Long and short scales explained

The long and short scales are two of several naming systems for integer powers of ten which use some of the same terms for different magnitudes.[1] [2]

Some languages, particularly in East Asia and South Asia, have large number naming systems that are different from both the long and short scales, such as the Indian numbering system and the Chinese, Japanese, or Korean numerals.

Much of the remainder of the world adopted either the short scale or the long scale for everyday counting powers of ten. Countries with the usage of the long scale include most countries in continental Europe and most that are French-speaking, German-speaking and Spanish-speaking.[3] Usage of the short scale is found in Brazil and in most English-speaking and Arabic-speaking countries.

For whole numbers smaller than 1,000,000,000 (109), such as one thousand or one million, the two scales are identical. For larger numbers, starting with 109, the two systems differ. For identical names, the long scale proceeds by powers of one million, whereas the short scale proceeds by powers of one thousand. For example, the short scale "one billion" (in many languages other than English called "one milliard", even on the short scale) means one thousand million (1,000,000,000), whereas in the long scale, "one billion" always means one million million (1,000,000,000,000). The long scale system employs additional terms for interleaved values, typically replacing the word ending "-ion" by "-iard". For numbers from 1012 and up, all the names in the short scale have different meanings than they have in the long scale.

The number systems of countries which use the short scale with "milliard" instead of "billion", do not have the word "billion", nor do they have any other words ending in "-iard". Thus, "milliard" can never mean anything other than 109, regardless of scale.

To avoid confusion resulting from the coexistence of the two scales, the International System of Units (SI) recommends using the metric prefix to indicate orders of magnitude, associated with physical quantities.

Scales

In both short and long scale naming, names are given each multiplication step for increments of the base-10 exponent of three, i.e. for each integer n in the sequence of multipliers 103n. Both systems use the same names for certain multipliers, including those for all numbers smaller than 109. The differences arise from the assignment of identical names to specific values of n, for numbers starting with 109, for which n=3. In the short scale system, the identical names are for n=3, 4, 5, ..., while the long scale places them at n=4, 6, 8, etc.

Short scale

In the short scale, a billion (in many countries called a "milliard") means a thousand millions (1,000,000,000, which is 109), a trillion means one thousand billions (1012), and so on. Thus, a short scale n-illion equals 103n+3.

Long scale

In the long scale, a billion means one million millions (1012) and a trillion means one million billions (1018), and so on. Therefore, a long scale n-illion equals 106n.

In some languages, the long scale uses additional names for the intermediate multipliers, replacing the ending -ion with -iard; for example, the next multiplier after million is milliard (109); after a billion it is billiard (1015). Hence, a long scale n-iard equals 106n+3.

Comparison

The relationship between the numeric values and the corresponding names in the two scales can be described as:

 Value in positional notation  Value in
scientific notation 
 Metric prefix  Short scale  Long scale 
PrefixSymbolNameLogicNameAlternative nameLogic
1100   oneone
10101deca datenten
100102hectohhundredhundred
1,000103kilokthousandthousand
1,000,000106megaMmillion1,000 × 1,0001million1,000,0001
1,000,000,000109gigaGbillion or milliard1,000 × 1,0002thousand millionmilliard1,000 × 1,000,0001
1,000,000,000,0001012teraTtrillion1,000 × 1,0003billion1,000,0002
1,000,000,000,000,0001015petaPquadrillion1,000 × 1,0004thousand billionbilliard1,000 × 1,000,0002
1,000,000,000,000,000,0001018exaEquintillion1,000 × 1,0005trillion1,000,0003
1,000,000,000,000,000,000,0001021zettaZsextillion1,000 × 1,0006thousand trilliontrilliard1,000 × 1,000,0003
1,000,000,000,000,000,000,000,0001024yottaYseptillion1,000 × 1,0007quadrillion1,000,0004
1,000,000,000,000,000,000,000,000,0001027ronnaRoctillion1,000 × 1,0008thousand quadrillionquadrilliard1,000 × 1,000,0004
1,000,000,000,000,000,000,000,000,000,0001030quettaQnonillion1,000 × 1,0009quintillion1,000,0005

The relationship between the names and the corresponding numeric values in the two scales can be described as:

 Name  Short scale  Long scale 
 Value in
scientific notation 
 Metric prefix Logic  Value in
scientific notation 
 Metric prefix Logic 
 Prefix Symbol Prefix Symbol
million106 megaM1,000 × 1,0001 106 megaM1,000,0001
billion109 gigaG1,000 × 1,0002 1012 teraT1,000,0002
trillion1012 teraT1,000 × 1,0003 1018 exaE1,000,0003
quadrillion1015 petaP1,000 × 1,0004 1024yottaY1,000,0004
quintillion1018 exaE1,000 × 1,0005 1030quettaQ1,000,0005
etc. For the next order of magnitude, multiply by 1,000For the next order of magnitude, multiply by 1,000,000

The root mil in million does not refer to the numeral, 1. The word, million, derives from the Old French, milion, from the earlier Old Italian, milione, an intensification of the Latin word, mille, a thousand. That is, a million is a big thousand, much as a great gross is a dozen gross or 12 × 144 = 1728.[4]

The word milliard, or its translation, is found in many European languages and is used in those languages for 109. However, it is not found in American English, which uses billion, and not used in British English, which preferred to use thousand million before the current usage of billion. The financial term yard, which derives from milliard, is used on financial markets, as, unlike the term billion, it is internationally unambiguous and phonetically distinct from million. Likewise, many long scale countries use the word billiard (or similar) for one thousand long scale billions (i.e., 1015), and the word trilliard (or similar) for one thousand long scale trillions (i.e., 1021), etc.[5] [6] [7] [8] [9]

History

Although this situation has been developing since the 1200s, the first recorded use of the terms short scale (French: échelle courte) and long scale (French: échelle longue) was by the French mathematician Geneviève Guitel in 1975.

The short scale was never widespread before its universal adoption in the United States. It has been taught in American schools since the early 1800s. It has since become common in other English-speaking nations and several other countries. For most of the 19th and 20th centuries, the United Kingdom largely used the long scale,[10] [11] whereas the United States used the short scale, so that the two systems were often referred to as British and American in the English language. After several decades of increasing informal British usage of the short scale, in 1974 the government of the UK adopted it,[12] and it is used for all official purposes.[13] [14] [15] [16] [17] [18] The British usage and American usage are now identical.

The existence of the different scales means that care must be taken when comparing large numbers between languages or countries, or when interpreting old documents in countries where the dominant scale has changed over time. For example, British English, French, and Italian historical documents can refer to either the short or long scale, depending on the date of the document, since each of the three countries has used both systems at various times in its history. Today, the United Kingdom officially uses the short scale, but France and Italy use the long scale.

The pre-1974 former British English word billion, post-1961 current French word billion, post-1994 current Italian word bilione, Spanish billón, German Billion, Dutch biljoen, Danish billion, Swedish biljon, Finnish biljoona, Slovenian bilijon, Polish bilion, and European Portuguese word bilião (with a different spelling to the Brazilian Portuguese variant, but in Brazil referring to short scale) all refer to 1012, being long-scale terms. Therefore, each of these words translates to the American English or post-1974 British English word: trillion (1012 in the short scale), and not billion (109 in the short scale).

On the other hand, the pre-1961 former French word billion, pre-1994 former Italian word bilione, Brazilian Portuguese word bilhão, and Welsh word biliwn all refer to 109, being short scale terms. Each of these words translates to the American English or post-1974 British English word billion (109 in the short scale).

The term billion originally meant 1012 when introduced.

Timeline
 Date Event
13th centuryThe word million was not used in any language before the 13th century. The monk and polymath Maximus Planudes (–1305) was among the first recorded users of the word to document Mediterranean trade between Constantinople and Italian states. Over the next two centuries, the term became widely accepted and was adopted by other Italian states, France and other European countries.
Late 14th century The word million entered the English language. One of the earliest references is William Langland's Piers Plowman (written –1387 in Middle English), with

Translation:

1475French mathematician Jehan Adam, writing in Middle French, recorded the words bymillion and trimillion as meaning 1012 and 1018 respectively in a manuscript Traicté en arismetique pour la practique par gectouers, now held in the Bibliothèque Sainte-Geneviève in Paris.[19] [20] [21]

Translation:

1484 French mathematician Nicolas Chuquet, in his article Le Triparty en la Science des Nombres par Maistre Nicolas Chuquet Parisien,[22] [23] [24] used the words byllion, tryllion, quadrillion, quyllion, sixlion, septyllion, ottyllion, and nonyllion to refer to 1012, 1018, ... 1054. Most of the work was copied without attribution by Estienne de La Roche and published in his 1520 book, L'arismetique. Chuquet's original article was rediscovered in the 1870s and then published for the first time in 1880.

Translation:

The extract from Chuquet's manuscript, the transcription and translation provided here all contain an original mistake: one too many zeros in the 804300 portion of the fully written out example: 745324'8043000 '700023'654321 ...

1516 French mathematician Budaeus (Guillaume Budé), writing in Latin, used the term milliart to mean "ten myriad myriad" or 109 in his book De Asse et partibus eius Libri quinque.[25]

Translation:

1549The influential French mathematician Jacques Pelletier du Mans used the name milliard (or milliart) to mean 1012, attributing the term to the earlier usage by Guillaume Budé
17th centuryWith the increased usage of large numbers, the traditional punctuation of large numbers into six-digit groups evolved into three-digit group punctuation. In some places, the large number names were then applied to the smaller numbers, following the new punctuation scheme. Thus, in France and Italy, some scientists then began using billion to mean 109, trillion to mean 1012, etc.[26] This usage formed the origins of the later short scale. The majority of scientists either continued to say thousand million or changed the meaning of the Pelletier term, milliard, from "million of millions" down to "thousand million". This meaning of milliard has been occasionally used in England, but was widely adopted in France, Germany, Italy and the rest of Europe, for those keeping the original long scale billion from Adam, Chuquet and Pelletier.
1676The first published use of milliard as 109 occurred in the Netherlands.[27]

Translation:

1729The short-scale meaning of the term billion had already been brought to the British American colonies. The first American appearance of the short scale value of billion as 109 was published in the Greenwood Book of 1729, written anonymously by Prof. Isaac Greenwood of Harvard College.
Late 18th centuryAs early as 1762 (and through at least the early 20th century), the dictionary of the Académie française defined billion as a term of arithmetic meaning a thousand millions.[28] [29] [30] [31]
Early 19th centuryFrance widely converted to the short scale, and was followed by the U.S., which began teaching it in schools. Many French encyclopedias of the 19th century either omitted the long scale system or called it "désormais obsolète", a now obsolete system. Nevertheless, by the mid 20th century France would officially convert back to the long scale.
1926 H. W. Fowler's A Dictionary of Modern English Usage noted

Although American English usage did not change, within the next 50 years French usage changed from short scale to long and British English usage changed from long scale to short.

1948The 9th General Conference on Weights and Measures received requests to establish an International System of Units. One such request was accompanied by a draft French Government discussion paper, which included a suggestion of universal use of the long scale, inviting the short-scale countries to return or convert.[32] This paper was widely distributed as the basis for further discussion. The matter of the International System of Units was eventually resolved at the 11th General Conference in 1960. The question of long scale versus short scale was not resolved and does not appear in the list of any conference resolutions.[33]
1960The 11th General Conference on Weights and Measures adopted the International System of Units (SI), with its own set of numeric prefixes.[34] SI is therefore independent of the number scale being used. SI also notes the language-dependence of some larger-number names and advises against using ambiguous terms such as billion, trillion, etc.[35] The National Institute of Standards and Technology within the US also considers that it is best that they be avoided entirely.[36]
1961The French Government confirmed their official usage of the long scale in the Journal officiel (the official French Government gazette).[37]
1974 British prime minister Harold Wilson explained in a written answer to the House of Commons that UK government statistics would from then on use the short scale, reported in Hansard for 20 December 1974:

The BBC and other UK mass media quickly followed the government's lead within the UK.

During the last quarter of the 20th century, most other English-speaking countries (Ireland, Australia, New Zealand, South Africa, Zimbabwe, etc.) either also followed this lead or independently switched to the short scale use. However, in most of these countries, some limited long scale use persists and the official status of the short scale use is not clear.

1975French mathematician Geneviève Guitel introduced the terms long scale (French: échelle longue) and short scale (French: échelle courte) to refer to the two numbering systems.
1994The Italian Government confirmed their official usage of the long scale.

As large numbers in natural sciences are usually represented by metric prefixes, scientific notation or otherwise, the most commonplace occurrence of large numbers represented by long or short scale terms is in finance. The following table includes some historic examples related to hyper-inflation and other financial incidents.

Timeline
 Date Event
1923

German hyperinflation in the 1920s Weimar Republic caused 'Eintausend Mark' (1000 Mark = 103 Mark) German banknotes to be over-stamped as 'Eine Milliarde Mark' (109 Mark). This introduced large-number names to the German populace.

The Mark or Papiermark was replaced at the end of 1923 by the Rentenmark at an exchange rate of

1 Rentenmark = 1 billion (long scale) Papiermark = 1012 Papiermark = 1 trillion (short scale) Papiermark

1946

Hyperinflation in Hungary in 1946 led to the introduction of the 1020 pengő banknote.

100 million b-pengő (long scale) = 100 trillion (long scale) pengő = 1020 pengő = 100 quintillion (short scale) pengő.

On 1 August 1946, the forint was introduced at a rate of

1 forint = 400 quadrilliard (long scale) pengő = 4 × 1029 pengő = 400 octillion (short scale) pengő.

1993Hyperinflation in Yugoslavia led to the introduction of 5 × 1011 dinar banknotes.

500 thousand million (long scale) dinars = 5 × 1011 dinar banknotes = 500 billion (short scale) dinars.

The later introduction of the new dinar came at an exchange rate of

1 new dinar = 1 × 1027 dinars = ~1.3 × 1027 pre 1990 dinars.

2009Hyperinflation in Zimbabwe led to banknotes of 1014 Zimbabwean dollars, marked "One Hundred Trillion Dollars" (short scale), being issued in 2009, shortly ahead of the currency being abandoned[38] [39] [40] after a final redenomination to the 'fourth dollar'. From 2013 to 2019 when the RTGS Dollar entered use, no new currency was announced, and so foreign currencies were used instead.

100 trillion (short scale) Zimbabwean dollars = 1014 Zimbabwean dollars = 100 billion (long scale) Zimbabwean dollars = 1027 pre-2006 Zimbabwean dollars = 1 quadrilliard (long scale) pre-2006 Zimbabwean dollars.

2022, the combined total public debt of the United States stood at $31.299 trillion.[41]

31 trillion (short scale) US Dollars = 3.1 × 1013 US Dollars = 31 billion (long scale) US Dollars

Current usage

Short scale users

English-speaking

Most English-language countries and regions use the short scale with 109 being billion. For example:

Arabic-speaking

Most Arabic-language countries and regions use the short scale with 109 being Arabic: مليار|rtl=yes Arabic: milyar, except for a few countries like Saudi Arabia and the UAE which use the word بليون Arabic: billion for 109. For example:

Other short scale

Other countries also use a word similar to trillion to mean 1012, etc. Whilst a few of these countries like English use a word similar to billion to mean 109, most like Arabic have kept a traditionally long scale word similar to milliard for 109. Some examples of short scale use, and the words used for 109 and 1012, are

Long scale users

The long scale is used by most Continental European countries and by most other countries whose languages derive from Continental Europe (with the notable exceptions of Albania, Greece, Romania, and Brazil). These countries use a word similar to billion to mean 1012. Some use a word similar to milliard to mean 109, while others use a word or phrase equivalent to thousand millions.

Dutch-speaking

Most Dutch-language countries and regions use the long scale with 109 = .[42] [43]

French-speaking

Most French-language countries and regions use the long scale with 109 = French: milliard, for example:

German-speaking

German-language countries and regions use the long scale with 109 = German: Milliarde.

Portuguese-speaking

With the notable exception of Brazil, a short scale country, most Portuguese-language countries and regions use the long scale with 109 = Portuguese: mil milhões or Portuguese: milhar de milhões.

Spanish-speaking

Most Spanish-language countries and regions use the long scale, for example:

Other long scale

Some examples of long scale use, and the words used for 109 and 1012, are

Using both

Some countries use either the short or long scales, depending on the internal language being used or the context.

Country or territoryShort scale usageLong scale usage
Canadian English (109 = billion, 1012 = trillion)Canadian French (109 = French: [[:fr:milliard|milliard]], 1012 = French: [[:fr:billion|billion]][45] or French: [[:fr:mille milliards|mille milliards]]).
English (109 = billion, 1012 = trillion)French (109 = French: milliard, 1012 = French: billion)
South African English (109 = billion, 1012 = trillion)Afrikaans (109 = Afrikaans: [[:af:miljard|miljard]], 1012 = Afrikaans: [[:af:biljoen|biljoen]])
Economic and technical (109 = Spanish; Castilian: billón, 1012 = Spanish; Castilian: trillón)Latin American export publications (109 = Spanish; Castilian: millardo or Spanish; Castilian: mil millones, 1012 = Spanish; Castilian: billón)

Using neither

The following countries use naming systems for large numbers that are not etymologically related to the short and long scales:

CountryNumber systemNaming of large numbers
,,,, Indian numbering systemTraditional system for everyday use, but short or long scale may also be in use
Traditional system
Traditional system
East Asian numbering system: Traditional myriad system for the larger numbers; special words and symbols up to 1068
Calque of the short scale Names of the short scale have not been loaned but calqued into Greek, based on the native Greek word for million, Greek, Modern (1453-);: εκατομμύριο Greek, Modern (1453-);: ekatommyrio ("hundred-myriad", i.e. 100 × 10,000):
  • Greek, Modern (1453-);: δισεκατομμύριο Greek, Modern (1453-);: disekatommyrio "bi+hundred-myriad" = 109 (short scale billion)
  • <
--τρισεκατομμύριο-->Greek, Modern (1453-);: τρισεκατομμύριο Greek, Modern (1453-);: trisekatommyrio "tri+hundred-myriad" = 1012 (short scale trillion)
  • Greek, Modern (1453-);: [[:el:τετράκις εκατομμύριο|τετράκις εκατομμύριο]] Greek, Modern (1453-);: tetrakis ekatommyrio "quadri+hundred-myriad" = 1015 (short scale quadrillion), and so on.[46]
Traditional system
Traditional myriad system for the larger numbers; special words up to 1067
Traditional systems
Traditional system based on millions
Traditional system(s) based on thousands

By continent

The long and short scales are both present on most continents, with usage dependent on the language used. Examples include:

ContinentShort scale usageLong scale usage
AfricaArabic (Egypt, Libya), South African EnglishFrench (Benin, Guinea), Portuguese (Mozambique)
North AmericaAmerican English, Canadian EnglishCanadian French, Mexican Spanish, U.S. Spanish
South AmericaBrazilian Portuguese, English (Guyana)American Spanish, Dutch (Suriname), French (French Guiana)
AntarcticaAustralian English, British English, New Zealand English, RussianAmerican Spanish (Argentina, Chile), French (France), Norwegian (Norway)
AsiaHebrew (Israel), Indonesian, Philippine EnglishPersian (Iran), Portuguese (East Timor, Macau)
EuropeBritish English, Russian, TurkishDutch, French, German, Italian, Portuguese, Spanish
OceaniaAustralian English, New Zealand EnglishFrench (French Polynesia, New Caledonia)

Notes on current usage

Neither long nor short scale

Alternative approaches

See also

External links

Notes and References

  1. Book: Guitel, Geneviève . Geneviève Guitel . Histoire comparée des numérations écrites . . 1975 . Paris . 51–52 . fr . 978-2-08-211104-1.
  2. Book: Guitel, Geneviève . Geneviève Guitel . Histoire comparée des numérations écrites . Flammarion . 1975 . Paris . 566–574 . "Les grands nombres en numération parlée (État actuel de la question)", i.e. "The large numbers in oral numeration (Present state of the question)" . fr . 978-2-08-211104-1.
  3. Web site: Authoritative Real Academia Española (RAE) dictionary: billón . 12 March 2015 . https://web.archive.org/web/20151104145435/http://lema.rae.es/drae/ . 4 November 2015 . dead .
  4. Book: Smith , David Eugene . David Eugene Smith . History of Mathematics . . II . first published 1925 . 1953 . 81 . 978-0-486-20430-7 .
  5. Web site: Wortschatz-Lexikon: Milliarde. Universität Leipzig

    Wortschatz-Lexikon

    . de. 19 August 2011. dead. https://web.archive.org/web/20110927052529/http://wortschatz.uni-leipzig.de/cgi-bin/wort_www.exe?site=1&Wort=Milliarde. 27 September 2011.
  6. Web site: Wortschatz-Lexikon: Billion. Universität Leipzig: Wortschatz-Lexikon. de. 19 August 2011. dead. https://web.archive.org/web/20110807144431/http://wortschatz.uni-leipzig.de/cgi-bin/wort_www.exe?site=1&Wort=Billion. 7 August 2011.
  7. Web site: Wortschatz-Lexikon: Billiarde. Universität Leipzig: Wortschatz-Lexikon. de. 28 July 2011. dead. https://web.archive.org/web/20110927052446/http://wortschatz.uni-leipzig.de/cgi-bin/wort_www.exe?site=1&Wort=Billiarde. 27 September 2011.
  8. Web site: Wortschatz-Lexikon: Trilliarde. Universität Leipzig: Wortschatz-Lexikon. de. 28 July 2011. dead. https://web.archive.org/web/20110927052502/http://wortschatz.uni-leipzig.de/cgi-bin/wort_www.exe?site=1&Wort=Trilliarde. 27 September 2011.
  9. Web site: Direttiva CEE / CEEA / CE 1994 n. 55, p.12 . it . . 21 November 1994 . 24 July 2011.
  10. Book: British-English usage of 'Billion vs Thousand million vs Milliard' . Google Books ngram viewer . Google Inc . 26 April 2014.
  11. Book: Fowler , H. W. . Henry Watson Fowler . A Dictionary of Modern English Usage . . 1926 . Great Britain . 52–53 . 978-0-19-860506-5.
  12. Web site: "BILLION" (DEFINITION) — HC Deb 20 December 1974 vol 883 cc711W–712W . Hansard Written Answers . . 20 December 1972 . 2 April 2009.
  13. News: O'Donnell . Frank . Britain's £1 trillion debt mountain – How many zeros is that? . . 30 July 2004 . 31 January 2008.
  14. News: Who wants to be a trillionaire?. 7 May 2007. BBC News. 11 May 2010. en-GB.
  15. Comrie . Bernard . Bernard Comrie . billion:summary . Linguist List . 24 March 1996 . 24 July 2011.
  16. Web site: Oxford Dictionaries: How many is a billion? . Oxford University Press . https://web.archive.org/web/20170112163426/https://en.oxforddictionaries.com/explore/how-many-is-a-billion . dead . 12 January 2017 . 7 May 2018.
  17. Web site: Oxford Dictionaries: Billion . Oxford University Press . https://web.archive.org/web/20110811080212/http://oxforddictionaries.com/definition/billion . dead . 11 August 2011 . 24 July 2011.
  18. Book: Nielsen , Ron . The Little Green Handbook . . 2006 . 290 . 978-0-312-42581-4 . registration .
  19. Book: Adam , Jehan . Jehan Adam . Traicté en arismetique pour la practique par gectouers... (MS 3143) . . Paris . 1475 . frm .
  20. Web site: HOMMES DE SCIENCE, LIVRES DE SAVANTS A LA BIBLIOTHÈQUE SAINTE-GENEVIÈVE, Livres de savants II . 2005 . Traicté en arismetique pour la practique par gectouers… . Bibliothèque Sainte-Geneviève . 25 October 2014 . fr . 25 October 2014 . https://web.archive.org/web/20141025140206/http://www-bsg.univ-paris1.fr/ExposVirtuelles/exposvirtuellesreserves/sciences/savants2.htm . dead .
  21. Thorndike . Lynn . Lynn Thorndike . The Arithmetic of Jehan Adam, 1475 A.D . The American Mathematical Monthly . 1926 . 1926 . January . 24–28 . . 10.2307/2298533 . 2298533 .
  22. Chuquet . Nicolas . Nicolas Chuquet . Le Triparty en la Science des Nombres par Maistre Nicolas Chuquet Parisien . Bulletino di Bibliographia e di Storia delle Scienze Matematische e Fisische . XIII . 1880 . 593–594 . . Bologna . 1880 . written 1484 . frm . 1123-5209 . 17 July 2011.
  23. Web site: Chuquet . Nicolas . Nicolas Chuquet . Le Triparty en la Science des Nombres par Maistre Nicolas Chuquet Parisien . miakinen.net . frm . 1 March 2008 . 1880 . written 1484.
  24. Flegg . Graham . Tracing the origins of One, Two, Three. . . 72 . 1032 . 747 . . 23–30 December 1976 . 0262-4079 . 17 July 2011 .
  25. Book: Budaeus , Guilielmus . Guillaume Budé . De Asse et partibus eius Libri quinque . 1516 . folio 93 . la .
  26. Book: Littré, Émile . Émile Littré . 1873–1874 . Dictionnaire de la langue française . Paris, France . L. Hachette . 347 . Ce n'est qu'au milieu du XVIIe siècle qu'il fut réglé que les tranches, au lieu d'être de six en six chiffres, seraient de trois en trois chiffres; ce qui revint à diviser par 1000 l'ancien billion, l'ancien trillion, etc. [It was only in the middle of the 17th century that it was settled that the slices, instead of being from six to six digits, would be from three to three digits; which resulted in dividing by 1000 the old billion, the old trillion, and so on.].
  27. Book: Houck . 1676 . Arithmetic . Netherlands . 2 .
  28. Book: . Dictionnaire de l'académie françoise . Paris, France . Institut de France . 177 . 1762 . 4th.
  29. Book: . Dictionnaire de l'Académie française . Paris, France . 189 . 1835 . 6th.
  30. Book: . Dictionnaire de l'Académie française . Paris, France . Institut de France . 182 . 1877 . 7th.
  31. Book: . Dictionnaire de l'Académie française . Paris, France . Institut de France . 144 . 1932–1935. 8th.
  32. Web site: Resolution 6 of the 9th meeting of the CGPM (1948) . . 7 August 2011.
  33. Web site: Resolution 6 of the 10th meeting of the CGPM (1954) . BIPM . 23 June 2012.
  34. Web site: Resolution 12 of the 11th meeting of the CGPM (1960) . BIPM . 28 July 2011.
  35. Book: The International System of Units (SI) . BIPM . 8 . May 2006 . 134 / 5.3.7 Stating values of dimensionless quantities, or quantities of dimension one . 92-822-2213-6 . https://web.archive.org/web/20110725080532/http://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf . 25 July 2011 . 24 July 2011 . dead .
  36. Thompson . Ambler . Taylor . Barry N. . 30 March 2008 . Guide for the Use of the International System of Units (SI), NIST SP – 811 . NIST . US . National Institute of Standards and Technology . 21 . 13 September 2014. 8 March 2021. https://web.archive.org/web/20210308121012/https://www.nist.gov/publications/guide-use-international-system-units-si?pub_id=200349. live.
  37. Décret 61-501. Journal Officiel. 4587 note 3a, and erratum on page 7572. French Government. 11 August 1961. fr. 31 January 2008. commissioned 3 May 1961 published 20 May 1961. dead. https://web.archive.org/web/20100120005624/http://www.ensmp.net/pdf/1961/decr-61-0501.pdf. 20 January 2010.
  38. News: BBC News: Zimbabweans play the zero game . BBC . 23 July 2008 . 13 July 2012.
  39. News: BBC News: Zimbabwe rolls out Z$100tr note . BBC . 16 January 2009 . 24 July 2010.
  40. News: BBC News: Zimbabwe abandons its currency . BBC . 29 January 2009 . 13 July 2012.
  41. Web site: US National Debt Clock . 24 Nov 2022.
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