Long and short scales

The long and short scales are two different numerical systems used throughout the world:

Short scale is the English translation of the French term échelle courte.^[1] It refers to a system of numeric names in which every new term is 1 000 times greater than the previous term: "billion" means "a thousand millions" (10⁹), "trillion" means "a thousand billions" (10¹²), and so on.

Long scale is the English translation of the French term échelle longue. It refers to a system of numeric names in which every new term is 1 000 000 times greater than the previous term: "billion" means "a million squared" (10¹²), "trillion" means "a million to the third power" (10¹⁸), and so on.

Note that the difference between the two scales grows as numbers get larger. The long-scale billion is a thousand times larger than the short-scale billion, but the long-scale trillion is a million times larger than the short-scale trillion, and so on. For most of the 19th and 20th centuries, the United Kingdom uniformly used the long scale, while the United States of America used the short scale, so that the two systems were often referred to as "British" and "American" usage respectively. Today, the UK uses the short scale exclusively in official and mass media usage and, although some long-scale usage still continues, the terms "British" and "American" no longer reflect usage.

Contents 1 Comparison 2 Historical and language context 3 History 4 Current usage 4.1 Short scale countries 4.1.1 English language-speaking countries 4.1.2 Other languages and countries 4.1.3 Short scale use with long scale milliard 4.1.4 Short scale use but with other terminology 4.2 Long scale countries 4.3 Both long and short scales countries 4.4 Neither short nor long scale countries 5 Notes on current usage 5.1 English language countries 5.1.1 US usage 5.1.2 UK usage 5.1.3 Australian usage 5.1.4 Indian usage 5.2 Italian language usage 5.3 Esperanto language usage 5.4 Use of "thousand milliard" 6 Alternative approaches 7 See also 8 References 9 External links 9.1 Modern UK usage 9.2 Traditional UK usage

Comparison

For a more extensive table, see names of large numbers.

Value		Short Scale	Short Scale Logic	Long Scale	Long Scale Logic
10 0  =	1	one	1,000 1 - 1	one	1,000,000  0.0
10 3  =	1,000	thousand	1,000 1 + 0	thousand	1,000,000  0.5
10 6  =	1,000,000	million	1,000 1 + 1	million	1,000,000  1.0
10 9  =	1,000,000,000	billion	1,000 1 + 2	thousand million  (or milliard)	1,000,000  1.5
1012  =	1,000,000,000,000	trillion	1,000 1 + 3	billion	1,000,000  2.0
1015  =	1,000,000,000,000,000	quadrillion	1,000 1 + 4	thousand billion  (or billiard)	1,000,000  2.5
1018  =	1,000,000,000,000,000,000	quintillion	1,000 1 + 5	trillion	1,000,000  3.0

Since bi refers to 2 and tri refers to 3, the logic in the names is:

• short scale:  Billion is thousand times thousand ²  =  10⁹.  Trillion is thousand times thousand ³  =  10¹².

To get from one named order of magnitude to the next, multiply by a thousand.

• long scale:  Million is million¹  =  10⁶.  Billion is million²  =  10¹².  Trillion is million³  =  10¹⁸.

To get from one named order of magnitude to the next, multiply by a million. In other words, a billion (bi / two) has twice as many zeros as a million, and a trillion (tri / three) has three times as many zeros as a million, etc.

The old word "milliard", also found in many other languages, can be used for 10⁹ but is unknown in American English and not used in British English - however, 'Yard', which derives from 'milliard', is used on financial markets, as unlike 'billion', it is unambiguous.

Historical and language context

The existence of the different scales means that care must be taken when comparing large numbers between languages or countries, or when using old documents in countries where the dominant scale has changed over time. For example, British English documents from 1900 used long scale values, which are different from current British short scale usage. Both scales were used in France and Italy at various times in their history, but these countries (and most other European countries) now officially use long scale. In addition, apparently identical words in different languages may mean different values. For example, the French word 'billion' (10¹²) translates to the English word 'trillion' (usually 10¹²), not 'billion' (usually 10⁹). However, the German word 'Billion' and the Dutch word 'biljoen' both refer to 10¹². See Current usage below.

History

1475	French mathematician Jehan Adam recorded the words "bymillion" and "trimillion" as meaning 1012 and 1018 respectively.
1484	French mathematician Nicolas Chuquet, in his article "Triparty en la science des nombres"[2][3], used the words byllion, tryllion, quadrillion, quyllion, sixlion, septyllion, ottyllion, and nonyllion to refer to 1012, 1018, etc. Chuquet's work was not published until the 1870s, but most of it was copied without attribution by Estienne de La Roche and published in his 1520 book, L'arismetique.
1549	Jacques Peletier used the name milliard (“milliart”) for "Million de Millions", i.e. 1012. He attributed this meaning to earlier usage by Guillaume Budé (1467-1540), a French scholar.
During 1600s	The traditional six-digit-groups were split up. Therefore, in France and Italy, some scientists began using "billion" to mean 109. The majority either continued to say "thousand million", even with the three-digit-groups, or started using the Peletier term, milliard, as a synonym for "thousand million". This word was used in England but was widely adopted in France, Germany, Italy and the rest of Europe, for those keeping Chuquet's original long scale billion.
Mid 1700s	The short-scale meaning of the term "billion" was brought to the British American colonies.
Early 1800s	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 "a now obsolete system".
1926	H. W. Fowler's Modern English Usage 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."
1948	The 9th General Conference on Weights and Measures proposed the universal use of the long scale, inviting the short scale countries to return or convert. The proposal was considered but not adopted.
1960	The 11th General Conference on Weights and Measures adopted the International System of Units (SI), 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. [4].
1961	France confirmed their official usage of the long scale in the Journal Officiel (the official French Government gazette). (Décret 61-501, page 4587, and note 3 and erratum on page 7572. (French) [5])
1974	British prime minister Harold Wilson explained to the House of Commons that UK government statistics would from then on use the short scale [6][7]. During the last quarter of 20th century, most other English-speaking countries (Ireland, Australia, New Zealand, and South Africa) followed this lead and switched to the short scale use. However, in all 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 échelle courte (short scale) and échelle longue (long scale) to refer to the two numbering systems.

Current usage

Short scale countries

10⁶ = one million, 10⁹ = one billion, 10¹² = one trillion, etc.

English language-speaking countries

Most English-language countries use the short scale. For example:

Other languages and countries

 Brazil, where 10⁹ = bilhão, 10¹² = trilhão, etc.

Short scale use with long scale milliard

Some countries adopted the short scale for the seldom-occurring higher numbers (such as 10¹²), but kept the traditional word "milliard" instead of the short-scale "billion". Countries that adopt this usage include:

Short scale use but with other terminology

 Greece, which despite using the word εκατομμύριο ("hundred-myriad") for 10⁶ continues with terms for 10⁹ (δισεκατομμύριο, "bi-hundred-myriad"), 10¹² (τρισεκατομμύριο, "tri-hundred-myriad"), 10¹⁵ (τετράκις εκατομμύριο, "tetra-hundred-myriad"), and so on.^[8]

Long scale countries

10⁶ = one million, 10⁹ = one milliard / thousand million, 10¹² = one billion, 10¹⁵ = one billiard / thousand billion, 10¹⁸ = one trillion, etc.

Most countries and languages in the world use the traditional long scale, with many using a word similar to milliard to mean 10⁹, and/or a word similar to billion to mean 10¹². Some examples of long scale use, and the words used for 10⁹ and 10¹², are:

Both long and short scales countries

 Puerto Rico, a Spanish-speaking US Commonwealth country, generally uses short scale (10⁹ = billón, 10¹² = trillón) in economic and technical matters, but the long scale is used in publications intended for a Latin American audience outside Puerto Rico.

Neither short nor long scale countries

The following countries have their own numbering systems and use neither short nor long scales:

 China - see Chinese large numbers. - which features symbols for all the myriads up to 10 ⁴⁴.

 India - see Indian numbering system - which is commonly used. For Indian English speakers see below.

 Japan - see Japanese numerals: powers of 10 - which uses myriads as in Chinese.

 South Korea  North Korea - see Korean numerals - which uses a traditional myriad system for the larger numbers, with special words and symbols up to 10 ⁴⁸.

Notes on current usage

English language countries

Apart from the United States, the long scale was used for centuries in many English language countries before being superseded in recent times by short scale usage. Because of this history, some long scale use persists and the official status of the short scale in these countries is sometimes obscure.

US usage

In the United States of America, the short scale has been taught in school since the early 19th century. It is therefore used exclusively.

UK usage

"Billion" has meant 10⁹ in most sectors of official published writing for many years now. The UK government, BBC, and most other broadcast or published mass media, use the short scale exclusively in all contexts^[6]. Anyone using billion to mean 10¹² in British English may be misunderstood. However, this short scale usage is not uniformly accepted and the "traditional usage" of a billion to mean 10¹² is not unknown.

The long scale term "milliard", for 10⁹, is obsolete in British English (though its derivation "yard" is still used as slang in the London money, foreign exchange and bond markets). Before the recent widespread use of "billion" for 10⁹, UK usage generally referred to thousand million rather than milliard.

Australian usage

In Australia, some documents use the term thousand million for 10⁹ in cases where two amounts are being compared using a common unit of one 'million'. As of 1999, the Australian Government's financial department did not consider short scale to be standard, but used it occasionally ^[9]. The current recommendation by the Australian Department of Finance and Administration (formerly known as AusInfo), and the legal definition, is the short scale. Education, media outlets, and literature all use the short scale in line with other English-speaking countries.

Indian usage

Like the other English speaking countries, India also is currently strongly influenced by the standard U.S. short scale use. However (outside of financial media) the use of "billion" by Indian English speakers highly depends on their educational background. Some Indians may continue to use the traditional British long scale. In everyday life, Indians largely use their own system - for instance, Indian English commonly use the words lakh to denote 100 thousand, and crore to denote ten million (i.e. 100 lakhs).

Italian language usage

Italy – with France – was one of the two European countries partially converted to the short scale during the 19th century, but returned to the original long scale in 20th century. In Italian, the word bilione officially means 10¹². Colloquially, bilione can mean both 10⁹ and 10¹²; trilione both 10¹² and (rarer) 10¹⁸ and so on. Therefore, in order to avoid ambiguity, they are seldom used. Forms such as mille miliardi (a thousand milliards) for 10¹², un milione di miliardi for 10¹⁵, un miliardo di miliardi for 10¹⁸, mille miliardi di miliardi for 10²¹ are much more common ^[10].

Esperanto language usage

The official Esperanto words biliono, triliono etc. are ambiguous, and the inherently international nature of Esperanto communication compounds the problem by preventing any national presumption in favour of long or short scale. Ambiguity may be avoided by the use of the unofficial but generally-recognised suffix -iliono, whose function is analogous to the long scale, i.e. it is appended to a numeral indicating the power of a million, e.g. duiliono (from du meaning "two") = (10⁶)² = 10¹², triiliono = 10¹⁸, etc. Miliardo is an unambiguous term for 10⁹.

Use of "thousand milliard"

In those countries using the term milliard, the term "thousand milliard" is occasionally used, but only in budgetary contexts. One milliard currency units has become the major budgetary unit, as in the national debt of Germany at the end of 2004 was about 1418 milliard euros. In all other contexts in these same countries, 10¹² is always termed "billion" and not "thousand milliard".

Alternative approaches

Unambiguous ways of identifying large numbers include:

• In written communications, the simplest solution for moderately large numbers is simply to write the full amount- i.e., 1,000,000,000 rather than 1 milliard or 1 billion.
• Combinations of the unambiguous word 'million', for example: 10⁹ = "one thousand million"; 10¹² = "one million million". This becomes rather unwieldy for numbers above 10¹².
• Combination of numbers with more than 3 digits with million, as in 15,300 million.
• Scientific notation (standard form), including its engineering notation variant, for example 10⁹, 10¹², or in writing using the computer programming notation (1e9, 1e12, etc). This is the most common practice among scientists and mathematicians, and is both unambiguous and convenient.
• SI prefixes, for example, giga for 10⁹ and tera for 10¹². The International System of Units (SI) is independent of whichever scale is being used. In information technology contexts, these SI prefixes are sometimes used as powers of 2¹⁰ (= 1024) instead of powers of 10³ (= 1000), although there is a binary prefix system that can eliminate this ambiguity.

See also

References

1. ^ The first recorded use of the terms échelle courte and échelle longue was by the French mathematician Geneviève Guitel in 1975.
   See pp. 51–52 of Histoire comparée des numérations écrites, Geneviève Guitel, Éd. Flammarion, Paris, 1975 and also the chapter
   "Les grands nombres en numération parlée (État actuel de la question)", i.e. "The large numbers in oral numeration (Present state of the question)", pp. 566–574.
2. ^ Nicolas Chuquet's manuscript   published by www.miakinen.net. (French)
3. ^ Nicolas Chuquet's chapter  The names of great numbers in "Triparty en la science des nombres" (ISSN 9012-9458), transcription by Florencetime.net. (Widely in French.)
4. ^ http://www.bipm.org/en/si/si_brochure/chapter5/5-3-7.html
5. ^ Décret 61-501, page 4587, i.e. page 14 of 15 in this reference pdf and note 3 and erratum on page 7572, the last page of the pdf. (French)
6. ^ ^{a b} The Scotsman newspaper - 30 July 2004 article referencing Harold Wilson's 1974 decision to change numbering systems
7. ^ "Names for large numbers", Russ Rowlett, University of North Carolina, 1 Nov. 2001 - also contains a reference to Harold Wilson's 1974 decision. Otherwise, this article is an unadopted proposal for new names in the short scale system: it replaces billion by "gillion", cf. giga, then it uses Greek prefixes. Rowlett's proposal used the ambiguous terms American and European instead of Short and Long Scale respectively. Note that North America uses both scales depending on the language (American English and Canadian English vs Canadian-French); South America uses both scales (Brazilian Portuguese vs Spanish) and Europe uses both scales (British English and Irish English vs most languages of continental Europe).
8. ^ Greek Numbers and Numerals (Ancient and Modern), Harry Foundalis, part of a Greek tutorial at foundalis.com; accessed May 20, 2007.
9. ^ http://www.finance.gov.au/publications/pdf/prelims.pdf
10. ^ Federico Peiretti ("Archimede e i grandi numeri") (Italian)

External links

Modern UK usage

• Who wants to be a trillionaire?. - a 2007 BBC page on common modern usage of trillion.
• Linguist List 7.451.- A 1996 email list message, comparing the two scales usage in different languages.

Traditional UK usage

• AskOxford: How many is a billion?. - a traditional UK viewpoint.
• Million, Billion, Trillion .... - a 1999 webpage comparing the long and short scale numbers, which only considers traditional UK numbering.