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 the short scales. Such numbering systems include the Indian numbering system and Chinese, Japanese, and Korean numerals. Much of the remainder of the world has 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-speaking and Arabic-speaking countries, most Eurasian post-communist countries, and Brazil.
For powers of ten less than 9, 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, whereas the short scale grows by multiples of one thousand. For example, the short scale billion is one thousand million, whereas in the long scale, billion is one million million, making the word 'billion' a false friend 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 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, 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 ; after a billion it is billiard. 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, 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 or 1 billion. This method becomes unwieldy for very large numbers.Another way is to combine unambiguous words: ten, hundred, thousand, and million. For example: one thousand million and one million million, though this also becomes unwieldy with longer numbers.
Methods that are better at longer numbers include:
- Scientific notation, or its engineering notation variant, or the computing variant E notation. This is the most common practice among scientists and mathematicians, but can be cumbersome in spoken word.
- SI metric prefixes. For example, giga for 109 and tera for 1012 can give gigawatt and terawatt. These prefixes can be used unambiguously even with non-SI units. For example: giga-dollars, megabucks, k€, and M€. instead of 1000 bytes or using megabyte to refer to 1,048,576 bytes
History
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 all refer to 1012, being long-scale terms. Therefore, each of these words translates to the American English or post-1974 British English word: trillion, and not billion.
On the other hand, the pre-1961 former French word billion, pre-1994 former Italian word bilione, Brazilian Portuguese word bilhão, and Welsh word biliwn all refer to 109, being short scale terms. Each of these words translates to the American English or post-1974 British English word billion.
The term billion originally meant 1012 when introduced. 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 word million derives from the Old French milion from the earlier Old Italian milione, an intensification of the Latin word, mille, a thousand. That is, a million is a big thousand, much as a great gross is a dozen gross or 12 × 144 = 1728.
The word milliard, or its translation, is found in many European languages and is used in those languages for 109. However, it is not found in American English, which uses billion, and not used in British English, which preferred to use thousand million before the current usage of billion. The financial term yard, which derives from milliard, is used on financial markets, as, unlike the term billion, it is internationally unambiguous and phonetically distinct from million. Likewise, many long scale countries use the word billiard for one thousand long scale billions, and the word trilliard for one thousand long scale trillions, etc.
;Timeline
| Date | Event |
| 13th century | The word million was not used in any language before the 13th century. The monk and polymath Maximus Planudes 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|thumb|upright|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, with Translation: |
| 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: |
| 1484 | French mathematician Nicolas Chuquet, in his article Le Triparty en la Science des Nombres par Maistre Nicolas Chuquet Parisien, used the words byllion, ''tryllion, quadrillion, quyllion, sixlion, septyllion, ottyllion, and nonyllion to refer to 1012, 1018,... 1054. Most of the work was copied without attribution by Estienne de La Roche and published in his 1520 book, L'arismetique. Chuquet's original article was rediscovered in the 1870s and then published for the first time in 1880. Translation: The extract from Chuquet's manuscript, the transcription and translation provided here all contain an original mistake: one too many zeros in the 804300 portion of the fully written out example: 745324'8043000 '700023'654321... |
| 1516 | French mathematician Budaeus, writing in Latin, used the term milliart to mean "ten myriad myriad" or 109 in his book De Asse et partibus eius Libri quinque. Translation: |
| 1549 | The influential French mathematician Jacques Pelletier du Mans used the name milliard to mean 1012, attributing the term to the earlier usage by Guillaume Budé |
| 17th 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: |
| 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, 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|thumb|upright|A Dictionary of Modern English Usage by H. W. Fowler H. W. Fowler's A Dictionary of Modern English Usage noted Although American English usage did not change, within the next 50 years, French usage changed from short scale to long, and British English usage changed from long scale to short. |
| 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, 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. |
| 1974 | File:Lord Harold Wilson Allan Warren.jpg|thumb|upright|British prime minister Harold Wilson 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: The BBC and other UK mass media quickly followed the government's lead within the UK. During the last quarter of the 20th century, most other English-speaking countries either also followed this lead or independently switched to the short scale use. However, in most of these countries, some limited long scale use persists and the official status of the short scale use is not clear. |
| 1975 | French mathematician Geneviève Guitel introduced the terms long scale and short scale'' to refer to the two numbering systems. |
| 1994 | The Italian Government confirmed their official usage of the long scale. |