Long and short scales


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.
For whole numbers smaller than 1,000,000,000, 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, one billion is one thousand millions in the short scale, while it is one million millions in the long scale. The long scale system introduces new terms for the intervening values, typically replacing the word ending -ion with -iard.

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. For certain multipliers, including those for all numbers smaller than 109, both systems use the same names. 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, billion means a thousand millions, trillion means one thousand billions, and so on. Thus, an n-illion equals 103n+3.

Long scale

In the long scale, billion means 1,000,000,000,000 and trillion means one million billions, and so on. Therefore, an n-illion equals 106n. In some languages, the long scale introduces new names for the interleaving multipliers, replacing the ending -ion with -iard, for example, the next multiplier after million is milliard, after billion it is billiard. Hence, an n-iard equals 106n+3.

Use

Countries with usage of the long scale include most countries in continental Europe, and most that are French-speaking, Spanish-speaking and Portuguese-speaking countries.
The short scale is used in most English-speaking and Arabic-speaking countries, in Brazil, and in several other countries.
Number names are rendered in the language of the country, but are similar due to shared etymology. Some languages, particularly in East Asia and South Asia, have large number naming systems that are different from both the long and short scales, for example the Indian numbering system.
For most of the 19th and 20th centuries, the United Kingdom largely used the long scale, 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, and it is used for all official purposes. With very few exceptions, the British usage and American usage are now identical.
The first recorded use of the terms short scale and long scale was by the French mathematician Geneviève Guitel in 1975.
To avoid confusion resulting from the coexistence of short and long term in any language, the International System of Units recommends using the metric prefix to indicate orders of magnitude, but it is only relevant to scientific applications, and not to finance. Unlike words like billion and million, metric prefixes keep the same meaning regardless of the country and the language.

Comparison

The relationship between the numeric values and the corresponding names in the two scales can be described as:
The relationship between the names and the corresponding numeric values in the two scales can be described as:
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.
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 for one thousand long scale billions, and the word trilliard for one thousand long scale trillions, etc.

History

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, German Billion; Dutch biljoen; Swedish biljon; Finnish biljoona; Danish billion; Polish bilion, Spanish billón; Slovenian bilijon and the European Portuguese word bilião 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, and not billion.
On the other hand, the pre-1961 former French word billion, pre-1994 former Italian word bilione, Brazilian Portuguese word bilhão and the 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.
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. Maximus Planudes was among the first recorded users.
Late 14th century, a 17th-century copy of the original 14th-century allegorical narrative poem by William Langland The word million entered the English language. One of the earliest references is William Langland's Piers Plowman, 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.
Translation:
1484 French mathematician Nicolas Chuquet, in his article Le Triparty en la Science des Nombres par Maistre Nicolas Chuquet Parisien, 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, writing in Latin, used the term milliart to mean "ten myriad myriad" or 109 in his book De Asse et partibus eius Libri quinque.
Translation:
1549The influential French mathematician Jacques Pelletier du Mans used the name milliard 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. 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.
Translation:
18th centuryThe short-scale meaning of the term billion was brought to the British American colonies. As early as 1762, the dictionary of the Académie française defined billion as a term of arithmetic meaning a thousand millions.
1729The 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
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
by H. W. Fowler 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. 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.
1960The 11th General Conference on Weights and Measures adopted the International System of Units, with its own set of numeric prefixes. 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. The National Institute of Standards and Technology within the US also considers that it is best that they be avoided entirely.
1961The French Government confirmed their official usage of the long scale in the Journal officiel.
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. Hansard, for 20 December 1974, reported it
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 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 and short scale 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' German banknotes to be over-stamped as 'Eine Milliarde 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 Papiermark = 1012 Papiermark = 1 trillion Papiermark
1946
Hyperinflation in Hungary in 1946 led to the introduction of the 1020 pengo banknote.
100 million b-pengo = 100 trillion pengo = 1020 pengo = 100 quintillion pengo.
On 1 August 1946, the forint was introduced at a rate of
1 forint = 400 quadrilliard pengo = 4 × 1029 pengo = 400 octillion pengo.
1993 banknotes from 1993
Hyperinflation in Yugoslavia led to the introduction of 5 × 1011 dinar banknotes.
500 thousand million dinars = 5 × 1011 dinar banknotes = 500 billion 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.
2009s banknote from 2009
Hyperinflation in Zimbabwe led to banknotes of 1014 Zimbabwean dollars, marked "One Hundred Trillion Dollars", being issued in 2009, shortly ahead of the currency being abandoned., a new currency has yet to be announced – so foreign currencies are being used instead.
100 trillion Zimbabwean dollars = 1014 Zimbabwean dollars = 100 billion Zimbabwean dollars = 1027 pre-2006 Zimbabwean dollars = 1 quadrilliard pre-2006 Zimbabwean dollars.
2013, the combined total public debt of the United States stood at $17.078 trillion.
17 trillion US Dollars = 1.7 × 1013 US Dollars = 17 billion 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 مليار milyar, except for a few countries like Saudi Arabia and the UAE which use the word بليون 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 traditional 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 traditional long scale is used by most Continental European countries and by most other countries whose languages derive from Continental Europe. 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.

Spanish-speaking

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

French-speaking

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

Portuguese-speaking

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

Dutch-speaking

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

German-speaking

German-language countries and regions use the long scale with 109 = Milliarde, 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 Canadian French .
; ; English French

South African English Afrikaans
Economic & technical Latin American export publications

Using neither

The following countries use naming systems for large numbers that are not etymologically related to the short and long scales:
Country Number system Naming of large numbers
, , , Indian Numbering System For everyday use, but short or long scale may also be in use
Dzongkha numeralsTraditional system
Khmer numeralsTraditional system
Traditional myriad system for the larger numbers; special words and symbols up to 1088
Calque of the short scaleNames of the short scale have not been loaned but calqued into Greek, based on the native Greek word for million, εκατομμύριο ekatommyrio :
δισεκατομμύριο disekatommyrio "bi+hundred-myriad" = 109 ;
τρισεκατομμύριο trisekatommyrio "tri+hundred-myriad" = 1012 ;
:el:τετράκις εκατομμύριο|τετράκις εκατομμύριο tetrakis ekatommyrio "quadri+hundred-myriad" = 1015, and so on.
Lao numeralsTraditional system
, Traditional systems
Mongolian numeralsTraditional myriad system for the larger numbers; special words up to 1067
Thai numeralsTraditional system based on millions
Vietnamese numeralsTraditional system based on thousands

;Presence on most continents
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, English, South African EnglishAfrikaans, French, Portuguese
North AmericaAmerican English, Canadian EnglishU.S. Spanish, Canadian French, Mexican Spanish
South AmericaBrazilian Portuguese, English American Spanish, Dutch, French
AntarcticaAustralian English, British English, New Zealand English, RussianAmerican Spanish, French, Norwegian
AsiaBurmese, Hebrew, Indonesian, Malaysian English, Philippine English, Kazakh, Uzbek, KyrgyzPortuguese, Persian
EuropeBritish English, Welsh, Estonian, Greek, Latvian, Lithuanian, Russian, Turkish, UkrainianDanish, Dutch, Finnish, French, German, Icelandic, Italian, Norwegian, Polish, Portuguese, Spanish, Swedish and most other languages of continental Europe
OceaniaAustralian English, New Zealand EnglishFrench

Short scale

Long scale

Both long and short scale

Neither long nor short scale

Alternative approaches

Unambiguous ways of identifying large numbers include: