The long and short scales are two powers-of-ten number-naming systems that are consistent with each other for smaller numbers, but are distinct for larger numbers. Much 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, as well as most Eurasian Post-Soviet countries, Israel, and Brazil.
For smaller powers of ten (one, ten, hundred, thousand, and million), the short and long scales are identical; they however differ for larger powers of ten. The long scale grows between names 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 '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 (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.
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.
Naming 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 the first few short and long-scale magnitudes. Notice how billion, trillion, and quadrillion are in both scales but have different sizes.
Quantity Short scale Long scale106millionmillion 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 10 deca ten 102 hecto hundred 103 kilo thousand 106 mega 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 million106 106 billion109 1012 trillion1012 1018 quadrillion1015 1024 quintillion1018 1030 ... ... n-illion103n+3 106n
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.
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 (for example 1.2), or its engineering notation variant (for example 12, where the exponent–here 9–is a multiple of 3), or the computing variant E notation (for example 1.2e10). These are the most common practices 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 (109 W) and terawatt (1012 W). These prefixes can be used unambiguously even with non-SI units. For example: giga-dollars, megabucks, k€, and M€. An exception to this is digital storage, where it is still common practice to use prefixes incorrectly, such as using kilobyte to refer to 1024 bytes (a kibibyte) instead of 1000 bytes or using megabyte to refer to 1,048,576 bytes (a mebibyte) instead of 1,000,000.
History Although this situation has been developing since the 17th century, the first recorded use of the terms short scale () and long scale () was by the French mathematician Geneviève Guitel in 1975.
Etymology The term billion originally meant 1012 when introduced: etymologically it derived from bymillion (a double million), just as trillion derives from trimillion (a triple million). 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 (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.
Evolution 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).
Timeline Date Event 13th century The word million was not used in any language before the 13th century. The monk and polymath Maximus Planudes (–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 The word million entered the English language. One of the earliest references is William Langland's Piers Plowman (written –1387 in Middle English), with
Translation:1475French 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 (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: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. 1676The first published use of milliard as 109 occurred in the Netherlands.