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.^[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.^[1][2]

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.

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.

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.

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.^[4]

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.^[1][2]

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.

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.^[5]

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.^{[6][7][8][9][10]}

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.^[1][2]

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.^[5] 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,^[4][11] whereas the United States used the short scale,^[11] 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.^[5]

• In long scale countries, milliard was defined to its current value of 10⁹, leaving billion at its original 10¹² value and so on for the larger numbers.^[5] Some of these countries, but not all, introduced new words billiard, trilliard, etc. as intermediate terms.^{[6][7][8][9][10]}
• In some short scale countries, milliard was defined to 10⁹ and billion dropped altogether, with trillion redefined down to 10¹² and so on for the larger numbers.^[5]
• In many short scale countries, milliard was dropped altogether and billion was redefined down to 10⁹, adjusting downwards the value of trillion and all the larger numbers.

Timeline

Date	Event
13th century	The word million was not used in any language before the 13th century. The monk and polymath Maximus Planudes (c. 1260–1305) was among the first recorded users of the word to document Mediterranean trade between Constantinople and Italian states.[5] 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 c. 1360–1387 in Middle English),[5] with Coueyte not his goodes For millions of moneye Translation: Covet not his goods for millions of money
1475	French 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] ... item noctes que le premier greton dembas vault ung, le second vault dix, le trois vault cent, le quart vult [sic] mille, le Ve vault dix M, le VIe vault cent M, le VIIe vault Milion, Le VIIIe vault dix Million, Le IXe vault cent Millions, Le Xe vault Mil Millions, Le XIe vault dix mil Millions, Le XIIe vault Cent mil Millions, Le XIIIe vault bymillion, Le XIIIIe vault dix bymillions, Le XVe vault cent mil [sic] bymillions, Le XVIe vault mil bymillions, Le XVIIe vault dix Mil bymillions, Le XVIIIe vault cent mil bymillions, Le XIXe vault trimillion, Le XXe vault dix trimillions ... Translation: ... Likewise, note that the first counter from the bottom is worth one, the 2nd is worth ten, the 3rd is worth one hundred, the 4th is worth one thousand, the 5th is worth ten thousand, the 6th is worth one hundred thousand, the 7th is worth a million, the 8th is worth ten millions, the 9th is worth one hundred millions, the 10th is worth one thousand millions, the 11th is worth ten thousand millions, the 12th is worth one hundred thousand million, the 13th is worth a bymillion, the 14th is worth ten bymillions, the 15th is worth one [hundred] bymillions, the 16th is worth one thousand bymillions, the 17th is worth ten thousand bymillions, the 18th is worth hundred thousand bymillions, the 19th is worth a trimillion, the 20th is worth ten trimillions ...
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.[22] Chuquet's original article was rediscovered in the 1870s and then published for the first time in 1880. ...[preder s'] Item l'on doit savoir que ung million vault mille milliers de unitez, et ung byllion vault mille milliers de millions, et [ung] tryllion vault mille milliers de byllions, et ung quadrillion vault mille milliers de tryllions et ainsi des aultres : Et de ce en est pose ung exemple nombre divise et punctoye ainsi que devant est dit, tout lequel nombre monte 745324 tryllions 804300 byllions 700023 millions 654321. Exemple : 745324'8043000'700023'654321 ... [sic] Translation: ...likewise, one should know that a million is worth a thousand thousand units, and a byllion is worth a thousand thousand millions, and tryllion is worth a thousand thousand byllions, and a quadrillion is worth a thousand thousand tryllions, and so on for the others. And an example of this follows, a number divided up and punctuated as previously described, the whole number being 745324 tryllions, 804300 byllions 700023 millions 654321. Example: 745324'8043000'700023'654321 ... [sic] 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] .. hoc est decem myriadum myriadas:quod vno verbo nostrates abaci studiosi Milliartum appellant:quasi millionum millionem Translation: .. this is ten myriad myriads, which in one word our students of numbers call Milliart, as if a million millions
1549	The 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[25]
17th century	With 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".[5] This meaning of milliard has been occasionally used in England,[11] 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.
1676	The first published use of milliard as 109 occurred in the Netherlands.[5][27] .. milliart/ofte duysent millioenen.. Translation: ..milliart / also thousand millions..
1729	The 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.[5]
Late 18th century	As 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 century	France 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[11] noted It should be remembered that "billion" does not mean in American use (which follows the French) what it means in British. For to us it means the second power of a million, i.e. a million millions (1,000,000,000,000); for Americans it means a thousand multiplied by itself twice, or a thousand millions (1,000,000,000), what we call a milliard. Since billion in our sense is useless except to astronomers, it is a pity that we do not conform. 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.
1948	The 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.[32][33]
1960	The 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]
1961	The 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,[13] reported in Hansard for 20 December 1974:[12] Mr. Maxwell-Hyslop asked the Prime Minister whether he would make it the practice of his administration that when Ministers employ the word 'billion' in any official speeches, documents, or answers to Parliamentary Questions, they will, to avoid confusion, only do so in its British meaning of 1 million million and not in the sense in which it is used in the United States of America, which uses the term 'billion' to mean 1,000 million. The Prime Minister: No. The word 'billion' is now used internationally to mean 1,000 million and it would be confusing if British Ministers were to use it in any other sense. I accept that it could still be interpreted in this country as 1 million million and I shall ask my colleagues to ensure that, if they do use it, there should be no ambiguity as to its meaning. 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.
1975	French mathematician Geneviève Guitel introduced the terms long scale (French: échelle longue) and short scale (French: échelle courte) to refer to the two numbering systems.[1][2]
1994	The Italian Government confirmed their official usage of the long scale.[10]

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	10 Milliarden Mark (1010 mark) stamp 1000 Mark German banknote, over-stamped in red with "Eine Milliarde Mark" (109 mark) Using German banknotes as wallpaper following the 1923 hyperinflation 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	1020 Hungarian pengő banknote issued in 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.
1993	Hyperinflation 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.
2009	Hyperinflation 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	As of 24 November 2022[update], 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

Most English-language countries and regions use the short scale with 10⁹ being billion. For example:^{[shortscale note 1]}

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:^{[shortscale note 5][46][47]}

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

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,^[49] 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.

Most Dutch-language countries and regions use the long scale with 10⁹ = miljard.^[50][51]

Most French-language countries and regions use the long scale with 10⁹ = milliard, for example:^{[longscale note 1][52][53]}

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

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

Most Spanish-language countries and regions use the long scale, for example:^{[longscale note 2][55][56]}

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

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

Country or territory	Short scale usage	Long scale usage
Canada[shortscale longscale note 1]	Canadian English (109 = billion, 1012 = trillion)	Canadian French (109 = milliard, 1012 = billion[61] or mille milliards).
Mauritius  Seychelles  Vanuatu	English (109 = billion, 1012 = trillion)	French (109 = milliard, 1012 = billion)
Namibia  South Africa [shortscale longscale note 2]	South African English (109 = billion, 1012 = trillion)	Afrikaans (109 = miljard, 1012 = biljoen)
Puerto Rico	Economic and technical (109 = billón, 1012 = trillón)	Latin American export publications (109 = millardo or mil millones, 1012 = billón)

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
Bangladesh,  India,  Maldives,  Nepal,  Pakistan	Indian numbering system	Traditional system for everyday use, but short or long scale may also be in use [other scale note 1]
Bhutan	Dzongkha numerals	Traditional system
Cambodia	Khmer numerals	Traditional system
People's Republic of China  Republic of China  South Korea  North Korea  Japan	East Asian numbering system: Chinese numerals Korean numerals Japanese numerals	Traditional myriad system for the larger numbers; special words and symbols up to 1068
Greece	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, εκατομμύριο ekatommyrio ("hundred-myriad", i.e. 100 × 10,000): δισεκατομμύριο disekatommyrio "bi+hundred-myriad" = 109 (short scale billion) τρισεκατομμύριο trisekatommyrio "tri+hundred-myriad" = 1012 (short scale trillion) τετράκις εκατομμύριο tetrakis ekatommyrio "quadri+hundred-myriad" = 1015 (short scale quadrillion), and so on.[65]
Laos	Lao numerals	Traditional system
Mongolia	Mongolian numerals	Traditional myriad system for the larger numbers; special words up to 1067
Sri Lanka	Sinhala numerals Tamil numerals	Traditional systems
Thailand	Thai numerals	Traditional system based on millions
Vietnam	Vietnamese numerals	Traditional system(s) based on thousands

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

• In written communications, the simplest solution for moderately large numbers is to write the full amount, for example 1,000,000,000,000 rather than 1 trillion (short scale) or 1 billion (long scale).
• Combinations of the unambiguous word million, for example: 10⁹ = "one thousand million"; 10¹² = "one million million".^[66]
• Scientific notation (also known as standard form or exponential notation, for example 1×10⁹, 1×10¹⁰, 1×10¹¹, 1×10¹², etc.), or its engineering notation variant (for example 1×10⁹, 10×10⁹, 100×10⁹, 1×10¹², etc.), or the computing variant E notation (for example 1e9, 1e10, 1e11, 1e12, etc.). This is the most common practice among scientists and mathematicians, and is both unambiguous and convenient.
• SI prefixes in combination with SI units, for example, giga for 10⁹ and tera for 10¹² can give gigawatt (=10⁹ W) and terawatt (=10¹² W). The International System of Units (SI) is independent of whichever scale is being used.^[34] Use with non-SI units (e.g. "giga-dollars", "megabucks") is possible. k€ and M€ is frequently encountered.

• BBC News article: "Is trillion the new billion?"
• Live-Counter.com: How to visualize large numbers:"