for a series denoted R_n where n = 5, 10, 20, or 40, each term is derived from a constant common ρ = 10^{1/n}, ensuring that successive terms approximate a by this fixed factor. For the R5 series, ρ ≈ 1.5849, while for R10, R20, and R40, the ratios are approximately 1.2589, 1.1220, and 1.0593, respectively. This choice of ρ guarantees that applying the ratio n times yields exactly 10, as ρ^n = (10^{1/n})^n = 10. The unrounded terms of the series follow the equation r_k = 10^{k/n} for k ranging from 0 to n, starting at r_0 = 1 and ending at r_n = 10. These values are then rounded to two to produce practical while preserving ratios close to ρ; for example, the R5 series yields 1.0, 1.6, 2.5, 4.0, 6.3, and 10 after rounding. The full series extends indefinitely by multiplying by powers of 10 to cover all magnitudes, forming infinite geometric sequences in each . The geometric progression is formally proven by examining the ratios on a logarithmic scale: \log(r_{k+1} / r_k) = \log(10^{(k+1)/n} / 10^{k/n}) = (1/n) \log 10, which remains constant for all k, confirming uniform logarithmic spacing equivalent to an arithmetic progression in the logarithms of the terms. This property ensures that the series provides consistent relative increments across scales, independent of absolute magnitude. A key advantage of this geometric basis is its ability to cover each with precisely n steps, offering an efficient that minimizes the number of distinct values needed for broad coverage while allowing for controlled rounding deviations from the ideal ratios. The rounding strategy keeps these deviations small, with denser series like R40 achieving maximum ratio deviations below 10% relative to ρ, thereby maintaining the progression's integrity for precise applications.

Logarithmic Distribution and Coverage

The Renard series achieve uniform distribution on a by spacing the base-10 logarithms of their terms equally at intervals of k/nk/n, where k=0,1,…,nk = 0, 1, \dots, n and nn denotes the series designation (e.g., 5 for R5). This placement ensures that the points lie evenly across the log-decade from 0 to 1, making the series particularly suitable for scenarios with multiplicative tolerances, where relative variations are more relevant than absolute ones. The coverage of the series is evaluated using the maximum relative error metric ϵ=max⁡∣log⁡10(x/r)∣/(1/n)\epsilon = \max \left| \log_{10}(x / r) \right| / (1/n), where xx is an arbitrary value and rr is the nearest preferred number; for the ideal geometric construction, ϵ≈0.5\epsilon \approx 0.5, confirming optimal even spacing and minimal peak deviation on the logarithmic scale. Precision levels vary across series, with the relative tolerance defined as t=100.5/n−1t = 10^{0.5/n} - 1. The R5 series provides coverage with approximately ±24% tolerance (theoretical ≈ 26%), suitable for coarse approximations in early design phases, while the R40 series offers finer coverage at approximately ±3% tolerance (theoretical ≈ 2.9%), ideal for precise component selection.

References

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  4. https://themetricmaven.com/preferred-numbers-and-the-preferred-measurement-system/
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  12. https://asmedigitalcollection.asme.org/books/chapter-pdf/2793814/802612_ch4.pdf
  13. https://cadem.com/preferred-numbers-and-sizes-in-engineering/
  14. https://www.cobanengineering.com/Tolerances/ANSI-PreferedNumbers.asp
  15. https://www.audio-talk.co.uk/phpBB3/viewtopic.php?t=1927
  16. https://www.vishay.com/docs/28372/e-series.pdf
  17. https://www.mainasrl.it/wp-content/uploads/2020/01/RENK_Slide_Bearings_Type_E-Series_EG_ER.pdf
  18. https://www.finesz.com/pic/ISO286-1.pdf
  19. https://e-magnetica.pl/doku.php/enamelled_wire/iec_dimensions
  20. https://www.mathscinotes.com/2016/05/standard-resistor-values/
  21. https://eepower.com/resistor-guide/resistor-standards-and-codes/resistor-values/
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  23. https://boltport.com/specifications/astm-f568m/
  24. https://www.portlandbolt.com/technical/faqs/imperial-astm-equivalents-for-metric-grades/
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