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{{short description|Standard guidelines for choosing exact product dimensions within a given set of constraints}}
{{Use dmy dates|date=May 2019|cs1-dates=y}}
{{Use Oxford spelling|date=December 2011}}
In [[industrial design]], '''preferred numbers''' (also called '''preferred values''' or '''preferred series''') are standard [[guideline]]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]]. 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]] 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]]. 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.<ref name="GOVPUB-C13"/>
[[File:preferred_numbers.svg|thumb|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==
{{Main|Renard series}}

In 1870 [[Charles Renard]] proposed a set of preferred numbers.<ref name="Sizes_2014"/> His system was adopted in 1952 as international standard [[ISO 3]].<ref name="ISO_3_1973"/> 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==
{{Main|E series of preferred numbers}}
[[File:E12 values graph.png|thumb|right|Graph of two decades of E12 series resistor values, which gives resistor values from 1 to 82 [[ohm]]s (Ω)]]

The E series is another system of preferred numbers. It consists of the [[E1 series (preferred numbers)|E1]], [[E3 series (preferred numbers)|E3]], [[E6 series (preferred numbers)|E6]], [[E12 series (preferred numbers)|E12]], [[E24 series (preferred numbers)|E24]], [[E48 series (preferred numbers)|E48]], [[E96 series (preferred numbers)|E96]] and [[E192 series (preferred numbers)|E192 series]]. Based on some of the existing manufacturing conventions, the [[International Electrotechnical Commission]] (IEC) began work on a new international standard in 1948.<ref name="IEC_60063_1952"/> The first version of this [[IEC&nbsp;63]] (renamed into [[IEC&nbsp;60063]] in 2007) was released in 1952.<ref name="IEC_60063_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 [[electronics|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 [[IEC 60228|wires]]).

=={{anchor|1-2-5 series}} 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 (log scale)|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 [[oscilloscope#Graticule|oscilloscopes]].

The [[denomination (currency)|denominations]] of most modern [[Currency|currencies]], notably the [[euro]] and [[Pound sterling|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 dinar|Iraqi]] and [[Jordanian dinar|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==
{{Main|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|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&nbsp;mm.<ref name="GOVPUB-C13"/>

== 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&nbsp;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.<ref>{{Cite web|url=http://webshop.ds.dk/Files/Files/Products/18315_attachPV.pdf|title=ISO 266: Acoustics—Preferred frequencies}}</ref>

For example, a set of nominal center frequencies for use in audio tests and audio test equipment is:
{| class="wikitable"
|+Frequencies to be used in one-third octave analyzers in the audible range<ref>{{Cite book|title=Software-Based Acoustical Measurements|last=Miyara|first=Federico|publisher=Springer Nature|year=2017|isbn=978-3-319-55870-7|pages=21}}</ref>
!Nominal Center Frequency (Hz)
|-
|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
|}

== 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 ...

{| class="wikitable" align="right"
|+ Preferred aspect ratios
! !! 16: !! 15: !! 12:
<!--| (2²)² || 3 × 5 || 2² × 3-->
|-
! :8 <!--|| 2<sup>3</sup>-->
| 2:1 ||     || 3:2
|-
! :9<!--|| 3²-->
|16:9 || 5:3 || 4:3
|-
! :10<!--|| 2 × 5-->
| 8:5 || 3:2 || <!--not preferred-->
|-
! :12<!--|| 2² × 3-->
| 4:3 || 5:4 || 1:1
|}
In [[computer graphics]], widths and heights of [[Raster graphics|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 ratio]]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 ==
{{Main|Paper size}}
Standard metric paper sizes use the [[square root]] of two ({{sqrt|2}}) as factors between neighbouring dimensions rounded to the nearest mm ([[Georg Christoph Lichtenberg|Lichtenberg]] series, [[ISO 216]]). An A4 sheet for example has an aspect ratio very close to {{sqrt|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 {{sqrt|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&nbsp;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-number|f-stop]]s: {{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 encoding|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, {{frac|1|2}}, {{frac|1|4}}, {{frac|1|8}}, {{frac|1|15}}, {{frac|1|30}}, {{frac|1|60}}, {{frac|1|125}}, {{frac|1|250}}, {{frac|1|500}}, {{frac|1|1000}} 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<ref name="EEC_106"/>). It restricts the list of allowed wine-bottle sizes to 0.1, 0.25 ({{frac|1|4}}), 0.375 ({{frac|3|8}}), 0.5 ({{frac|1|2}}), 0.75 ({{frac|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 {{frac|2|3}} or {{frac|3|4}}, in some cases even {{frac|1|2}}, {{frac|4|5}}, or some other ratio of two small integers.

== See also ==
* [[Convenient number]]
* [[Nominal impedance]]
* [[Nominal size]]
* [[Preferred metric sizes]]

==References==
{{reflist|refs=
<ref name="GOVPUB-C13">{{cite web |author-first=Hans J. |author-last=Milton |id=NBS Technical Note 990 (Code: NBTNAE) |title=The Selection of Preferred Metric Values for Design and Construction |date=December 1978 |location=Washington, USA |publisher=[[The National Bureau of Standards]] (NBS) |website=U.S. Government Printing Office |url=https://www.gpo.gov/fdsys/pkg/GOVPUB-C13-f5fea679df4c3a1c2e3e1dd63488707c/pdf/GOVPUB-C13-f5fea679df4c3a1c2e3e1dd63488707c.pdf |access-date=2017-11-01 |url-status=live |archive-url=https://web.archive.org/web/20171101233709/https://www.gpo.gov/fdsys/pkg/GOVPUB-C13-f5fea679df4c3a1c2e3e1dd63488707c/pdf/GOVPUB-C13-f5fea679df4c3a1c2e3e1dd63488707c.pdf |archive-date=2017-11-01}}</ref>
<ref name="Sizes_2014">{{cite web |title=preferred numbers |publisher=Sizes, Inc. |date=2014-06-10 |orig-year=2000 |url=http://sizes.com/numbers/preferred_numbers.htm |access-date=2017-11-01 |url-status=live |archive-url=https://web.archive.org/web/20171101233141/https://sizes.com/numbers/preferred_numbers.htm |archive-date=2017-11-01}}</ref>
<ref name="EEC_106">{{cite web |title=COUNCIL DIRECTIVE of 19 December 1974 on the approximation of the laws of the Member States relating to the making-up by volume of certain prepackaged liquids (75/106/EEC) |date=2004-05-01 |orig-year=1974-12-19 |url=http://eur-lex.europa.eu/LexUriServ/site/en/consleg/1975/L/01975L0106-20040501-en.pdf |url-status=dead |archive-url=https://web.archive.org/web/20130516040109/http://eur-lex.europa.eu/LexUriServ/site/en/consleg/1975/L/01975L0106-20040501-en.pdf |archive-date=2013-05-16}}</ref>
<ref name="IEC_60063_1952">{{cite book |title=IEC 60063:1952 - Series of preferred values and their associated tolerances for resistors and capacitors |date=2007 |orig-year=1952-01-01 |edition=1.0 |publisher=[[International Electrotechnical Commission]] (IEC) |url=https://webstore.iec.ch/publication/12583 |access-date=2017-07-11 |url-status=live |archive-url=https://web.archive.org/web/20171101160911/https://webstore.iec.ch/webstore/webstore.nsf/xpPublication.xsp?Open&pubid=12583&test&SessionID=EV3AV7TCL8 |archive-date=2017-11-01}}</ref>
<ref name="ISO_3_1973">{{cite book |title=ISO 3:1973-04 - Preferred numbers - Series of preferred numbers |date=April 1973 |publisher=[[International Standards Organization]] (ISO) |url=https://www.iso.org/standard/3564.html |access-date=2017-11-02 |url-status=live |archive-url=https://web.archive.org/web/20171102224515/https://www.iso.org/standard/3564.html |archive-date=2017-11-02}} (Replaced: {{cite book |title=ISO Recommendation R3-1954 - Preferred Numbers - Series of Preferred Numbers |date=July 1954}} (1953))</ref>
}}

==Further reading==
* {{cite journal |author-first1=Clarence Floyd |author-last1=Hirshfeld |author-link1=Clarence Floyd Hirshfeld |author-first2=C. H. |author-last2=Berry |title=Size Standardization by Preferred Numbers |publisher=[[The American Society of Mechanical Engineers]] |journal=Mechanical Engineering |location=New York, USA |volume=44 |number=12 |date=1922-12-04 |pages=791– |url=https://archive.org/stream/sizestandardizat00hirsiala#page/n1/mode/1up}} [https://www.forgottenbooks.com/en/download/SizeStandardizationbyPreferredNumbers_10296778.pdf<!-- https://web.archive.org/web/20171102005125/https://www.forgottenbooks.com/en/download/SizeStandardizationbyPreferredNumbers_10296778.pdf -->]
* {{cite journal |author-first=Louis Alan |author-last=Hazeltine |author-link=Louis Alan Hazeltine |title=Preferred Numbers |date=January 1927 |orig-year=December 1926 |journal=[[Proceedings of the Institute of Radio Engineers]] |publisher=[[Institute of Radio Engineers]] (IRE) |volume=14 |issue=4 |doi=10.1109/JRPROC.1926.221089 |issn=0731-5996 |pages=785–787}}
* {{cite journal |author-first=Arthur F. |author-last=Van Dyck |title=Preferred Numbers |journal=[[Proceedings of the Institute of Radio Engineers]] |publisher=[[Institute of Radio Engineers]] (IRE) |volume=24 |issue=2 |pages=159–179 |date=February 1936 |issn=0731-5996 |doi=10.1109/JRPROC.1936.228053 |s2cid=140107818 |quote=[…] choice of series is influenced by the fact that these units are sold with different standard tolerances, namely five, ten and twenty per cent, and there is a desire to have every unit manufactured, regardless of what its value may be, fall into some standard size and tolerance […]}}
* {{cite book |title=Reference Data for Radio Engineers |editor-first1=Harold H. |editor-last1=Buttner
|editor-first2=H. T. |editor-last2=Kohlhaas |publisher=[[Federal Telephone and Radio Corporation]] (FTR) |edition=1 |date=1943 |pages=37–38 |url=https://books.google.com/books?id=bGUuAQAAIAAJ&q=%22preferred+values+of+resistance%22 |access-date=2020-01-03 }} (NB. This 1943 publication already shows a list of new "preferred values of resistance" following what was adopted by the [[International Electrotechnical Commission|IEC]] for standardization since 1948 and standardized as the [[E series of preferred numbers]] in IEC&nbsp;63:1952. For comparison, it also lists "old standard resistance values" as follows: 50, 75, 100, 150, 200, 250, 300, 350, 400, 450, 500, 600, 750, {{val|1000}}, {{val|1200}}, {{val|1500}}, {{val|2000}}, {{val|2500}}, {{val|3000}}, {{val|3500}}, {{val|4000}}, {{val|5000}}, {{val|7500}}, {{val|10000}}, {{val|12000}}, {{val|15000}}, {{val|20000}}, {{val|25000}}, {{val|30000}}, {{val|40000}}, {{val|50000}}, {{val|60000}}, {{val|75000}}, {{val|100000}}, {{val|120000}}, {{val|150000}}, {{val|200000}}, {{val|250000}}, {{val|300000}}, {{val|400000}}, {{val|500000}}, {{val|600000}}, {{val|750000}}, 1&nbsp;Meg, 1.5&nbsp;Meg, 2.0&nbsp;Meg, 3.0&nbsp;Meg, 4.0&nbsp;Meg, 5.0&nbsp;Meg, 6.0&nbsp;Meg, 7.0&nbsp;Meg, 8.0&nbsp;Meg, 9.0&nbsp;Meg, 10.00&nbsp;Meg.)
* {{cite book |title=Reference Data for Radio Engineers |editor-first1=Harold H. |editor-last1=Buttner
|editor-first2=H. T. |editor-last2=Kohlhaas |editor-first3=F. J. |editor-last3=Mann |publisher=[[Federal Telephone and Radio Corporation]] (FTR) |edition=2 |date=1946 |pages=53–54 |url=http://www.tubebooks.org/Books/FTR_ref_data.pdf |access-date=2020-01-03 |url-status=live |archive-url=https://web.archive.org/web/20180516201551/http://www.tubebooks.org/Books/FTR_ref_data.pdf |archive-date=2018-05-16}} (NB. Shows a list of "old standard resistance values" vs. new "preferred values of resistance" following the later standardized [[E series of preferred numbers]].)
* {{cite journal |author-first=Arthur F. |author-last=Van Dyck |title=Preferred Numbers |journal=[[Proceedings of the Institute of Radio Engineers]] |publisher=[[Institute of Radio Engineers]] (IRE) |volume=39 |issue=2 |orig-year=February 1951 |date=March 1951 |page=115 |issn=0096-8390 |doi=10.1109/JRPROC.1951.230759 |quote=[…] For example, some years ago, the [[Radio-Television Manufacturers Association]] found it desirable to standardize the values of resistors. The [[ASA Preferred Numbers Standard]] was considered, but judged not to suit the manufacturing conditions and the buying practices of the resistor field at the moment, whereas a special series of numbers suited better. The special series was adopted and, since it was an official RTMA list, it has been utilized by later RTMA committees for other applications than resistors, although adopted originally because of seeming advantages for resistors. Ironically, the original advantages have largely disappeared through changes in resistor manufacturing conditions. But the irregular standard remains… […]}}
* {{cite book |title=ISO 17:1973-04 - Guide to the use of preferred numbers and of series of preferred numbers |date=April 1973 |publisher=[[International Standards Organization]] (ISO) |url=https://www.iso.org/standard/3602.html |access-date=2017-11-02 |url-status=live |archive-url=https://web.archive.org/web/20171102224012/https://www.iso.org/standard/3602.html |archive-date=2017-11-02}} (Replaced: {{cite book |title=ISO Recommendation R17-1956 - Preferred Numbers - Guide to the Use of Preferred Numbers and of Series of Preferred Numbers |date=1956}} (1955) and {{cite book |title=ISO R17/A1-1966 - Amendment 1 to ISO Recommendation R17-1955 |date=1966}})
* {{cite book |title=ISO 497:1973-05 - Guide to the choice of series of preferred numbers and of series containing more rounded values of preferred numbers |date=May 1973 |publisher=[[International Standards Organization]] (ISO) |url=https://www.iso.org/standard/4548.html |access-date=2017-11-02 |url-status=live |archive-url=https://web.archive.org/web/20171102224132/https://www.iso.org/standard/4548.html |archive-date=2017-11-02}} (Replaced: {{cite book |title=ISO Recommendation R497-1966 - Preferred Numbers - Guide to the Choice of Series of Preferred Numbers and of Series Containing More Rounded Values of Preferred Numbers |date=1966}})
* {{cite book |title=ANSI Z17.1-1973 - American National Standard for Preferred Numbers |date=1973-09-05 |publisher=[[American National Standards Institute]] (ANSI)}} (9 pages) (Replaced: {{cite book |title=ASA Z17.1-1958 - American National Standard for Preferred Numbers |date=1958}} Reaffirmed as USASI Z17.1-1958 in 1966 and named ANSI Z17.1-1958 since 1969.)
* {{cite book |title=Logarithmen, Normzahlen, Dezibel, Neper, Phon - natürlich verwandt! |language=de |trans-title=Logarithms, preferred numbers, decibel, neper, phon - naturally related! |author-first=Eugen |author-last=Paulin |date=2007-09-01 |url=http://www.rechenschieber.org/Normzahlen.pdf |access-date=2016-12-18 |url-status=live |archive-url=https://web.archive.org/web/20161218223050/http://www.rechenschieber.org/Normzahlen.pdf |archive-date=2016-12-18}}
* {{cite book |title=Normungszahlen |language=de |trans-title=Preferred numbers |author-first=Otto Helmut |author-last=Kienzle |author-link=:de:Otto Kienzle (Ingenieur) |volume=2 |series=Wissenschaftliche Normung |publisher=[[Springer-Verlag OHG]] |location=Hannover, Germany |publication-place=Berlin / Göttingen / Heidelberg, Germany |edition=reprint of 1st |date=2013-10-04 |orig-year=1950 |isbn=978-3-642-99831-7 |url=https://books.google.com/books?id=-8q1BgAAQBAJ |access-date=2017-11-01}} (340 pages)
* {{cite book |author-first=Fritz |author-last=Bergtold |title=Mathematik für Radiotechniker und Elektroniker |language=de |trans-title=Mathematics for Radio and Electronics Technicians |edition=3 |date=1965 |publisher=[[Franzis-Verlag]] |location=München, Germany}}
* {{cite book |editor-first=Horst |editor-last=Bauer |title=Kraftfahrtechnisches Taschenbuch |language=de |edition=22 |date=1995 |isbn=3-18419122-2 |publisher=[[Robert Bosch GmbH|Bosch]], {{ill|VDI-Verlag|de}} |location=Düsseldorf, Germany}}
* {{cite book |author-first=Clemens |author-last=Ries |title=Normung nach Normzahlen |language=de |trans-title=Standardization by preferred numbers |publisher={{ill|Duncker & Humblot Verlag|de}} |location=Berlin, Germany |date=1962 |edition=1 |isbn=3-42801242-9}} (135 pages)
* {{cite book |author-first=Siegfried |author-last=Berg |title=Angewandte Normzahl - Gesammelte Aufsätze |language=de |trans-title=Applied preferred number - Collected papers |publisher=Beuth-Vertrieb GmbH |location=Berlin / Köln, Germany |date=1949 |url=https://books.google.com/books?id=5PBgAAAAMAAJ |access-date=2017-11-01}} (191 pages)
* {{cite journal |author-first1=Karl |author-last1=Tuffentsammer |author-first2=P. |author-last2=Schumacher |title=Normzahlen – die einstellige Logarithmentafel des Ingenieurs |language=de |trans-title=Preferred numbers - the engineer's single-digit logarithm table |journal=Werkstattechnik und Maschinenbau |volume=43 |number=4 |date=1953 |page=156}}
* {{cite journal |author-last=Tuffentsammer |author-first=Karl |title=Das Dezilog, eine Brücke zwischen Logarithmen, Dezibel, Neper und Normzahlen |language=de |trans-title=The decilog, a bridge between logarithms, decibel, neper and preferred numbers |journal=VDI-Zeitschrift |volume=98 |date=1956 |pages=267–274}}
* {{cite book |author-first=Wilhelm |author-last=Strahringer |title=Zauberwelt der Normzahlen |language=de |trans-title=Magic world of preferred numbers |publisher=Verlags- und Wirtschaftsgesellschaft der Elektrizitätswerke m.b.H. (VWEW) |location=Frankfurt a. Main, Germany |date=1952}} (95 pages)

[[Category:Numbers]]
[[Category:Industrial design]]
[[Category:Logarithmic scales of measurement]]