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The long and short scales are two of several large-number naming systems for integer powers of ten, that use the same words with different meanings:
Contents
- Comparison
- History
- English speaking
- Arabic speaking
- Other short scale
- Long scale users
- Spanish speaking
- French speaking
- Portuguese speaking
- Dutch speaking
- Other long scale
- Using both
- Using neither
- Alternative approaches
- References
For whole numbers less than a thousand million (< 109) the two scales are identical. From a thousand million up (≥ 109) the two scales diverge, using the same words for different numbers; this can cause misunderstanding.
Countries where the long scale is currently used include most countries in continental Europe and most French-speaking, Spanish-speaking, and Portuguese-speaking countries except Brazil. The short scale is now used in most English-speaking and Arabic-speaking countries, in Brazil, in former Soviet Union and several other countries. Number names are rendered in the language of the country, but are similar everywhere due to shared etymology (e.g., billion is billón in Spanish). Some languages, particularly in East Asia and South Asia, have large number naming systems that are different from both the long and short scales, for example the Indian numbering system.
For most of the 19th and 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 purposes including official. With very few exceptions, the British usage and American usage are now identical.
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.
Comparison
At and above a thousand million (≥ 109) the same names are used to refer to numbers differing by a factor of an integer power of 1,000.
Each scale has a logical justification to explain the use of each such differing numerical name and value within that scale. The short-scale logic is based on powers of one thousand, whereas the long-scale logic is based on powers of one million. In both scales, the prefix bi- refers to 2 and tri- refers to 3, etc. However only in the long scale do the prefixes beyond one million indicate the actual power or exponent (of 1,000,000). In the short scale, the prefixes refer to one less than the exponent (of 1,000).
The relationship between the numeric values and the corresponding names in the two scales can be described as:
The relationship between the names and the corresponding numeric values in the two scales can be described as:
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 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, 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.
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 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, German Billion; Dutch biljoen; Swedish biljon; Finnish biljoona; Danish billion; Polish bilion, Spanish billón; Slovenian bilijon and the 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 the 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 terms billion and milliard both originally meant 1012 when introduced.
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:
Arabic-speaking
106, مَلْيُوْن malyoon: 109, مِلْيَار milyar; 1012, تِرِلْيُوْن trilyoon; etc.Most Arabic-language countries and regions use the short scale with 109 being مليار milyar. 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 traditional long scale word similar to milliard for 109. Some examples of short scale use, and the words used for 109 and 1012, are
Long scale users
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 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.
Spanish-speaking
106, millón; 109, mil millones or millardo; 1012, billón; etc.Most Spanish-language countries and regions use the long scale with 109 = mil millones, for example:
French-speaking
106, million; 109, milliard; 1012, billion; etc.Most French-language countries and regions use the long scale, for example:
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, for example:
Dutch-speaking
106, miljoen; 109, miljard; 1012, biljoen;Most Dutch-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; 1015, one billiard or one thousand billion; 1018, one trillion; etc.Some examples of long scale use, and the words used for 109 and 1012, are
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:
The long and short scales are both present on most continents, with usage dependent on the language used. Examples include:
Alternative approaches
Unambiguous ways of identifying large numbers include:
1e9, 1e10, 1e11, 1e12, etc.) This is the most common practice among scientists and mathematicians, and is both unambiguous and convenient.