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Long and short scales

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Long and short scales

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

Long scale
Every new term greater than million is one million times larger than the previous term. Thus, billion means a million millions (1012), trillion means a million billions (1018), and so on.
Short scale
Every new term greater than million is one thousand times larger than the previous term. Thus, billion means a thousand millions (109), trillion means a thousand billions (1012), and so on.

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.

  • In long scale countries, milliard was redefined down 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 redefined down 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.
    • Timeline

    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:

    Presence on most continents

    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:

  • In written communications, the simplest solution for moderately large numbers is to write the full amount, for example 1,000,000,000,000 rather than, say, 1 trillion (short scale) or 1 billion (long scale).
  • Combinations of the unambiguous word million, for example: 109 = "one thousand million"; 1012 = "one million million". This becomes rather unwieldy for numbers above 1012.
  • Combination of numbers of more than 3 digits with the unambiguous word million, for example 13,600 million
  • Scientific notation (also known as standard form or exponential notation, for example 1×109, 1×1010, 1×1011, 1×1012, etc.), or its engineering notation variant (for example 1×109, 10×109, 100×109, 1×1012, etc.), or the computing variant E notation (for example 1e9, 1e10, 1e11, 1e12, etc.) This is the most common practice among scientists and mathematicians, and is both unambiguous and convenient.
  • SI prefixes in combination with SI units, for example, giga for 109 and tera for 1012 can give gigawatt (=109 W) and terawatt (=1012 W), respectively. The International System of Units (SI) is independent of whichever scale is being used. Use with non-SI units (e.g. "giga-dollars", "giga-miles") is uncommon although "megabucks" is in informal use representing a large sum of money rather than exactly a million dollars.

    • References

    Long and short scales Wikipedia
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