Long and short scales
Different meanings for numbers
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Related on Lably
Through Power of 10, Order of magnitude and International System of Units
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.
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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".
The different sizes of the same name of the two scales can be described as:
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 1.2×1010), or its engineering notation variant (for example 12×109, 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 (French: échelle courte) and long scale (French: échelle longue) 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
Current usage
Short scale users
English-speaking
106, one million; 109, one billion; 1012, one trillion; etc.
Most English-language countries and regions use the short scale with 109 being billion. For example:
- Australia
- Canada (English-speaking) see Using both below
- Ireland (English-speaking, Irish: billiún, trilliún)
- United Kingdom
- United States
Arabic-speaking
106, مَلْيُوْن milyūn; 109, مِلْيَار milyar; 1012, تِرِلْيُوْن tirilyūn; etc.
Most Arabic-language countries and regions use the short scale with 109 being مليار milyar, except for a few countries like Saudi Arabia and the UAE which use the word بليون billion for 109. For example:
Other short scale
106, one million; 109, one milliard or one billion; 1012, one trillion; etc.
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
- Brazil (bilhão, trilhão)
- Indonesia (miliar, triliun)
- Israel (Hebrew: מיליארד milyard, טריליון trilyon)
- Russia (миллиард milliard, триллион trillion)
- Turkey (milyar, trilyon)
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
106, miljoen; 109, miljard; 1012, biljoen; etc.
Most Dutch-language countries and regions use the long scale with 109 = miljard.
French-speaking
106, million; 109, milliard; 1012, billion; etc.
Most French-language countries and regions use the long scale with 109 = milliard, for example:
- Canada (Canadian French) see Using both below
- France
German-speaking
106, Million; 109, Milliarde; 1012, Billion; etc.
German-language countries and regions use the long scale with 109 = Milliarde.
Portuguese-speaking
106, milhão; 109, mil milhões or milhar de milhões; 1012, bilião
With the notable exception of Brazil, a short scale country, most Portuguese-language countries and regions use the long scale with 109 = mil milhões or milhar de milhões.
Spanish-speaking
106, millón; 109, mil millones or millardo; 1012, billón; etc.
Most Spanish-language countries and regions use the long scale, for example:
Other long scale
106, one million; 109, one milliard or one thousand million; 1012, one billion; etc.
Some examples of long scale use, and the words used for 109 and 1012, are:
- Czech Republic (miliarda, bilion)
- Denmark (milliard, billion)
- Esperanto (miliardo, duiliono)
- Finland (miljardi, biljoona; Swedish: miljard, biljon)
- Iran (Persian: میلیارد miliyard, بیلیون billion, تریلیون trillion)
- Italy (miliardo, bilione)
- Norway (Bokmål: milliard, billion; Nynorsk: milliard, billion)
- Poland (miliard, bilion)
- Romania (miliard, bilion). There are ambiguities for numbers above 1012.
- Sweden (miljard, biljon)
- Switzerland (French: milliard, billion; German: Milliarde, Billion; Italian: miliardo, bilione; Romansh: milliarda, billiun)
Using both
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.
Using neither
The following countries use naming systems for large numbers that are not etymologically related to the short and long scales:
By continent
The long and short scales are both present on most continents, with usage dependent on the language used. For example: