Long and short scales

The long and short scales ^[1] are two of several different large number naming systems used throughout the world for integer powers of ten (10). Many countries, including most in continental Europe, use the long scale whereas most English-speaking countries use the short scale. In all such countries, the number names are translated into the local language, but retain a name similarity due to shared etymology. Some languages, particularly in East Asia, have large number naming systems that are neither long nor short scale.

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

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

Up to but excluding a thousand million (< 10⁹), the two scales are identical. At and above a thousand million (≥ 10⁹), the two scales diverge by using the same words for different number values. These false friends^[2] can be a source of misunderstanding.

For most of the 19th and 20th centuries, the United Kingdom uniformly used the long scale,^[3] while the United States of America used the short scale,^[3] so that usage of the two systems was often referred to as British and American respectively. In 1974, the government of the UK switched to the short scale, a change that is reflected in its mass media and official usage.^[4][5][6][7] Although some residual usage of the long scale continues in the UK,^[8] the phrases British usage and American usage are no longer accurate nor helpful characterizations.

Usage of the two systems can be a subject of controversy. Differences in opinion as to which system should be used can evoke resentment between adherents, while national differences of any kind can acquire jingoistic overtones.^[9]

Contents 1 Comparison 2 History 2.1 Timeline 3 Current usage 3.1 Short scale countries and regions 3.1.1 English language-speaking countries and regions 3.1.2 Other short scale languages and countries 3.1.3 Short scale use but with other terminology 3.2 Long scale countries 3.3 Both long and short scales countries 3.4 Neither short nor long scale countries 4 Notes on current usage 4.1 English language countries 4.1.1 UK usage 4.1.2 US usage 4.1.3 Australian usage 4.1.4 Canadian usage 4.1.5 South African usage 4.1.6 Indian, Pakistani and Bangladeshi usage 4.2 Indonesian usage 4.3 Italian language usage 4.4 Spanish language usage 4.5 Esperanto language usage 4.6 Presence on most continents 5 Alternative approaches 6 See also 7 References 8 External links 8.1 Current UK usage 8.2 Traditional UK usage

Comparison

The same numerical value has two different names, depending on whether the value is being expressed in the long or short scale. Equivalently, the same name has two different numerical values depending on whether it is being used in the long or short scale.

Each scale has a logical justification to explain the use of each such differing numerical name and value within each scale - the short scale logic being based on powers of one thousand; the long scale logic being based on powers on one million. In both scales, bi- refers to 2 and tri- refers to 3, etc.

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

Value in Scientific notation	Value in numerals	Short Scale		Long Scale
		Name	Logic	Name	Logic
10 0  =	1	one		one
10 3  =	1,000	thousand		thousand
10 6  =	1,000,000	million		million
10 9  =	1,000,000,000	billion		milliard = thousand million
1012  =	1,000,000,000,000	trillion		billion
1015  =	1,000,000,000,000,000	quadrillion		billiard = thousand billion
1018  =	1,000,000,000,000,000,000	quintillion		trillion
1021  =	1,000,000,000,000,000,000,000	sextillion		trilliard = thousand trillion
etc.	etc.	etc.	etc.	etc.	etc.

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	Logic	Value in Scientific notation	Logic
Million	106		106
Billion	109		1012
Trillion	1012		1018
etc.	etc.	etc.	etc.	etc.
	To get from one named order of magnitude to the next: multiply by a thousand.		To get from one named order of magnitude to the next: multiply by a million.

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

The word "milliard", or its translation, is found in many European languages and is used in those languages for 10⁹. However, it is unknown 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 it is unambiguous, unlike "billion". Likewise, many long scale countries use the word billiard (or similar) for a thousand long scale billions (i.e. 10¹⁵), and the word trilliard for a thousand long scale trillions (i.e. 10²¹), 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 used both systems at various times in their history. Nowadays, the United Kingdom officially uses the short scale, but France and Italy (and most other European countries) 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; Spanish billón and the European Portuguese word bilião (note the alternate spelling to the Brazilian Portuguese variant) all refer to 10¹², being long scale terms. Therefore, each of these words translates to the American English or post-1974 modern 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 the Welsh word biliwn all refer to 10⁹, being short scale terms. Each of these words translates to the American English or post-1974 modern British English word "billion" (10⁹ in the short scale).

The terms "billion" and "milliard" both originally meant 10¹² when introduced.

In long scale countries, "milliard" was redefined down to its current value of 10⁹, leaving "billion" at its original 10¹² value and so on for the larger numbers. New terms: "billiard", "trilliard", etc. were introduced.

In some short scale countries, "milliard" was redefined down to 10⁹, with "trillion" redefined down to 10¹² and so on for the larger numbers.

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.

See Current usage below.

Timeline

Date	Event
13th century	The word "million" was not used before the 13th century, and Maximos Planoudes (cir 1260-1330) was among the first recorded users [10].
1475	French mathematician Jehan Adam recorded the words "bymillion" and "trimillion" as meaning 1012 and 1018 respectively.[11] "... item noctes que le premier greton dembas vault ung, le second vault [...here some words seem to be omitted...] cent, le quart vult 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 Mill Millions, Le XIe vault dix mill Millions, Le XIIe vault Cent mil Millions, Le XIIIe vault bymillion, Le XIIIIe vault dix bymillions, Le XVe vault mil 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: "... Also note that the first counter from the bottom stands for one, the 2nd stands for [...] one hundred, the 4th stands for one thousand, the 5th stands for ten thousand, the 6th stands for one hundred thousand, the 7th stands for a million, the 8th stands for ten millions, the 9th stands for one hundred millions, the 10th stands for one thousand millions, the 11th stands for ten thousand millions, the 12th stands for one hundred thousand million, the 13th stands for a billion, the 14th stands for ten billions, the 15th stands for one hundred [the "mil" in the original is an obvious error] billions, the 16th stands for one thousand billions, the 17th for ten thousand billions, the 18th stands for hundred thousand billions, the 19th stands for a trimillion, the 20th stands for ten trimillions ..."
1484	French mathematician Nicolas Chuquet, in his article "Triparty en la science des nombres",[12] [13] 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. [11] "... Item lon 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 ..." [12] Translation: "...Item: one should know that a million is worth a thousand thousand units, and a byllion is worth a thousand thousand millions, and tryillion 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 seven hundred forty-five thousand three hundred and twenty-four tryllions, 804300 byllions 700023 millions 654321. Example: 745324'8043000'700023'654321 ..."
1549	Jacques Pelletier du Mans 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.
17th century	The traditional six-digit groups were split up into three-digit groups. 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 new meaning of "milliard" was occasionally 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. The first published use of milliard as 109 occurred in 1676 in the Netherlands (Houck's Arithmetic) [10].
Early 18th century	The short-scale meaning of the term "billion" was brought to the British American colonies, and first appeared in published form in the Greenwood book of 1729, written anonymously by Prof. Isaac Greenwood of Harvard[10]
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 "a now obsolete system".
1926	H. W. Fowler's A Dictionary of 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." [3]
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. [14].
1961	The French Government confirmed their official usage of the long scale in the Journal Officiel (the official French Government gazette). [15]
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 the 20 December 1974, reported it: "Mr. Maxwell-Hyslop asked the Prime Minister whether he will 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."[4][5][16] During the last quarter of the 20th century, most other English-speaking countries (Ireland, Australia, New Zealand, South Africa, Zimbabwe, etc.) either 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 échelle courte (short scale) and échelle longue (long scale) to refer to the two numbering systems. [1]
1994	The Italian government confirmed their official usage of the long scale [17].

The origin of the word "million" seems to derive from the Old French Milion, thought to derive from Old Italian milione, an intensification of mille, a thousand. That is, a "million" is a "big thousand", much as "1728" (a dozen gross) is a "great gross".

Current usage

Short scale countries and regions

English language-speaking countries and regions

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

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

Other short scale languages and countries

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

Other countries also use a word similar to trillion to mean 10¹², etc. While a few of these countries use a word similar to billion to mean 10⁹, most 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:

Short scale use but with other terminology

 Greece (εκατομμύριο ekatommyrio "hundred-myriad" = 10⁶; δισεκατομμύριο disekatommyrio "bi+hundred-myriad" = 10⁹; τρισεκατομμύριο trisekatommyrio "tri+hundred-myriad" = 10¹²; τετράκις εκατομμύριο tetrakis ekatommyrio "quadri+hundred-myriad" = 10¹⁵, and so on.) ^[21]

Long scale countries

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

The traditional 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 Greece, Romania, Bulgaria 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 something equivalent to thousand millions. Some examples of long scale use, and the words used for 10⁹ and 10¹², are:

Both long and short scales countries

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

10⁶ = one million, 10⁹ = EITHER one billion (Short Scale) OR one milliard / thousand million (Long Scale), 10¹² = EITHER one trillion (Short Scale) OR one billion (Long Scale), etc.

Country	Short scale usage	Long scale usage
Canada	Canadian English (billion = 109, trillion = 1012)	Canadian French (milliard = 109, billion = 1012).
Puerto Rico	economic & technical (billón = 109, trillón = 1012)	Latin american export publications (millardo or mil millones = 109, billón = 1012)
South Africa [22] [23]	South African English (billion = 109, trillion = 1012)	Afrikaans (miljard = 109, biljoen = 1012)

Neither short nor long scale countries

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

Country	Main article	Notes
Bangladesh,  India,  Myanmar (Burma),  Nepal,  Pakistan	Indian numerals	For everyday use, but short or long scale may also be in use; see below.
China (People's Republic),  Japan, North Korea,  South Korea	Chinese numerals, Japanese numerals, Korean numerals	All use a traditional myriad system for the larger numbers; special words and symbols up to 1088.

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 anglophone countries other than the UK and US, is sometimes obscure.

UK usage

"Billion" has meant 10⁹ in most sectors of official published writing for many years now. The UK government, the BBC, and most other broadcast or published mass media, have used the short scale in all contexts since the mid-1970s.^[4][5][6][7]

Before the widespread use of "billion" for 10⁹, UK usage generally referred to thousand million rather than milliard.^[8][24] The long scale term "milliard", for 10⁹, is obsolete in British English, though its derivative, "yard", is still used as slang in the London money, foreign exchange and bond markets.

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.

Australian usage

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

Canadian usage

Canada uses both the long scale (in Canadian French) and the short scale (in Canadian English).

South African usage

South Africa uses both the long scale (in Afrikaans) and the short scale (in South African English). Some Afrikaans publications briefly attempted usage of the "American System" but that has led to comment in the papers ^[22] and has been disparaged by the "Taalkommissie" (The Afrikaans Language Commission of the South African Academy of Science and Art) ^[23] and has thus, to most appearances, been abandoned.

Indian, Pakistani and Bangladeshi usage

Outside of financial media, the use of "billion" by Bangladeshi, Indian and Pakistani English speakers highly depends on their educational background. Some may continue to use the traditional British long scale. In everyday life, Bangladeshis, Indians and Pakistanis largely use their own common number system, commonly referred to as the Indian numbering system – for instance, Bangladeshi, Pakistani, and Indian English commonly use the words lakh to denote 100 thousand, crore to denote ten million (i.e. 100 lakhs) and arab to denote one billion.

Indonesian usage

Indonesia officially employs the term milyar (derived from Dutch Miljard) for the number 10⁹, with no exception. For the number 10¹², the long-scale billion the term seribu milyar (a thousand milliards) or rarely sejuta juta (a million millions) is used. The high number of English literate speakers however are able to comprehend the short scale billion as being defined as 10⁹, an Indonesian milyar. Occasionally the English word 'billion' (understood as meaning milyar) may be used in conversation.

Italian language usage

Italy – with France – was one of the two European countries which partially converted to the short scale during the 19th century, but returned to the original long scale in the 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. ^[25][26] Therefore, in order to avoid ambiguity, they are seldom used. Forms such as miliardo for 10⁹, 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 more common. ^[17][27]

Spanish language usage

"Millardo" (milliard) is sometimes used in Spanish-speaking countries for 10⁹, but "mil millones" (a thousand millions) is used more frequently. The word "Billón" is sometimes incorrectly used in those countries more influenced by the United States but it is considered unacceptable^[28].

Esperanto language usage

The Esperanto words biliono, triliono etc. used to be ambiguous, and both long or short scale were used and presented in dictionaries. The current edition of the main Esperanto dictionary PIV however recommends the long scale meanings, as does the grammar PMEG ^[29]. 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") = biliono (10¹²); triiliono = triliono (10¹⁸), etc. Miliardo is an unambiguous term for 10⁹.

Presence on most continents

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

Continent	Short scale usage	Long scale usage
Africa	South African English	Afrikaans
North America	American English and Canadian English	Canadian French and American Spanish
South America	Brazilian Portuguese	American Spanish
Australia	Australian English	Indonesian
Europe	British English, Hiberno-English, Scottish English, Welsh English, Welsh, Bulgarian, Estonian, Latvian, Lithuanian, Romanian, Turkish, Ukrainian	French, German, Spanish, Portuguese, Italian, Dutch, Danish and most other languages of continental Europe

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., 1,000,000,000,000 rather than, say, 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". 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 (also known as standard form or exponential notation, for example 1x10⁹, 1x10¹⁰, 1x10¹¹, 1x10¹², etc.), or its engineering notation variant (for example 1x10⁹, 10x10⁹, 100x10⁹, 1x10¹², etc.), or computer programming notation (1e9, 1e10, 1e11, 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. Use with non-SI units (e.g. "giga-dollars", "giga-miles") is uncommon.

See also

References

External links

Current UK usage

• "BILLION (DEFINITION)". Hansard. http://hansard.millbanksystems.com/written_answers/1974/dec/20/billion-definition#S5CV0883P0_19741220_CWA_439.  – the official Hansard report of 20 Dec 1974 when Harold Wilson defined 'Billion' for the UK Government.
• "Linguist List 7.451". http://www.linguistlist.org/issues/7/7-451.html. - A 1996 email list message, comparing the two scales usage in different languages.
• "Who wants to be a trillionaire?". BBC. 2007-05-07. http://news.bbc.co.uk/1/hi/programmes/more_or_less/6625545.stm. Retrieved 2010-05-11.  – a 2007 BBC page on actual common media usage of trillion.

Traditional UK usage

• "Million, Billion, Trillion ...". http://www.jimloy.com/math/billion.htm.  – a 1999 webpage comparing the long and short scale numbers, which only considers traditional UK numbering.
• "AskOxford: How many is a billion?". http://www.askoxford.com/asktheexperts/faq/aboutwords/billion?view=uk.  – a current UK webpage with traditional usage viewpoint (featuring 'UK' and 'American' systems).