Long and short scales

The long and short scales are two powers of ten number naming systems that are consistent with each other for smaller numbers, but are contradictory for larger numbers. Other numbering systems, particularly in East Asia and South Asia, have large number naming that differs from both the long and short scales. Such numbering systems include the Indian numbering system and Chinese, Japanese, and Korean numerals. Much of the remainder of the world adopted either the short or long scale. Countries using the long scale include most countries in continental Europe and most that are French-speaking, German-speaking and Spanish-speaking. Use of the short scale is found in most English and Arabic speaking countries, most Eurasian post-communist countries and Brazil.

For powers of ten less than 9 (one, ten, hundred, thousand, and million), the short and long scales are identical, but for larger powers of ten, the two systems differ in confusing ways. For identical names, the long scale grows by multiples of one million (10⁶), whereas the short scale grows by multiples of one thousand (10³). For example, the short scale billion is one thousand million (10⁹), whereas in the long scale, billion is one million million (10¹²), making the word false friends between long and short scale languages. The long scale system includes additional names for interleaved values, typically replacing the word ending "-ion" with "-iard".

To avoid confusion, the International System of Units (SI) recommends using the metric prefixes to indicate magnitude. For example, giga is always 10⁹, which is billion in short scale but milliard in long scale.

Definition

In both scales, names are given to orders of magnitude at increments of 1000. Both systems use the same names for magnitudes less than 10⁹. Differences arise from the use of identical names for larger magnitudes. For the same magnitude name (n-illion), the value is 10³ⁿ⁺³ in the short scale but 10⁶ⁿ in the long scale for positive integers n.

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-illiard equals 10⁶ⁿ⁺³.

The following table shows the size of first few short and long scale magnitudes. Notice how billion and trillion are in both scales but have different sizes.

Quantity	Short scale	Long scale
106	million	million
109	billion	milliard
1012	trillion	billion
1015	quadrillion	billiard
1018	quintillion	trillion
1021	sextillion	trilliard
1024	septillion	quadrillion
1027	octillion	quadrilliard

Comparison

The following tables show the corresponding names and values of the two scales.

Note that instead of using an intermediate long scale word (illiard), a quantity is sometimes specified in terms of the smaller illion word. For example, "thousand billion" instead of "billiard".

Value	Metric prefix	Short scale	Long scale
1		one	one
10	deca	ten	ten
102	hecto	hundred	hundred
103	kilo	thousand	thousand
106	mega	million	million
109	giga	billion	milliard
1012	tera	trillion	billion
1015	peta	quadrillion	billiard
1018	exa	quintillion	trillion
1021	zetta	sextillion	trilliard
1024	yotta	septillion	quadrillion
1027	ronna	octillion	quadrilliard
1030	quetta	nonillion	quintillion

The different sizes of the same name of the two scales can be described as:

Name	Short scale	Long scale
million	106	106
billion	109	1012
trillion	1012	1018
quadrillion	1015	1024
quintillion	1018	1030
	. . .	. . .

Avoiding confusion

One way to avoid confusion between the two scales is to use positional notation. For example, 1,000,000,000,000 rather than 1 trillion (short scale) or 1 billion (long scale). This method becomes unwieldy for very large numbers.

Combinations of the unambiguous words: ten, hundred, thousand, and million. For example: one thousand million and one million million.

Scientific notation (for example 1Template:E), or its engineering notation variant (for example 10Template:E), or the computing variant E notation (for example 1e10). This is the most common practice among scientists and mathematicians.

SI metric prefixes. For example, giga for 10⁹ and tera for 10¹² can give gigawatt (10⁹ W) and terawatt (10¹² W). Use with non-SI units is unambiguous. For example, giga-dollars, megabucks, k€, and M€.

History

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

The short scale was never widespread before its general 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 early 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. 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. 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. Some of these countries, but not all, introduced new words billiard, trilliard, etc. as intermediate terms. 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. 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.

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 (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.

Timeline

Date	Event
13th century	The word million was not used in any language before the 13th century. The monk and polymath Maximus Planudes (Template:Circa–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	File:Piers Plowman.jpg Piers Plowman, 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 (written Template:Circa–1387 in Middle English), with 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. 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	File:Chuquet.gif Le Triparty en la Science des Nombres par Maistre Nicolas Chuquet Parisien an extract from Chuquet's original 1484 manuscript 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. [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	File:Guillaume Budé, by Jean Clouet.jpg Guilielmus Budaeus or Guillaume Budé (1467–1540) 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. 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
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. 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.
1676	The first published use of milliard as 109 occurred in the Netherlands. 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.
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.
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	File:FowlersModernEnglishUsageFrontispiece.jpg A Dictionary of Modern English Usage by H. W. Fowler 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. 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. 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.
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. The National Institute of Standards and Technology within the US also considers that it is best that they be avoided entirely.
1961	The French Government confirmed their official usage of the long scale in the Journal officiel (the official French Government gazette).
1974	File:Lord Harold Wilson Allan Warren.jpg British prime minister Harold Wilson (1916–1995) 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: 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 (Template:Langx) and short scale (Template:Langx) to refer to the two numbering systems.
1994	The Italian Government confirmed their official usage of the long scale.

Current usage

File:EScalas corta y larga.svg

Short and long scale usage throughout the worldTemplate:Div col

  Long scale

  Short scale

  Short scale with milliard instead of billion

  Both scales

  Other naming system

  No data

Template:Div col end

Short scale users

English-speaking

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

• (English-speaking) see Using both below
• (English-speaking, Template:Langx, trilliún)

Arabic-speaking

Most Arabic-language countries and regions use the short scale with 10⁹ being , except for a few countries like Saudi Arabia and the UAE which use the word بليون 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

• (Brazilian Portuguese: , )
• (, )
• (Hebrew: , )
• ( , )
• (, )

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

Template:Block indent Most Dutch-language countries and regions use the long scale with 10⁹ = .

French-speaking

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

• (Canadian French) see Using both below

German-speaking

Template:Block indent German-language countries and regions use the long scale with 10⁹ = .

Portuguese-speaking

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

Spanish-speaking

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

• ( or )
• ( or typ. )

Other long scale

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

• Template:Flagicon image Esperanto (, )
• (Persian: میلیارد , بیلیون , تریلیون )
• (, )
• (, )
• (, ). There are ambiguities for numbers above 10¹².
• (French: , ; German: , ; Italian: , ; Romansh: , )

Using both

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
	Canadian English (109 = billion, 1012 = trillion)	Canadian French (109 = , 1012 = or ).
Template:Unbulleted list	English (109 = billion, 1012 = trillion)	French (109 = , 1012 = )
Template:Unbulleted list	South African English (109 = billion, 1012 = trillion)	Afrikaans (109 = , 1012 = )
	Economic and technical (109 = , 1012 = )	Latin American export publications (109 = or , 1012 = )

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	Traditional system for everyday use, but short or long scale may also be in use
	Dzongkha numerals	Traditional system
	Khmer numerals	Traditional system
Template:Unbulleted list	East Asian numbering system: Template:Unbulleted list	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, ("hundred-myriad", i.e. 100 × 10,000): "bi+hundred-myriad" = 109 (short scale billion) "tri+hundred-myriad" = 1012 (short scale trillion) "quadri+hundred-myriad" = 1015 (short scale quadrillion), and so on.
	Lao numerals	Traditional system
	Mongolian numerals	Traditional myriad system for the larger numbers; special words up to 1067
	Template:Unbulleted list	Traditional systems
	Thai numerals	Traditional system based on millions
	Vietnamese numerals	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. For example:

Continent	Short scale usage	Long scale usage
Africa	Arabic (Egypt, Libya), South African English	French (Benin, Guinea), Portuguese (Mozambique)
North America	American English, Canadian English	Canadian French, Mexican Spanish, U.S. Spanish
South America	Brazilian Portuguese, English (Guyana)	American Spanish, Dutch (Suriname), French (French Guiana)
Antarctica	Australian English, British English, New Zealand English, Russian	American Spanish (Argentina, Chile), French (France), Norwegian (Norway)
Asia	Hebrew (Israel), Indonesian, Philippine English	Persian (Iran), Portuguese (East Timor, Macau)
Europe	British English, Russian, Turkish	Dutch, French, German, Italian, Portuguese, Spanish
Oceania	Australian English, New Zealand English	French (French Polynesia, New Caledonia)

See also

• Googol (number)
• Googolplex (number)
• Names of large numbers
• Names of small numbers
• Orders of magnitude (numbers)
• Hindu units of time which displays some similar issues
• Indian numbering system

External links

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